Testing Sparse Functions over the Reals

T esting Sparse F unctions o v er the Reals Vipul Arora ∗ 1 Arnab Bhattac haryya † 2 Philips George John 3 Sa yan tan Sen ‡ 3 1 National Univ ersity of Singapore. vipul@comp.nus.edu.sg . 2 Univ ersity of W arwic k. arnab.bhattacharyya@warwick.ac.uk . 3 CNRS@CREA TE & NUS. philips.george.john@u.nus.edu . 4 Cen tre for Quantum T ec hnologies, National Universit y of Singap ore. sayantan789@gmail.com . Abstract Ov er the last three decades, function testing has b een extensiv ely studied ov er Bo olean, finite fields, and discrete settings. How ever, to encode the real-w orld applications more succinctly , function testing o ver the reals (where the domain and range, b oth are reals) is of prime importance. Recen tly , there hav e b een some works in the direction of testing for algebraic representations of suc h functions: the w ork by Fleming and Y oshida (ITCS 20), Arora, Kelman, and Meir (SOSA 25) on line arity testing and the work of Arora, Bhattacharyy a, Fleming, Kelman, and Y oshida (SOD A 23) for testing low-de gr e e p olynomials . Our work follo ws the same av enue, wherein w e study three well-studied sparse representations of functions, ov er the reals, namely (i) k -line arity , (ii) k -sp arse p olynomials , and (iii) k -junta . In this setting, given approximate query access to some f : R n → R , we w ant to decide if the function s atisfies some prop ert y of interest, or if it is far from all functions that satisfy the prop ert y . Here, the distance is measured in the ℓ 1 -metric, under the assumption that we are dra wing samples from the Standard Gaussian distribution. W e present efficient testers and Ω( k ) lo wer b ounds for testing each of these three prop erties. 1 In tro duction Prop ert y testing [ BLR90 , RS96 , GGR96 ] is a rigorous framew ork for studying the global prop erties of large datasets by accessing only a few en tries of it. In particular, given query access to an unknown “h uge ob ject”, the goal of prop ert y testing is to verify some prop ert y of the ob ject by only insp ecting a small p ortion of it. F ormally , we can define prop ert y testing of functions, or function testing , by considering the “huge ob ject” as a function f o ver an underlying domain, which w e can query at a small num ber of p oin ts in order to verify a prop ert y of the function f in an approximate sense. F or example, consider the problem of line arity testing of functions f : F n → F , where F is a finite field. W e are giv en access to a query oracle for f ; i.e., on input x , the oracle returns f ( x ) . The goal is to distinguish with high probability b et ween t wo cases, viz. (i) f is a linear function, or (ii) f is “far” from all linear functions, while p erforming as few queries to the oracle as p ossible. The field of prop ert y testing w as initiated in the seminal work of [ BLR90 , BLR93 ], who studied the problem of self-testing of programs, where the goal w as to understand the correctness of a ∗ This w ork was done in part while the author w as visiting the Simons Institute for the Theory of Computing. † Researc h supp orted b y a start-up grant at the Univ ersity of W arwic k. ‡ Researc h supp orted b y the NRF Inv estigatorship aw ard (NRF-NRFI10-2024-0006) and CQT Y oung Researcher Career Dev elopment Gran t (25-YRCDG-SS). 1 program by verifying its outputs on a set of correlated input data. A fundamental problem they studied is that of testing whether f is line ar (or generally , a homomorphism of ab elian groups) or ε -far from linearit y , with the notion of ε -farness of f : F n → F from a function class P defined as: dist Unif ( F n ) ,ℓ 0 ( f , P ) = inf g ∈P Pr x ∼ Unif ( F n ) [ f ( x )  = g ( x )] ≥ ε, i.e., for an y function g that satisfies prop ert y P , f disagrees with g on at least an ε -fraction of the inputs x (uniformly drawn from F n ). In the con text of linearit y testing, P is the class of linear functions/homomorphisms (the ℓ 0 -distance definition given ab o ve can b e easily generalized in terms of function domain and range, input distribution etc). They sho wed a constant query (indep enden t of | F | and n ) linearit y tester, ep on ymously known as the BLR tester . Ov er the last three decades, prop ert y testing has b een extensiv ely studied in many settings, suc h as when the unkno wn ob ject is a function, a graph, or a probability distribution, with man y natural connections to real-world problems, e.g., in the context of probabilistically chec k able pro ofs [ ALM + 98 , AS98 , RS97 , Din07 ], P A C learning [ GGR96 ], program chec king [ RS92 , RS96 ], appro ximation algorithms [ GGR96 , FGL + 91 ] and man y more. In particular, starting from the very first w ork that initiated the field of prop ert y testing [ BLR90 ], o ver the last three decades, function prop ert y testing has b een extensively studied in several settings, culminating in a wide arra y of to ols and techniques. This includes settings where the function is defined ov er finite fields [ AKK + 05 , KR06 , JPRZ09 , FS95 , BKS + 10 , BFH + 13 , GOS + 11 , Sam07 ], o ver hyper-grids [ DGL + 99 , BR Y14b , CS13 , BCP + 17 ], etc. One may see the b o oks [ Gol17 , BY22 ] and the surv eys [ Fis04 , R + 08 , Ron09 ] for detailed references. Our problem setting Supp ose w e are given query access to an unkno wn function f : R n → R , and our goal is to distinguish whether (i) f satisfies some property P , or (ii) f is ε -far from all functions that satisfy P . W e define the notion of ε -farness ov er the reals in the following w ay: we fix a reference distribution D on R n , and w e say f is ε -far from P if the following holds 1 : dist D ,ℓ 1 ( f , P ) ≜ inf g ∈P  E x ∼D [ | f ( x ) − g ( x ) | ]  ≥ ε. Studying function testing ov er the reals often requires new tec hniques compared to testing on finite domains. A common reference distribution on R n is the standard n -dimensional Gaussian distribution N ( 0 , I n ) , whic h is appro ximately a uniform distribution on an ℓ 2 -sphere of radius √ n . And the notion of, exact queries (oracle giving f ( x ) ) is not realistic ov er the reals since one may require exp onen tially man y bits to represent the exact function v alue. Although the problem of function testing o v er the reals has several practical motiv ations, there hav e b een comparatively few works in this setting till no w (compared to testing ov er finite domains) such as testing surface areas [ Nee14 , KNO W14 ], testing halfspaces [ MORS10a , MORS10b , MORS09 ], linear separators [ BBBY12 ], high-dimensional con vexit y [ CFSS17 ], linear k -jun tas [ DMN19 ] etc. In terestingly , these w orks mostly fo cus on the setting when f is Bo olean v alued, i.e., f : R n → {± 1 } . The setting where the range of f is real has also b een studied, e.g., in the work of [ BR Y14a ], wherein they test prop erties of functions, defined ov er finite hyper-grids, with resp ect to L p -distances. In [ BKR23 ], functions o ver the h yp ercube, i.e., f : { 0 , 1 } n → R are studied for monotonicit y . Recen tly , [ FPJ23 , FPJ24 ] studied L p testing of monotonicit y of Lipschitz functions f : [0 , 1] n → R . 1 The notion of ℓ 1 -distance is more appropriate for real-v alued functions, rather than the commonly used Hamming ( ℓ 0 ) distance, esp ecially in the appr oximate query setting whic h we define later. F or example, consider the case if g = f + ε for some ε > 0 , then dist D ,ℓ 1 ( f , g ) = ε , whereas dist D ,ℓ 0 ( f , g ) = 1 . 2 [ FY20 ] studied the problem of linearity testing for real-v alued real-domain functions in full generalit y , i.e., for functions f : R n → R . Later, [ ABF + 23 ] studied the problem of testing low-degree p olynomials in this regime, follo wed b y the work of [ AKM25 ], whic h improv ed these results b y ac hieving query-optimality with resp ect to the pro ximity parameter ε . In this work, we expand the landscap e of testing v arious fundamental notions of sparsity to the general f : R n → R regime. A recent work of [ BDN + 25 ] studies a problem similar to sparsit y testing of low degree p olynomials, in a sample-based access mo del. How ev er, it assumes a promise that the unkno wn function f : R n → R is a multi-line ar p olynomial to b egin with, and needs a large gap for the sparsit y parameter b et w een the YES and NO cases. When the function f to b e tested is real v alued, tw o different mo dels can b e considered with resp ect to the query ac cur acy . In the first mo del, known as the arbitr ary pr e cision arithmetic or the exact testing mo del, we assume that the oracle representing f can giv e the exact v alue of f at an y p oin t of query x . How ev er, this mo del is unrealistic from an implementation, or even from a classical complexity- theoretic viewpoint. So, we instead consider a finite pr e cision arithmetic or appr oximate testing mo del: Definition 1.1 ( η - appr oximate query ) . The oracle, when queried for f ( x ) , outputs e f ( x ) suc h that | e f ( x ) − f ( x ) | ≤ η , for some small parameter η ∈ (0 , 1) , for every query p oin t x . It is clear that for any prop ert y P , an y tester for P in the approximat e model will also w ork in the exact mo del. The notion of approximate testing was studied in earlier works [ GLR + 91 , ABCG93 , EKR01 ]. In this context, η can b e thought of as the resolution limit of the computational machine, i.e., if the machine offers some α bits of precision, then η = 2 − α . η can also b e thought of as the noise reliabilit y threshold of a channel communicating reals, i.e., if some information a ∈ R is transmitted on a channel with a reliabilit y threshold of η , then the received observ able e a ∈ R satisfies | a − e a | ≤ η . When η = 0 , this is the exact query mo del. All our testers in this work are analyzed in the appro ximate query mo del for η > 0 . Our results also require appropriate upp er b ounds on η , which will b e stated in eac h context. But we use the exact query mo del for pro ving the low er b ounds, since low er b ounds for η = 0 also hold for an y η -appro ximate query mo del for η > 0 . Similar to the context of access to the unknown functions, there are some v ariations in terms of the error profile of the testing algorithms. A tester is said to ha ve two-side d err or if it can err in b oth the cases: when f ∈ P , and when f is ε -far from P . This is in contrast with one-side d err or testers, whic h alw ays decide correctly when f ∈ P , and can only err when f is ε -far from P . Lik ewise, a tester is said to b e adaptive if it p erforms queries based on the answers it obtained for the previous queries. On the other hand, a non-adaptive tester p erforms all its queries together in a single round. A tester o ver a finite domain is said to b e lo c al if the n umber of queries p erformed b y it is indep enden t of the domain size. In this w ork, we design testers in the appro ximate query model for three sparse function represen tations: k -linear, k -sparse lo w-degree p olynomials, and k -jun tas. Our testers for k -linearit y , and k -sparse lo w-degree p olynomials hav e tw o-sided error, whereas our k -jun ta tester has one-sided error. Our testers for k -linear functions and k -jun tas are adaptiv e, while our tester for k -sparse lo w-degree p olynomials is non-adaptive. Imp ortan tly , all our testers are lo c al : the num b er of queries p erformed by our testers is indep enden t of the domain dimension n , and dep ends only on the sparsity parameter k and the proximit y parameter ε (as well as the total degree d in the k -sparse low-degree p olynomial case). These prop erties are well-studied in the Bo olean and finite fields regime, and this work striv es to do the same ov er contin uous domains. W e b eliev e the new techniques we hav e dev elop ed to design these testers will b e of indep enden t interest. 3 Boundedness W e assume that the unkno wn functions are b ounded inside an ℓ 2 ball of suitable radius (t ypically O ( √ n ) ), i.e., for every x ∈ B ( 0 , O ( √ n )) , f ( x ) ≤ C ∥ x ∥ 2 for some fixed constant C (so f ( x ) ≤ C √ n in B ( 0 , 2 √ n ) ) 2 . Note that testing only via b ounded queries is imp ossible without suc h an assumption. F or example, consider an arbitrary “go od” function f : R n → R ( k -linear, k -sparse p olynomial, or k -jun ta), and c ho ose a suitably small region R at random (e.g. by c ho osing y uniformly from B( 0 , √ n ) and setting R to b e a small radius ball around y ), where 0 < µ n ( R ) ≪ 1 /n c for any c > 0 under the N ( 0 , I n ) Gaussian measure on R n . F urther, sub divide R equally into R 1 and R 2 , and for some A ≥ 2 ε/µ n ( R ) , define a function f ′ : R n → R as: f ′ ( x ) ≜      f ( x ) , if x ∈ R, A, if x ∈ R 1 , and − A, if x ∈ R 2 . As A can b e arbitrarily large, f ′ can b e mov ed arbitrarily far from the required prop ert y (in ℓ 1 distance o ver the Gaussian measure). How ever, since µ n ( R ) is arbitrarily small, no algorithm using b ounded queries can distinguish b et ween f and f ′ . Choice of Reference Distribution In this w ork (for the k -linear and k -sparse lo w-degree testers), w e hav e used an anti-concen tration result of Glazer and Mikulincer ( Theorem 4.16 ) whic h uses the Carb ery-W right anti-concen tration inequality ( Theorem 4.15 ), but with a more usable expression for the v ariance (in terms of the p olynomial co efficien ts). The form that we hav e used assumes that the distribution (on the v ariables) is log-concav e, isotropic, and non-discrete. W e hav e chosen N ( 0 , I n ) as represen tative of suc h a distribution, but we can also tak e P ⊗ n for a contin uous log-conca ve distribution P on R . A probabilistic upp er b ound for the Hank el matrix eigenv alues, which we use in the analysis of the k -sparsit y tester, also assumes the isotropic Gaussian distribution ( Theorem 2.11 ). 2 Our results In this work, we fo cus on three problems: testing (i) k -linearit y , (ii) k -sparse lo w-degree p olyno- mials, and (iii) k -jun tas. Throughout this w ork, we assume that our reference distribution D is the standard Gaussian N ( 0 , I n ) , unless otherwise stated. 2.1 T esting k -linear functions Definition 2.1 ( k -linearit y) . Let f : R n → R , and k ∈ N b e a parameter. f is a k -line ar function if there exists a set S ⊆ [ n ] : | S | ≤ k , and there exist { c i ∈ R : i ∈ S } , suc h that for any x ∈ R n , f ( x ) = X i ∈ S c i x i . This problem has b een extensiv ely studied ov er finite domains with exact query access. [ FKR + 04 ] designed the first tester for k -linearit y with query complexity e O ( k 2 ) by studying the related problem of testing k -jun tas. Later, [ Bla09 ] improv ed the b ound for testing k -jun tas to O ( k log k ) queries. Using the BLR test [ BLR90 ], along with this result gives a tester for k -linearit y with O ( k log k ) query complexit y , as done by [ Bsh23 ], who presented an optimal, tw o-sided error, non-adaptive algorithm for this problem o v er Boolean domains. The first lo wer b ounds for this problem w ere presented 2 This boundedness notion corresp onds to the b ounds we get for low-degree p olynomials ov er compact domains, e.g., if f ( x ) = a 1 x + . . . + a d x d is a p olynomial in x , then | f ( x ) | ≤ P d i =1 | a i | R i for an y x ∈ [ − R, R ] , for all R ≥ 0 . 4 b y [ FKR + 04 ], pro ving Ω( √ k ) non-adaptiv e, and Ω( log k ) adaptiv e queries are necessary for testing k -linearit y . These were first improv ed by [ Gol10 ], to Ω( k ) non-adaptiv e, and Ω( √ k ) adaptiv e query lo wer b ounds. This w as further improv ed by [ BBM12 , BK12 ] who prov ed Ω( k ) adaptive query low er b ound. Interestingly , the Ω( k ) adaptive query lo wer bound b y [ BBM12 ] was pro ved by showing a nov el connection to communic ation complexit y , whic h w e will also use later to pro ve our lo wer b ounds. Our result for testing k -linearity is summarized in the following theorem. Theorem 2.2 (Informal, see Theorem 5.1 ) . Supp ose f : R n → R is a function b ounde d in the b al l B ( 0 , 2 √ n ) and we ar e given η -appr oximate query ac c ess to f . L et k b e a p ositive inte ger and ε, η ∈ (0 , 2 / 3) such that η < min n ε, O  min i ∈ [ n ]: f ( e i )  =0 | f ( e i ) | ( nk ) 2 o , wher e e i denotes the i th standar d b asis ve ctor. Ther e exists an e O ( k log k + 1 / ε ) -query tester ( Algorithm 1 ) that distinguishes whether f is k -line ar, or is ε -far fr om al l k -line ar functions, with pr ob ability at le ast 2 / 3 . Remark 2.3. Note the ne c essity of sufficiently go o d machine pr e cision (smal l η ) to suc c essful ly test k -line arity. This is inevitable b e c ause if a line ar function has k “lar ge” c o efficients a 1 , . . . , a k ( ≥ η ) and a smal l c o efficient a k +1 (say < η /n 2 ), we c annot distinguish b etwe en such a function and a k -line ar function P k i =1 a i x i using η -appr oximate queries and b ounding the absolute differ enc e b etwe en the function values (as we do in Algorithm 1 ), sinc e the function values wil l differ by at most | a k +1 ( x k +1 ) | . Similar r estrictions apply to testing k -sp arse p olynomials, and k -juntas. In another sense, a qualitative dep endenc e of η on the function structur e is inevitable. Otherwise, if given an η -appr oximate or acle e f to f , e f ′ ≜ e f / 2 n (which c an b e c ompute d in O ( n ) time given e f , assuming a variable-length binary flo ating-p oint r epr esentation) would b e an η / 2 n -appr oximate or acle to f . But such a sc aling would not help with our r esults, sinc e the c o efficients of f , the non-zer o influenc es, etc., would b e similarly sc ale d-down. 2.2 T esting k -sparse lo w degree p olynomials Definition 2.4 ( k -sparsit y) . Let f : R n → R , and k ∈ N > 0 b e a parameter. Define x α ≜ Q n i =1 x α i i , for any x ≜ ( x 1 , . . . , x n ) ∈ R n , and α ≜ ( α 1 , . . . , α n ) ∈ N n . A p olynomial f ( x 1 , . . . , x n ) = P ℓ i =1 a i x d i , with a i  = 0 , and d i ∈ N n for ev ery i ∈ [ ℓ ] , is said to b e a k -sp arse p olynomial if ℓ ≤ k . Grigorescu et al. [ GJR10 ] studied this problem for functions on finite fields, i.e., f : F n q → F q , assuming that q is large enough, using the machinery of Hank el matrices asso ciated with a p olynomial: Definition 2.5 (Hankel Matrix for p olynomials [ GJR10 , BOT88 ]) . Consider any u ≜ ( u 1 , . . . , u n ) ∈ R n , and define u i ≜ ( u i 1 , . . . , u i n ) ∈ R n , ∀ i ∈ N . F or a function f : R n → R and an in teger t ∈ Z > 0 , define the t -dimensional Hankel matrix associated with f at u to b e the following: H t ( f , u ) ≜      f ( u 0 ) f ( u 1 ) . . . f ( u t − 1 ) f ( u 1 ) f ( u 2 ) . . . f ( u t ) . . . . . . . . . . . . f ( u t − 1 ) f ( u t ) . . . f ( u 2 t − 2 )      ∈ R t × t . F rom Ben-Or and Tiw ari [ BOT88 ]’s Observ ation 6.4 , they designed a tester with a query complexit y of O ( k ) (indep enden t of d ), assuming f to b e an individual-degree- d p olynomial (note that all functions F n q → F q are p olynomials of individual degree ≤ q ). Note that if we ha ve exact query access to f , the Hankel matrix H t ( f , u ) can b e computed using only 2 t − 1 queries to f for an y p oin t u ∈ R n . F or the problem of testing a p olynomial f : R n → R for sparsity , giv en exact query access, the tester of [ GJR10 ] works as it is. In fact, it achiev es p erfect 5 soundness and completeness (pro ved in Lemma 6.5 ). Ho wev er, the assumption of an exact query oracle is not realistic. Moreo ver, for general functions f : R n → R , the promise of p olynomialit y no longer holds, and therefore a preliminary step is needed to eliminate functions that are far from b eing lo w-degree p olynomials. [ ABF + 23 ] designed a lo c al , approximate query , low degree tester with a query complexit y of O ( d 5 ) , for p olynomials ov er R n with total de gr e e d , whic h we will use in our w ork. Th us, we restrict our attention to p olynomials of total de gr e e at most d . Theorem 2.6 (Informal, see Theorem 6.1 ) . L et k , d ∈ N , ε, η ∈ (0 , 1) b e p ar ameters such that η < min { ε, 1 / 2 2 n } , and f : R n → R b e b ounde d in B ( 0 , 2 d √ n ) , given via an η -appr oximate query ac c ess. Then ther e exists an e O ( d 5 + d 2 / ε + dk 3 ) -query tester ( Algorithm 4 ), that distinguishes whether f is a k -sp arse, de gr e e- d p olynomial, or is ε -far fr om al l such p olynomials, with pr ob ability ≥ 2 / 3 . Remark 2.7. W e note that Test- k -Sp arse ( Algorithm 4 ) also works for testing k -line ar functions (by setting the de gr e e d = 1 ). However, in this c ontext, we make the fol lowing r emarks. (i) Sinc e the query c omplexity of Test- k -Sp arse is e O ( d 5 + d 2 / ε + dk 3 ) , if we invoke it for k -line arity testing, the b ound would b e e O ( k 3 + 1 / ε ) which is worse than that of Test- k -Linear . (ii) The r estriction on the appr oximation p ar ameter η is 1 / 2 2 n for Test- k -Sp arse , as c omp ar e d to 1 / ( nk ) 2 for Test- k -Linear . Thus, for a wider r ange of p ar ameters when η is not to o smal l, invoking Test- k -Linear is b etter, c omp ar e d to Test- k -Sp arse . (iii) Test- k -Linear is an adaptive tester, while Test- k -Sp arse is non-adaptive. 2.3 T esting k -jun tas Definition 2.8 ( k -jun ta) . Let f : R n → R , and k ∈ N b e a parameter. A co ordinate i ∈ [ n ] is said to b e influential with resp ect to f , if for some x ∈ R n , changing the v alue of x i c hanges the v alue of f ( x ) . f is said to b e a k -junta , if there are at most k influen tial v ariables with resp ect to f . T esting whether a Bo olean function is a k -jun ta has b een extensively studied in the exact query mo del. The first result in this con text w as b y [ PRS02 ], which was follo wed b y the work of [ FKR + 04 ], who designed a e O ( k 2 ) -query tester. Later, [ DLM + 07 ] extended it to the finite range setting. [ Bla08 ] then gav e an e O ( k 3 / 2 ) -query non-adaptive tester for this problem, while for adaptiv e testers, [ Bla09 ] show ed e O ( k log k + k /ε ) queries suffice. It is imp ortan t to note that all these results use F ourier-analytic techniques. Notably , [ BWY15 ] designed a new algorithm for testing k -jun tas with similar optimal b ounds in the context of testing partial isomorphism of functions. Interestingly , this work deviates from the common F ourier analytic approach and instead presents a combinatorial approach to this problem. This algorithm from [ BWY15 ] will b e used in designing our tester for k -jun tas. Recen tly [ DMN19 ] studied the linear k -jun ta testing problem, where the function f is defined as f : R n → { +1 , − 1 } 3 . This is differen t from our setting of real-v alued functions. It is not clear to us if their techniques can b e generalized to our setting. In terms of low er b ounds, [ FKR + 04 ] show ed a low er b ound of Ω( √ k ) queries for non-adaptive testers, which w as improv ed to e Ω ( k 3 / 2 /ε ) by [ CST + 18 ]. F or adaptive testers, [ CG04 ] show ed an Ω( k ) low er b ound, whic h was then improv ed to Ω( k log k ) b y [ Sağ18 ]. Our result for testing k -jun tas is summarized in the following theorem. 3 A function f : R n → {− 1 , 1 } is said to be a line ar k-junta if there are k unit vectors u 1 . . . , u k ∈ R n and g : R k → {− 1 , 1 } such that f ( x ) = g ( ⟨ u 1 , x ⟩ , . . . , ⟨ u k , x ⟩ ) . 6 Theorem 2.9 (Informal, see Theorem 7.1 ) . L et k ∈ N , and ε, η ∈ (0 , 1) b e p ar ameters such that η < min { O ( ε/k 2 ) , O ( 1 / k 2 log 2 k ) } , and f : R n → R is b ounde d in B ( 0 , 2 √ n ) given via η -appr oximate query ac c ess. Ther e exists a one-side d err or e O  k log k ε  -query tester ( Algorithm 6 ), that distinguishes if f is a k -junta, or is ε -far fr om al l k -juntas, with pr ob ability at le ast 2 / 3 . 2.4 Lo w er b ounds No w we briefly mention our low er b ound results, which hold even for adaptiv e testers. F or all these three properties ( k -linearit y , k -sparse degree- d p olynomials, and k -jun tas), w e pro ve low er b ounds of Ω( max { k , 1 / ε } ) queries. A dditionally , for k -sparse degree- d p olynomials, we prov e a low er b ound of Ω( max { k , d, 1 / ε } ) queries. All our lo wer b ounds follo w from the general reduction from comm unication complexit y , introduced by [ BBM12 ], coupled with some folklore results. W e use Set-Disjointness as the hard instance to prov e our low er b ounds. Theorem 2.10. Given exact query ac c ess to f : R n → R , some k , d ∈ N and a distanc e p ar ameter ε ∈ (0 , 1) , Ω (max { k , 1 / ε } ) queries ar e ne c essary for testing the fol lowing pr op erties with pr ob ability at le ast 2 / 3 : (i) k -line arity. (ii) k -junta. (iii) k -sp arse de gr e e- d p olynomials. Mor e over, for testing k -sp arse de gr e e- d p olynomial, the lower b ound is impr ove d to Ω (max { d, k , 1 / ε } ) . Discussion In this work, we design efficient algorithms for the problem of testing real-v alued functions given via appro ximate queries, o ver contin uous domains, for three prop erties: (i) k -linearit y , (ii) k -sparse, lo w-degree p olynomials, and (iii) k -jun tas. Our work op ens directions to several interesting questions. • W e note that our results hav e constraints on the approximate query parameter η . The first op en question is whether these can b e improv ed. • F urthermore, our k -sparse degree- d p olynomial tester p erforms e O ( d 5 + d 2 / ε + dk 3 ) queries, and our low er b ound for this problem is Ω (max { d, k , 1 / ε } ) . The second op en question is whether the gaps in these b ounds (w.r.t. k , and d ) can be improv ed, e.g., by assuming additional structure on the underlying function, lik e Lipschitzness, etc. • Another interesting direction is to design tolerant testers [ PRR06 ] for these prop erties. This is differen t from the appro ximate query testing notion, as in tolerant testing, the decision b oundary is expanded to require that functions that are sufficien tly close to the prop ert y are also accepted with high probability , whereas in the appro ximate query mo del, we accept functions e f suc h that dist D , ∞ ( f , e f ) ≤ η (p oin t wise η -close) which is a stronger constraint compared to the exp ected ℓ 1 -distance dist D ,ℓ 1 ( f , g ) that we use b et ween functions. • In this work, we ha ve fo cused on optimizing the query complexity in terms of the sparsity parameter k and degree d (for lo w-degree p olynomial testing). It is an interesting problem to optimize the dep endence of η in our arguments. 7 • Only our k -jun ta tester has a one-sided error profile, while our k -linearit y/sparsity testers hav e t wo-sided errors. Designing one-sided error testers for these problems is an op en problem. • Finally , we use the standard Gaussian distribution as the reference distribution. It would b e in teresting to see if our results can b e extended to other concentrated distributions as well. Problem Upp er Bound Restriction k -linearit y e O ( k log k + 1 / ε ) η < min n ε, O  min i ∈ [ n ]: f ( e i )  =0 | f ( e i ) | ( nk ) 2 o k -sparsit y e O ( d 5 + d 2 / ε + dk 3 ) η < min { ε, 1 / 2 2 n } k -jun ta e O  k log k ε  η < min { O ( ε/k 2 ) , O ( 1 / k 2 log 2 k ) } T able 1: A comparison of the upp er b ounds, as w ell as the corresp onding restrictions, for the three sparse representation testing problems. The upp er b ounds, and the restrictions in the three rows follo w sequentially from Theorem 2.2 , Theorem 2.6 , and Theorem 2.9 , resp ectiv ely . 2.5 New T ec hnical Con tributions k -sparsit y T ester: A critical ingredient in proving the k -sparse low degree tester ( Theorem 2.6 ) is a new probabilistic upp er b ound on the maximum singular v alue of Hankel matrices asso ciated with sparse, low-degree p olynomials ( Definition 2.5 ). This may b e of indep enden t in terest, is sketc hed out here, and pro ved in Section 6.2 . Theorem 2.11 (Probabilistic Upp er Bound on σ max ) . L et f : R n → R , f ( x ) = P k i =1 a i M i ( x ) b e a k -sp arse, de gr e e- d p olynomial, wher e M i ’s ar e its non-zer o monomials, and σ max ( H t ( f , u )) denote the lar gest singular value of the t -dimensional Hankel matrix asso ciate d with f at a p oint u ∈ R n , H t ( f , u ) . Then, for any γ ∈ (0 , 1) , with a ≜ ( a 1 , · · · , a k ) ⊤ ∈ R k , Pr u ∼N ( 0 ,I n )   σ max ( H t ( f , u )) ≥ ∥ a ∥ 2 2 2 d/ 2 ⌈ d/ 2 ⌉ ! + s k γ 2 d/ 2 √ d ! ! 2 t   ≤ γ . Pr o of Sketch. F or any v ector v ∈ R n , let | v | ∈ R n ≥ 0 denote the vector with the absolute v alues of the co ordinates of v . Then, for any z , u ∈ R n , using the triangle inequalit y , and a V andermonde-like decomp osition of H u ≜ H t ( | f | , | u | ) , (from Observ ation 6.4 ), where | f | ( x ) ≜ P i ∈ [ k ] | a i | M i ( x ) , w e can upp er-bound | z ⊤ H u z | b y | z | ⊤ V ⊤ D V | z | = ∥ D 1 2 V | z |∥ 2 2 , where D = diag ( | a | ) , V is the V andermonde matrix V t ( | M 1 ( u ) | , . . . , | M k ( u ) | ) ∈ R k × t , and M ( | u | ) = | M ( u ) | for any monomial M ( · ) . By the Couran t-Fischer characterization, we hav e σ max ( H u ) ≤ max ∥ z ∥ 2 ≤ 1 ∥ D 1 2 V | z |∥ 2 2 ≤ max ∥ z ∥ 2 ≤ 1 ∥ D 1 2 V ∥ 2 op ∥ z ∥ 2 2 ≤ ∥ D 1 2 V ∥ 2 F . Using some results from [ Ela61 ] for the moments of the folded normal distribution, along with the prop erties of the Gamma function and the fact that all the monomials M i ha ve total degree ≤ d , w e can upp er b ound E u ∼N ( 0 ,I n ) [ | M i ( u ) | ] ≤ 2 d/ 2 ⌈ d/ 2 ⌉ ! , and V ar[ | M i ( u ) | ] ≤ 2 d · d ! . Finally , using Chebyshev’s inequality and the union b ound (ov er the k monomials) giv es the upp er b ound for σ max ( H u ) . 8 Remark 2.12. The or em 2.11 se ems to b e extendable to mor e gener al me an-zer o distributions of u with appr opriate c onc entr ation (e.g. sub gaussian, sub exp onential) sinc e the crux of the pr o of, in addition to the use of pr op erties of the Hankel matrix (which do not dep end on the distribution of u ), is to b ound al l the de gr e e d moments of the distribution. k -linearit y T ester: F or k -linearit y , our tester is similar to the algorithm proposed in [ Bsh23 ] for testing k -linearit y of functions f : F n 2 → F 2 , with the BLR test replaced by the linearity tester from [ ABF + 23 , AKM25 ]. The crucial difficulties in the analysis ov er the reals come from (i) the η -appro ximate query access, and (ii) the use of ℓ 1 -distance for farness. W e first reject all functions whic h are far from linearity by means of the linearity test, and, conditioned on the fact that the tester do es not reject with high probability , pro ceed to testing the self-corrected function instead (with closeness to linearity guaranteed). But we now hav e only appro ximate query access to the function, whic h w e test for k -linearit y b y splitting the v ariables in to O ( k 2 ) random buck ets as in [ Bsh23 ] and identify the influential ones. Ho wev er, there is no analogous result for real-v alued functions. Hence, we present a new analysis in Claim 5.11 for the FindInfBucket ( Algorithm 2 ) that p erforms a binary searc h ov er the buc kets, but with a different test for influential buck ets that accounts for η -appro ximate queries. The analysis with approximate queries inv olves the appropriate use of an ti-concentration results ( Theorem 4.15 ) for real linear p olynomials. k -jun ta tester: Our tester is inspired b y the junta testers used in [ Bla09 ] and [ BWY15 ], but our definition of influence ( Definition 7.2 ) is different ( ℓ 1 rather than Hamming), to work with appro ximate query oracle. Theorem 7.3 , Lemma 7.4 , and Lemma 7.5 , in our analysis are similar to that of [ BWY15 ]. The significan t difference in analysis for the ℓ 1 distance comes in our Claim 7.10 , and the subsequen t pro of of Theorem 7.1 . Organization of the pap er In Section 3 , we present an ov erview of our results and tech niques, follo wed by a discussion of the preliminaries required, in Section 4 . In Section 5 , we present our k -linearit y tester, whic h is follo wed b y our tester for k -sparse low-degree p olynomials in Section 6 , and our k -jun ta tester in Section 7 . Finally , in Section 8 , w e present the low er b ounds. Some asso ciated pro ofs and subroutines from prior w orks are mov ed to the app endix for brevity . 3 T ec hnical Ov erview W e present a brief ov erview of our techniques, starting with k -linearit y testing. T esting k -linearit y W e build up on the self-c orr e ct and test approac h of [ BLR90 , HK07 ]. Instead of directly testing if f is k -linear, w e construct a function g self-correct suc h that if f is k -linear, then g self-correct will also b e k -linear. Moreov er, we simulate queries to g self-correct using queries to f , and test this newly constructed function g self-correct . W e use the Gaussian distribution as the reference distribution, since there are no uniform distributions o ver contin uous domains with infinite supp ort. (This deviates from the self-correction approac h of [ BLR90 ].) In particular, w e ev aluate f on a set of points sampled from N ( 0 , I n ) to construct the self-corrected function g self-correct . T o deal with the fact that different p oin ts from N ( 0 , I n ) hav e different probabilit y masses, the idea is to radially pro ject the sampled p oin ts from N ( 0 , I n ) into a small Euclidean ball B ( 0 , r ) of a small (constant) radius ( r = 1 / 50 suffices) suc h 9 that the probability masses of all p oin ts sampled from that ball is roughly the same. Moreo ver, since w e work with an approximate oracle, we use the following notion of the self-corrected function, g ( p ) ≜ κ p · med x ∼N ( 0 ,I n )  f  p κ p − x  + f ( x )  where κ p : R n → R is a con traction factor, defined as: κ p ≜ ( 1 , if ∥ p ∥ 2 ≤ r ⌈∥ p ∥ 2 /r ⌉ , if ∥ p ∥ 2 > r so that p /κ p ∈ B ( 0 , r ) , and med denotes the median function. This definition of self-correction function w as used in [ FY20 , ABF + 23 , AKM25 ]. T o test k -linearit y ( Algorithm 1 ), we first test if f is p oin twise close to some additive (ak a. linear) function using Appro xima te Additivity Tester ( Algorithm 7 ). If it rejects f , w e also reject f . Ho wev er, if Algorithm 7 do es not reject, then the self-corrected function g is p oin t wise close to some linear function. As we only hav e access to an approximate oracle access to f , we can’t simulate g exactly . So, w e use Appro xima te- g , the approximate query oracle for g (part of Algorithm 8 ). The w ork of [ ABF + 23 , AKM25 ] prov es: (i) g and Appro xima te- g are p oin twise close in B ( 0 , r ) , and (ii) f and Appro xima te- g are also p oin twise close. So, f is p oin t wise close to Appro xima te- g . W e partition the n -v ariables [ n ] into k 2 buc kets uniformly at random. As a result of the partition, if f is k -linear, the influen tial v ariables will b e separated into different buck ets w.h.p. W e can then detect them via the subroutine FindInfBuck et ( Algorithm 2 ), which recursively isolates the v ariables in a buc ke t and returns only the influential ones. W e then reject if the n umber of buc kets with influen tial v ariables (v ariables whose v alues determine the v alue of f ) is more than k . T esting k -sparsit y Our k -sparse, degree- d p olynomial tester ( Algorithm 4 ) adopts a similar approac h. W e first test if f is a lo w-degree p olynomial, using the Appro xLowDegreeTester ( Algorithm 9 ) from [ ABF + 23 , AKM25 ]. If it rejects f , we also reject f . Ho wev er, if Algorithm 9 do es not reject f , then f is p oin t-wise close to a low-degree p olynomial. As in k -linearit y testing, we use a self-corrected function g from [ ABF + 23 , AKM25 ]: F or p oin ts p ∈ B ( 0 , r ) , g ( p ) is the (w eighted) median v alue of g q ( p ) ≜ P d +1 i =1 ( − 1) i +1  d +1 i  f ( p + i q ) , w eighted according to the probabilit y of q ∼ N ( 0 , I n ) , i.e., g ( p ) ≜ med q ∼N ( 0 ,I n ) [ g q ( p )] . In tuitively , g q ( p ) is the v alue that f should take if, when restricted to the line L p , q ≜ { p + t q , t ∈ R } , f w ould be a degree- d univ ariate polynomial. T aking the w eighted median ov er all directions q ∼ N ( 0 , I n ) , ensures that the self-correction prop ortionately resp ects the v alues of f , in a lo cal neigh b orho od of p . F or p ∈ B ( 0 , r ) , g is defined via radial extrap olation from within B ( 0 , r ) along the radial line L 0 , p . Giv en exact query access to f , w e can sim ulate query access to the self-corrected function g . As we only hav e approximate query access to f , we use Appr oxQuer y- g ( Algorithm 10 , the approximate oracle to the self-corrected function asso ciated with f ), which was prov ed to b e p oin t wise close to g . As a result, f will b e p oin t wise close to Appro xQuer y- g as well. Once we hav e that f is close to a low-degree p olynomial, we use a Hankel matrix ( Definition 2.5 ) based characterization for sparse p olynomials. [ GJR10 , BOT88 ] prov ed Observ ation 6.4 : a p olynomial f (o ver finite fields) is k -sparse, if and only if its asso ciated Hankel matrix has a non-zero determinant. 10 This can b e efficiently tested with only 2 k + 1 queries to f . F or p olynomials ov er the reals with exact-queries, the idea from [ GJR10 ] can b e extended to giv e a zero-error (p erfect soundness and completeness), zero-gap (do es not require ε -farness for ε > 0 in the No case) tester, making 2 k + 1 queries for deciding whether (i) f is k -sparse, or (ii) f is not k -sparse (pro ved in Lemma 6.5 ). This result is even stronger than what [ GJR10 ] gets for finite fields, since the zero set of any p olynomial o ver the reals has zero Gaussian (or Leb esgue) measure, whereas the corresp onding probabilit y needs to b e b ounded in terms of the degree of the p olynomials ov er finite fields (e.g. b y the Sc hw artz-Zipp el lemma). Unfortunately , since we only ha ve approximate query access to f , this technique no longer w orks, as determinants of sums of matrices do not b eha ve nicely . So we take a probabilistic approach, sho wing that the noisy Hank el matrix constructed from the appro ximate query to f is not to o far from the exact Hankel matrix (prov ed in Observ ation 6.6 ). Finally , we show in Theorem 6.7 using W eyl’s inequality ( Theorem 4.14 ), that if f is a k -sparse, lo w-degree p olynomial, then the smallest eigen v alue of the noisy Hank el matrix asso ciated with Appr oxQuer y- g is not to o large. Combining them all, our main result is prov ed in Section 6.3 . Since, we only hav e access to the Hank el matrix of e f = Appr oxQuer y- g whic h is η ′ -close to f (on all the query p oin ts, whp), we can express e H u ≜ H t ( e f , u ) (for t = k + 1 ) of such a e f as the sum of H u ≜ H t ( f , u ) and a Hankel-structured error matrix with b ounded sp ectral/op erator norm ( Observ ation 6.6 ). F rom this, and W eyl’s inequality , we get completeness as long as we Accept when the smallest singular v alue σ min ( e H u ) ≤ η ′ ( k + 1) in all Θ( d ( k + 1) 2 ) rounds. T o prov e soundness of the tester, we need to show that σ min ( e H u ) > η ′ ( k + 1) in some round with probability ≥ 2 / 3 , as long as Q ( u ) ≜ det ( H u ) is a non-zero p olynomial. This analysis (in Theorem 6.7 ) follo ws by inv oking the probabilistic upp er b ound on σ max ( H u ) , for u ∼ N ( 0 , I n ) . T esting k -jun ta Finally , we discuss our algorithm for testing k -jun tas ( Algorithm 6 ). Our approach is to first randomly partition the n -v ariables in to k 2 buc kets. If f is a k -jun ta, the k influen tial v ariables will b e separated in to distinct buck ets w.h.p (by the birthday parado x). No w we run O ( k /ε ) iterations to find if there exists any influential v ariable in any buck et, using the subroutine FindInfBuck et ( Algorithm 2 ). After these iterations, if w e find more than k influen tial v ariables in f , w e reject it. Otherwise, w e accept f . Our analysis follows a combinatorial style similar to [ BWY15 ]. W e w ould like to note that although we use FindInfBuck et for k -linearit y testing as well, the analysis here significan tly deviates from that of k -linearit y testing and is presen ted in Claim 7.10 . The main result is formally prov ed in Section 7.1 . 4 Preliminaries Notations Throughout this w ork, we use b oldface letters to represent ve ctors of length n and normal face letters for v ariables. Sp ecifically , e i ≜ (0 , . . . , 0 , 1 i , 0 , . . . , 0) denotes the i th standard unit v ector. F or n ∈ N , let [ n ] denote the set { 1 , . . . , n } . F or a matrix A , let ∥ A ∥ ∞ , ∥ A ∥ op , and ∥ A ∥ F denote the suprem um, op erator, and F rob enious norms of A , resp ectiv ely . See [ HJ12 ] for the formal definitions. F or concise expressions and readabilit y , we use the asymptotic complexit y notion of e O ( · ) , where we hide p oly-logarithmic dep endencies of the parameters. F or any f : R n → R , let ∥ f ∥ −∞ ( / ∞ ) ,C denote the infim um(resp. suprem um) v alue of f o ver some C ⊆ R n . Let Π n b e the class of functions R n → R satisfying some particular first-order prop ert y and let Π = ∪ n ≥ 1 Π n . Definition 4.1 ( ℓ 1 -distance) . Let D = {D n } n ≥ 1 is family of distributions with D n b eing a distribution on R n . F or tw o arbitrary functions f , g : R n → R , the ℓ 1 -distance b et w een f and g is defined as: dist D ,ℓ 1 ( f , g ) ≜ E x ∼D n [ | f ( x ) − g ( x ) | ] . 11 W e also define the ℓ p -distance of f to the class Π n , and hence the class Π , by dist D ,ℓ p ( f , Π) ≜ dist D ,ℓ p ( f , Π n ) ≜ inf g ∈ Π n dist D ,ℓ p ( f , g ) . Remark 4.2. Sinc e our distributions ar e supp orte d over c ontinuous sp ac es, the distanc es we c onsider ar e also define d over such c ontinuous sp ac es, i.e., dist D ,ℓ j ( · , · ) ≡ dist D ,L j ( · , · ) , for al l j . W e hav e dist D ,ℓ p ( f , Π) = 0 if and only if there exists a function g ∈ Π n whic h agrees with f almost everywhere with respect to D n (in the measure-theoretic sense). W e will only concern ourselv es with p ∈ { 0 , 1 } since w e are dealing with scalar-v alued functions (ak a functionals) and hence ∥ f ( x ) − g ( x ) ∥ p = | f ( x ) − g ( x ) | for all p > 0 . The notion of ℓ 1 -distance makes more sense in the case of real-v alued functions, esp ecially if the ev aluation of f is not exact. T o see this, consider the case when g = f + ε for some ε > 0 , then dist D ,ℓ 1 ( f , g ) = ε whereas dist D ,ℓ 0 ( f , g ) = 1 . W e hav e the following observ ation that sho ws that o ver finite fields, any ( k + 1) -linear function is alw ays far from any k -linear function in ℓ 0 -distance. Observ ation 4.3. F or 1 ≤ k < n , for any ( k + 1) -line ar function f : F n 2 → F 2 , we have dist ℓ 0 ( f , g ) = 1 / 2 for any k -line ar function g : F n 2 → F 2 . Pr o of. Let f ( x ) = P i ∈ [ n ] c i x i , and S ≜ { i ∈ [ n ] : c i  = 0 } with | S | = k + 1 , since f is ( k + 1) -linear. F or any k -linear g with g ( x ) = P i ∈ [ n ] a i x i , there is at least one i ∗ ∈ S for which a i ∗ = 0 . Then dist ℓ 0 ( f , g ) = Pr x ← F n 2 [ f ( x )  = g ( x )] = Pr x ← F n 2 [ f ( x ) + g ( x ) = 1] (Note c i ∗ = 1 , a i ∗ = 0 ) = Pr x ← F n 2 " n X i =1 ( a i + c i ) x i = 1 # = Pr x ← F n 2   x i ∗ + X i  = i ∗ ( a i + c i ) x i = 0   = 1 2 . Unfortunately , there is no general analogous result in the real case with the ℓ 1 -distance, but there is an upp er b ound, provided the distribution D n has some concen tration. Claim 4.4. If f , g : R n → R ar e line ar functions with f ( 0 ) = g ( 0 ) , then ∥ f − g ∥ −∞ , supp ( D ) ≤ dist D ,ℓ 1 ( f , g ) ≤ ∥ f − g ∥ 2 · E x ∼D n ∥ x ∥ 2 . Pr o of. Supp ose f ( x ) = P i ∈ [ n ] a i x i and g ( x ) = P i ∈ [ n ] b i x i for some a , b ∈ R n . Let h = f − g , so that h ( x ) = P i ∈ [ n ] ( a i − b i ) x i = ⟨ a − b , x ⟩ . W e hav e, by Cauch y-Sc hw arz inequality , dist D ,ℓ 1 ( f , g ) ≜ E x ∼D n | h ( x ) | ≤ ∥ a − b ∥ 2 · E x ∼D n ∥ x ∥ 2 . and the fact that E [ | h ( x ) | ] ≥ inf x ∼D | h ( x ) | giv es the low er b ound. Our definitions hold for the general reference distributions D whic h are suitably concen trated. Later, we w ork with the standard Gaussian distribution N ( 0 , I n ) , which also has this desired concen tration prop ert y . Definition 4.5 (Concentrated distribution) . Let ε ∈ (0 , 1) , R ≥ 0 , and c ∈ R n . A distribution D supp orted on R n is ( ε, R, c ) - c onc entr ate d if most of its mass is con tained in a ball of radius R cen tered at some p oin t c ∈ R n , i.e., Pr p ∼D [ p ∈ B( c , R )] ≥ 1 − ε. 12 F act 4.6 ([ BHK20 , Theorem 2.9]) . The standard Gaussian distribution N ( 0 , I n ) is (0 . 01 , 2 √ n, 0 ) - concen trated. F or brevity , we ma y omit the third en try corresp onding to the center of the ball, when the center is the origin. F or example, (0 . 01 , 2 √ n ) -concen trated means (0 . 01 , 2 √ n, 0 ) -concen trated. Definition 4.7 (Prop ert y T ester) . Let P b e a real function prop ert y . An algorithm is said to b e a tester for P with resp ect to distance measure dist ( · , · ) ; with proximit y parameter ε > 0 , completeness error c ∈ (0 , 1) , and soundness error s ∈ (0 , 1) if, given query access (either exact, or η -appro ximate query access) to a function f : R n → R , and sample access to a reference distribution D n on R n , the algorithm p erforms q ( n, d, k , ε, c, s ) queries to f and: (i) Outputs A ccept with probability ≥ 1 − c (o ver the randomness of the algorithm), if f ∈ P . (ii) Outputs Reject with probability ≥ 1 − s , if dist ( f , g ) ≥ ε for all g ∈ P . 4.1 Preliminaries on P olynomials and Linear Algebra W e briefly discuss some notions, and prop erties of p olynomials: Definition 4.8 (Monomials) . Supp ose x 1 , . . . , x n are indeterminates. A monomial in these inde- terminates is a pro duct of the form x α 1 1 x α 2 2 · · · x α n n where α ≜ ( α 1 , . . . , α n ) ∈ N n , whic h we also denote b y x α . The total degree of monomial x α is ∥ α ∥ 1 ≜ α 1 + · · · + α n . The degree of x α in the v ariable x i is α i . The individual degree of x α is ∥ α ∥ ∞ ≜ max i ∈ [ n ] α i . Over a field F , each monomial x M = x M 1 1 · · · x M n n for M ∈ N n corresp onds to a function M : F n → F , M ( z ) ≜ z M 1 1 · · · z M n n . Definition 4.9 (Polynomials) . A p olynomial in x 1 , . . . , x n (ak a an n -v ariate p olynomial) ov er a field F is a finite F -linear com bination of monomials in x 1 , . . . , x n . That is, an n -v ariate p olynomial ov er F is of the form P ( x ) = P m i =1 a i x M i where m ≥ 0 , M 1 , . . . , M m ∈ N n and a 1 , . . . , a m ∈ F \ { 0 } . The set of all suc h p olynomials is denoted as F [ x 1 , . . . , x n ] . Definition 4.10 (Polynomial sparsity) . The sp arsity of a p olynomial P ( x ) = P m i =1 a i x M i , denoted ∥ P ∥ 0 , is the num b er of non-zero co efficien ts a i in its monomial-basis representation. The unique p olynomial P with ∥ P ∥ 0 = 0 is the zer o p olynomial , denoted by P ≡ 0 . Definition 4.11 (T otal and individual degrees) . The total de gr e e deg ( P ) of a non-zero p olynomial P ( x ) = P i ∈ [ m ] a i x M i is the maximum total degree of its monomials, i.e., deg ( P ) ≜ max i ∈ [ m ] ∥ M i ∥ 1 . Similarly , the individual de gr e e ideg ( P ) of P is the maximum individual degree of its monomials, i.e., ideg( P ) ≜ max i ∈ [ m ] ∥ M i ∥ ∞ . The total, as well as the individual degree of the zero p olynomial P ≡ 0 is defined to b e −∞ . In this w ork, w e will b e primarily w orking on the total degree. So, we will use degree to represent total degree when it is clear from the context. When F is an infinite field, each p olynomial P ( x 1 , . . . , x n ) = P i ∈ [ m ] a i x M i o ver F corresp onds to a unique F -v alued function P : F n → F with P ( z ) = P m i =1 a i M i ( z ) = P m i =1 a i z M i, 1 1 · · · z M i,n n . Suc h functions are referred to as p olynomial functions . Unlik e with finite fields, the equality of tw o real p olynomial functions implies the equality of the formal p olynomials. Th us, we use the formal p olynomial P ∈ R [ x 1 , . . . , x n ] and the p olynomial function P : R n → R interc hangeably . 13 Definition 4.12. Let P tot n,d,k denote the class of all n -v ariate real p olynomials with total de gr e e ≤ d and sparsit y ≤ k . P tot n,d,k ≜ ( P ∈ R [ x 1 , . . . , x n ] | P = k X i =1 a i x M i : ∀ i ∈ [ k ] , a i ∈ R , M i ∈ N n : ∥ M i ∥ 1 ≤ d ) . Similarly , let P ind n,d,k denote the class of all n -v ariate real p olynomials with individual de gr e e ≤ d in all v ariables and sparsity ≤ k . P ind n,d,k ≜ ( P ∈ R [ x 1 , . . . , x n ] | P = k X i =1 a i x M i : ∀ i ∈ [ k ] , a i ∈ R , M i ∈ {{ 0 } ∪ [ d ] } ⊗ n ) . W e will use the notion of V andermonde matrices in this work. Definition 4.13 (V andermonde matrix and determinan t) . F or an y m, n > 0 and x 1 , . . . , x m ∈ ( R, + , · ) , the m × n V andermonde matrix with no des x 1 , . . . , x m denoted b y V n ( x 1 , . . . , x m ) is: V n ( x 1 , . . . , x m ) ≜      1 x 1 · · · x n − 1 1 1 x 2 · · · x n − 1 2 . . . . . . . . . . . . 1 x m · · · x n − 1 m      . If m = n , det ( V n ( x 1 , . . . , x n )) = Q 1 ≤ i 0 , Pr x ∼N ( 0 ,I n ) [ | f ( x ) − t | ≤ ε ] ≤ O ( d ) ε 1 /d . A recen t result of Glazer and Mikulincer [ GM22 ] pro vided a more suitable form for applying Carb ery-W right to p olynomials, where the v ariance is low er b ounded in terms of the co efficien ts. Theorem 4.16 ([ GM22 , Corollary 4]) . If f : R n → R is a p olynomial f ( x ) = P k i =1 a i x M i of de gr e e d , then ther e exists an absolute c onstant C > 0 such that for any t ∈ R and ε > 0 , Pr x ∼N ( 0 ,I n ) [ | f ( x ) − t | ≤ ε ] ≤ C d  ε co eff d ( f )  1 /d , wher e co eff 2 d ( f ) ≜ X i ∈ [ k ]: ∥ M i ∥ 1 = d a 2 i . 14 The P aley-Zygmund anti-concen tration result will b e useful for analyzing our junta tester. Theorem 4.17 ([ PZ32 ]) . F or a r andom variable X ≥ 0 with finite varianc e, for al l θ ∈ [0 , 1] , Pr[ X ≥ θ E [ X ]] ≥ (1 − θ ) 2 E 2 [ X ] E [ X 2 ] . 5 k -linearit y T esting In this section, w e present and analyze our algorithm Test- k -Linear ( Algorithm 1 ). Theorem 5.1 (Generalization of Theorem 2.2 ) . L et k ∈ N , f : R n → R is a function b ounde d in the b al l B ( 0 , 2 √ n ) , and ε, η ∈ (0 , 2 / 3) b e such that η < min n ε, O  min i ∈ [ n ]: f ( e i )  =0 | f ( e i ) | ( nk ) 2 o , wher e e i denotes the i th standar d unit ve ctor. Given η -appr oximate query ac c ess to f , ther e exists a tester Test- k -Linear ( Algorithm 1 ) that p erforms e O ( k log k + 1 / ε ) queries and guar ante es: • Completeness: If f is a k -line ar function, then Algorithm 1 A ccepts with pr ob ability at le ast 2 / 3 . • Soundness: If f is ε -far fr om al l k -line ar functions, then A lgorithm 1 Rejects with pr ob ability at le ast 2 / 3 . Algorithm 1: Test- k -Linear 1 Inputs: η -appro ximate query oracle e f for f : R n → R , k ∈ N , pro ximity parameter ε ∈ (0 , 1) . 2 Output: Returns Accept iff f is a k -linear function. 3 Run Appr oxima te Additivity Tester ( f , N ( 0 , I n ) , η , ε, 2 √ n ) ( Algorithm 7 ). 4 If Algorithm 7 rejects, return Reject . 5 Set r = O ( k 2 ) . 6 Buc ket each i ∈ [ n ] into B = { B 1 , . . . , B r } , uniformly at random. ▷ [ n ] = ⊎ r i =1 B i . 7 InfBuck ets = FindInfBuckets ( Appr o xima te- g , B , [ r ]) . ▷ FindInfBuck ets = Algorithm 3 8 if | InfBuckets | > k then 9 return Reject . 10 return A ccept . Algorithm 1 uses tw o subroutines FindInfBuck et ( Algorithm 2 ) and FindInfBuck ets ( Algorithm 3 ). W e will first present and analyze them in Section 5.1 , and then analyze Algorithm 1 in Section 5.2 . A closely related (and efficiently testable ) notion of additivity ma y b e noted here. Definition 5.2 (Additiv e function) . A function f : A → B is additive , if for all x, y ∈ A, f ( x ⊕ A y ) = f ( x ) ⊕ B f ( y ) , where ⊕ A and ⊕ B denote the bit wise-xor op erations in A and B , resp ectiv ely . Ov er finite domains, additivity implies linearity . But ov er contin uous domains, this isn’t alwa ys the case. How ev er, for contin uous functions, testing additivity suffices from the following fact: F act 5.3 ([ Kuc09 , Section 5.2]) . F or con tinuous functions, additivity is equiv alen t to linearity . Algorithm 1 uses a query oracle to the self-corrected function, Appro xima te- g (presen ted in Algorithm 10 ). W e present a brief discussion of self-correction for additivity testing: 15 Self-Correction W e use the definition of self-correction from [ ABF + 23 , AKM25 ], who studied the problem of testing additiv e (linear) functions, giv en appro ximate oracle access. Let r b e a sufficien tly small rational; r ≜ 1 / 50 suffices. Define the v alue of the self-corrected function g at a point p ∈ B ( 0 , r ) as the (w eighted) median v alue of g x ( p ) ≜ f ( p − x ) + f ( x ) , eac h weigh ted according to its probability mass under x ∼ N ( 0 , I n ) . F or p oin ts p ∈ B ( 0 , r ) , w e pro ject them into the ball b y scaling by a sufficiently large contraction factor that dep ends on the magnitude of p . g ( p ) ≜ κ p · med x ∼N ( 0 ,I n )  g x  p κ p  = κ p · med x ∼N ( 0 ,I n )  f  p κ p − x  + f ( x )  , (1) where κ p : R n → R is defined as κ p ≜ ( 1 , if ∥ p ∥ 2 ≤ r ⌈∥ p ∥ 2 /r ⌉ , if ∥ p ∥ 2 > r , so that p /κ p ∈ B( 0 , r ) . W e note the following results from [ ABF + 23 , AKM25 ] ab out their approximate additivity tester. Theorem 5.4 ([ ABF + 23 , Theorem D.1], and [ AKM25 , Theorem 3.3]) . L et α, ε > 0 , for L > 0 , supp ose f : R n → R is a function that is b ounde d in the b al l B (0 , L ) and D b e an unknown ( ε/ 4 , R ) - c onc entr ate d distribution. Ther e exists a one-side d err or, O ( 1 / ε ) -query tester ( Algorithm 7 ) which with pr ob ability at le ast 99 / 100 , distinguishes when f is p ointwise α -close to some additive function and when, for every additive function h , Pr p ∼D [ | f ( p ) − h ( p ) | > O ( Rn 1 . 5 α )] > ε . Lemma 5.5 ([ ABF + 23 , Lemma D.3 and D.6]) . If TestAdditivity ( f , 3 α ) ( A lgorithm 8 ) ac c epts with pr ob ability at le ast 1 / 3 , then g is a 42 α -additive function inside the smal l b al l B ( 0 , r ) , and furthermor e, for every p ∈ B( 0 , r ) it holds that Pr x ∼N ( 0 ,I n ) [ | g ( p ) − g x ( p ) | ≥ 12 α ] < 12 / 125 . Lemma 5.6 ([ ABF + 23 , Lemma D.4 and D.5]) . If TestAdditivity ( f , 3 α ) ac c epts with pr ob ability at le ast 1 / 3 , then for every p , q ∈ B( 0 , r ) with ∥ p + q ∥ 2 ≤ r , it holds that | g ( p + q ) − g ( p ) − g ( q ) | ≤ 42 α. The follo wing lemma gives us a wa y to scale the closeness for degree- d p olynomials. Lemma 5.7 ([ ABF + 23 , Lemma 4.19]) . L et R > r ′ > 0 b e any r e al numb ers. If g is p ointwise η -close to a de gr e e- d p olynomial in B ( 0 , r ′ ) , then g is p ointwise (12 R/r ′ ) d η -close to a de gr e e- d p olynomial on al l p oints in B( 0 , R ) . Using Lemma 5.5 , Lemma 5.7 and the structure of Appr oxima te- g subroutine, they b ound the distance b et w een g and Appr oxima te- g . Claim 5.8. If TestAdditivity ( f , 3 α ) ( Algorithm 8 ) ac c epts with pr ob ability at le ast 1 / 3 , then Appr oxima te- g is p ointwise 6 α -close to g , in the b al l B( 0 , r ) with high pr ob ability; i.e., Pr p ∼D [ | g ( p ) − Appr oxima te- g ( p ) | ≤ 6 α (12 R /r ) | p ∈ B( 0 , R )] ≥ 1 − ε/ 4 . The following observ ation directly follows using the triangle inequality , along with the ab o ve claim and the notion of η -appro ximate queries. Observ ation 5.9. If Appro xima te Additivity Tester do es not r eje ct f with pr ob ability at le ast 2 / 3 , then fol lowing the description of Appro xima te Additivity Tester , we have Pr p ∼D [ | f ( p ) − Appro xima te- g ( p ) | ≤ 750 R n 1 . 5 α | p ∈ B( 0 , R )] ≥ 1 − ε 4 . W e will inv ok e them with D = N ( 0 , I n ) , implying R = 2 √ n (from F act 4.6 ). 16 5.1 Analyses of Subroutines Algorithm 2: FindInfBuck et ( f , B , S ) 1 Inputs: η -approximate query oracle e f for f : R n → R , Random Buck eting B = { B 1 , . . . , B r } of [ n ] , ∅  = S ⊆ [ r ] 2 Output: Either ∅ , or an influential buck et B j (with j ∈ S ) of f . B j is an influential buck et if x v is an influen tial v ariable of f for some v ∈ B j . 3 Let V ≜ { i ∈ [ n ] : i ∈ B j for some j ∈ S } . ▷ The #v ariables in V will b e | V | ≈ | S | n r . 4 Sample t wo indep enden t Gaussian vectors x , y ∼ N ( 0 , I n ) . 5 if | e f ( x V y V ) − e f ( y ) | ≤ 2 η then 6 return ∅ . 7 else 8 if S = { j } for some j then 9 return B j . 10 else 11 P artition S into tw o parts: S ( L ) and S ( R ) with 1 ≤ | S ( R ) | ≤ | S ( L ) | ≤ | S ( R ) | + 1 . 12 ret L ← FindInfBuck et ( f , B , S ( L ) ) 13 if ret L = ∅ then 14 return FindInfBuck et ( f , B , S ( R ) ). 15 else 16 return ret L . The subroutine FindInfBuck et is presented in Algorithm 2 . T o analyze it, we need: Definition 5.10 (Influen tial buc ket for linear functions) . F or a function f : R n → R , a buck et B ⊆ [ n ] is said to b e an influential bucket if there exists at least one v ariable x i , i ∈ B , whic h is influen tial with resp ect to f (i.e., the v alue of f ( x ) changes with a c hange in x i ). F or a linear f , with f ( x ) = P i ∈ [ n ] a i x i , x i is influen tial w.r.t f if and only if a i  = 0 , and hence B is an influen tial buc ket iff a i  = 0 for some i ∈ B . Claim 5.11 (Correctness of FindInfBuck et ) . If f : R n → R , with f ( x ) = P i ∈ [ n ] a i x i is given via an η -appr oximate query or acle ˜ f , wher e η ≤ 1 100 k 2 min k ∈ [ n ]: a k  =0 | a k | , B = { B 1 , . . . , B r } is a p artition of [ n ] , and ∅  = S ⊆ [ r ] , then FindInfBucket ( f , B , S ) ( A lgorithm 2 ) guar ante es the fol lowing: 1. If none of the buckets { B i , i ∈ S } ar e influential, FindInfBuck et ( f , B , S ) always r eturns ∅ and p erforms exactly 2 queries to f . 2. Otherwise, with pr ob ability at le ast 1 − 8 ⌈ log | S |⌉ 2 / 10 k 2 , FindInfBuck et ( f , B , S ) r eturns B j for some j ∈ S which is an influential bucket, and p erforms ≤ 8 ⌈ lg( | S | ) ⌉ 2 queries to f . Pr o of. Let x and y b e the Gaussian random vectors sampled by Algorithm 2 , and w ≜ y V x V ∈ R n . (P art 1) Since no B j is influen tial for j ∈ S , we ha ve a i = 0 for all i ∈ V , giving us f ( w ) − f ( y ) = X j ∈ V a j ( w j − y j ) | {z } = y j − y j =0 + X j ∈ V a j |{z} =0 ( w j − y j ) | {z } = x j − y j = 0 . 17 Hence, | e f ( w ) − e f ( z ) | = | e f ( w ) − f ( w ) − ( e f ( z ) − f ( z )) | ≤ ≤ η z }| { | e f ( w ) − f ( w ) | + ≤ η z }| { | e f ( z ) − f ( z ) | ≤ 2 η b y the triangle inequality , ensuring the chec k in Line 5 ( Algorithm 2 ) alw ays holds. So in this case, FindInfBuck et ( f , B , S ) will alw ays return ∅ , and make exactly 2 queries to f . (P art 2) It only remains to prov e the second part of the claim (when there exists some j ∈ S with B j b eing influential, i.e., there exists k ∈ B j suc h that a k  = 0 ). W e do so by strong induction on the size of S . When | S | = 1 , we ha ve S = { j } and V = B j , with a k  = 0 for some k ∈ B j . Then f ( w ) − f ( y ) = P k ∈ B j a k ( x k − y k ) . This implies Pr x , y ∼N ( 0 ,I n ) [ f ( w ) = f ( z )] = 0 , since P k ∈ B j a k ( x k − y k ) ≡ 0 , and an y non-zero p olynomial ov er reals v anishes only on a set (of its zero es) of measure zero. Moreo ver, Pr x , y ∼N ( 0 ,I n ) [ | e f ( w ) − e f ( y ) | > 2 η ] ≥ Pr x , y ∼N ( 0 ,I n ) [ | f ( w ) − f ( y ) | > 4 η ] = 1 − Pr x , y ∼N ( 0 ,I n )      X k ∈ B j a k ( x k − y k )    ≤ 4 η   ≥ 1 − Pr x , y ∼N ( 0 ,I n )   max k ∈ B j | a k |    X k ∈ B j ( x k − y k )    ≤ 4 η   = 1 − Pr x , y ∼N ( 0 ,I n ) " | P k ∈ B j ( x k − y k ) | p 2 | B j | ≤ 2 √ 2 η p | B j | max k ∈ B j | a k | # . As x , y ∼ N ( 0 , I n ) are independent, we hav e P k ∈ B j x k − y k √ 2 | B j | ≡ P k ∈ B j x k − y k √ 2 | B j | ∈ R [ ∪ k ∈ B j { x k , y k } ] , Cov [ x i , x j ] = Cov [ y i , y j ] = 0 , for all i  = j ∈ B j , and Cov [ x i , y j ] = 0 , for all i, j ∈ S , giving us V ar x , y ∼N ( 0 ,I n )   X k ∈ B j x k − y k p 2 | B j |   = 1 2 | B j | X k ∈ B j  V ar x k ∼N (0 , 1) [ x k ] + V ar y k ∼N (0 , 1) [ y k ]  = 1 . No w, applying Theorem 4.15 on P k ∈ B j x k − y k √ 2 | B j | , (with d = 1 , n = 2 | B j | , and t = 0 ) w e get Pr x , y ∼N ( 0 ,I n ) " | P k ∈ B j ( x k − y k ) | p 2 | B j | ≤ 2 √ 2 η p | B j | max k ∈ B j | a k | # ≤ O 2 √ 2 η p | B j | max k ∈ B j | a k | ! ≪ 1 10 k 2 . The last inequality follows b y the assumption that η ≤ 1 100 k 2 min k ∈ [ n ]: a k  =0 | a k | ≤ √ | B j | 100 k 2 max k ∈ B j | a k | . Hence, with probabilit y at least 1 − 1 / 10 k 2 , Algorithm 2 will return S = { j } in L i ne 9. No w, let | S | = k > 1 . Using a similar argument as in the base case, FindInfBuck et ( f , B , S ) will return ∅ with low probability , since f ( w ) − f ( y ) = P j ∈ V a j ( x j − y j ) will b e 0 only when x V − y V lies in a ( | V | − 1) -dimensional hyperplane. Ho wev er, Pr x , y ∼N ( 0 ,I n ) [ | ˜ f ( w ) − ˜ f ( y ) | ≤ 2 η ] ≤ Pr x , y ∼N ( 0 ,I n ) [ | f ( w ) − f ( y ) | ≤ 4 η ] = Pr x , y ∼N ( 0 ,I n )      X j ∈ V a j ( x j − y j )     ≤ 4 η  ≤ Pr x , y ∼N ( 0 ,I n )  max j ∈ V | a j |     X j ∈ V ( x j − y j )     ≤ 4 η  ≤ Pr x , y ∼N ( 0 ,I n )         X j ∈ V x j − y j p 2 | V |       ≤ 2 √ 2 η p | V | · max j ∈ V | a j |   ≜ ( ⋆ ) . 18 As earlier, w e may observe: as x , y ∼ N ( 0 , I n ) are sampled indep enden tly , we hav e X j ∈ V x j − y j p 2 | V | ≡ X j ∈ V x j − y j p 2 | V | ∈ R [ ∪ j ∈ V { x j , y j } ] , and with Cov [ x i , x j ] = Cov [ y i , y j ] = 0 , for all i  = j ∈ V , and Cov [ x i , y j ] = 0 , for all i, j ∈ V , we get V ar x , y ∼N ( 0 ,I n )   X j ∈ V x j − y j p 2 | V |   = 1 2 | V | | V | X j =1  V ar x j ∼N (0 , 1) [ x j ] + Va r y j ∼N (0 , 1) [ y j ]  = 1 . Applying Theorem 4.15 on P j ∈ V x j − y j √ 2 | V | , (with d = 1 , n = 2 | V | , and t = 0 ) we get ( ⋆ ) = Pr x , y ∼N ( 0 ,I n )         X j ∈ V x j − y j p 2 | V |       ≤ 2 √ 2 η p | V | · max j ∈ V | a j |   ≤ O 2 √ 2 η p | V | · max j ∈ V | a j | ! ≪ 1 10 k 2 . The last inequalit y again follows by η ≤ 1 100 k 2 min k ∈ [ n ]: a k  =0 | a k | ≤ √ | V | 100 k 2 max k ∈ V | a k | . Th us, with probability at least 1 − 1 / 10 k 2 , the condition in Line 5 will not hold, ensuring Algorithm 2 reac hes the recursive step (Lines 8–13). By construction, S = S ( L ) ⊔ S ( R ) , with 1 ≤ | S ( L ) | ≤ ⌈| S | / 2 ⌉ and 1 ≤ | S ( R ) | ≤ ⌊| S | / 2 ⌋ . Either, (i) a j = 0 for all j ∈ S ( L ) , and there exists S R ⊆ S ( R ) : S R  = ∅ and a j  = 0 for all j ∈ S R , or (ii) there exists S L ⊆ S ( L ) suc h that S L  = ∅ and a j  = 0 for all j ∈ S L . In Case (i), FindInfBuck et ( f , B , S ( L ) ) will alw ays return ∅ and p erform 2 queries (by Part 1) and hence the return v alue of FindInfBuck et ( f , B , S ) will b e FindInfBuck et ( f , B , S ( R ) ) . Using the strong induction hypothesis w e get, with probabilit y at least 1 − 4 ⌈ log | S |⌉ / 10 k 2 , this return v alue will b e { j } for some j ∈ S R , and the total num b er of queries made will b e ≤ 2 + 2 + 4 ⌈ lg( ⌊| S | / 2 ⌋ ) ⌉ ≤ 4 + 4( ⌈ lg | S |⌉ − 1) = 4 ⌈ lg | S |⌉ . In Case (ii), again using strong induction hypothesis, with probability at least 1 − 4 ⌈ log | S |⌉ / 10 k 2 , FindInfBuck et ( f , B , S ( L ) ) will return { j } for some j ∈ S L , and th us the algorithm will return { j } (line 13). The n umber of queries made will b e ≤ 2 + 4 ⌈ lg( ⌈| S | / 2 ⌉ ) ⌉ ≤ 2 + 4( ⌈ lg | S |⌉ − 1 2 ) = 4 ⌈ lg | S |⌉ . W e ma y note, if w e run FindInfBuck et ( f , B , S ) on some set S of buck et-indices with | S | > 1 , eac h recursive step (irresp ectiv e of whether it falls in case (i) or case (ii) ab o ve) will succeed with probabilit y ≥ 1 − 4 ⌈ log | S |⌉ / 10 k 2 . F or the top lev el call to succeed, all the recursive calls (at most 2 ⌈ lg | S |⌉ in n umber) must also succeed. So, w e do a union b ound ov er all the failure even ts to upp er b ound the total failure probabilit y by 8 ⌈ log | S |⌉ 2 / 10 k 2 , and the query complexity by 8 ⌈ lg | S |⌉ 2 . No w we are ready to describ e and analyze the subroutine FindInfBuckets ( Algorithm 3 ). Claim 5.12 (Correctness of FindInfBuck ets ) . If f : R n → R , with f ( x ) = P i ∈ [ n ] a i x i , is given via the η -appr oximate query or acle ˜ f , wher e η ≤ 1 100 k 2 min k ∈ [ n ]: a k  =0 | a k | , and ∅  = X ⊆ [ n ] , then FindInfBuck ets ( f , B , X ) ( Algorithm 3 ) p erforms at most 64 k ⌈ log( | X | ) ⌉ 2 queries, and guar ante es: (i) If f is ℓ -line ar function for some ℓ > 8 k , then Algorithm 3 wil l r eturn a set of 8 k influential buckets in f with pr ob ability at le ast 1 − 64 ⌈ log | X |⌉ 2 / 10 k . (ii) If f is ℓ -line ar function for some ℓ ≤ 8 k , then Algorithm 3 wil l r eturn the set of al l influential buckets in f with pr ob ability at le ast 1 − 64 ⌈ log | X |⌉ 2 / 10 k . 19 Algorithm 3: FindInfBuck ets ( e f , B , X ) 1 Inputs: η -approximate query oracle e f for f : R n → R , Buck eting B = { B 1 , . . . , B r } , ∅  = X ⊆ [ r ] . 2 Output: A set of influential buck ets InfBuckets ⊆ { B j : j ∈ X } with resp ect to f . 3 InfBuck ets ← ∅ 4 for j=1 to 8k do 5 RetV al = FindInfBucket ( f , B , X ) 6 if RetVal = B i for some i then 7 InfBuck ets ← InfBuckets ∪ { B i } 8 X ← X \ { i } 9 return InfBuck ets . Pr o of. Note, Algorithm 3 calls Algorithm 2 . So, we use Claim 5.11 , and pro ceed case-wise: (i) Consider the case when f is ℓ -linear for some ℓ > 8 k . This implies that the set of indices X giv en as input to Algorithm 3 contains more than 8 k influen tial v ariables. Thus, in every iteration of the for lo op starting in Line 4 of Algorithm 3 , Algorithm 2 in Line 5 will return an influen tial buck et, say B i , with probability at least 1 − 8 ⌈ log | X |⌉ 2 / 10 k 2 , follo wing Claim 5.11 (ii). Line 7 then computes V al = a i , follo wed b y Line 9 up dating f to f − a i x i , and X to X \ { i } , remo ving { i } from an y future iterations. Since the for lo op in Algorithm 3 runs for 8 k iterations, and the num b er of influential v ariables is more than 8 k , using a union b ound o ver all the iterations, with probability at least 1 − 64 ⌈ log | X |⌉ 2 / 10 k , Algorithm 3 returns a set of 8 k influential v ariables. (ii) When f is ℓ -linear for some ℓ ≤ 8 k . As b efore, this implies that the set of indices X giv en as input to Algorithm 3 contains at most 8 k influen tial v ariables. F ollowing the same argument, as in the ab o v e case, we may claim, the 8 k iterations of the f or lo op in Algorithm 3 returns all the ℓ ≤ 8 k influen tial v ariables in f , with probability at least 1 − 64 ⌈ log | X |⌉ 2 / 10 k . Query Complexity: Since each call of FindInfBuck et ( f , B , X ) mak es ≤ 8 ⌈ lg( | X | ) ⌉ 2 queries to f , and there are at most 8 k such calls, the total num b er of queries to f is ≤ 64 k ⌈ lg( | X | ) ⌉ 2 . 5.2 Analysis of the k -linearit y tester ( Algorithm 1 ) W e are now ready to prov e the main theorem of this section: Pr o of of The or em 5.1 . Completeness: Since f is a k -linear function, following Theorem 5.4 , we ha ve: Appr oxima te Additivity Tester accepts f , and hence by Lemma 5.6 , g is p oin twise 42 η -close to linearity in B ( 0 , r ) . Com bined with Lemma 5.7 , we get g is p oin t wise 42 η (12 R/r ) -close to linearity in B ( 0 , R ) . Now from Claim 5.8 , we know that g and Appro xima te- g are p oin twise 6 η (12 R/r ) -close in B ( 0 , R ) with probability at least 1 − ε/ 4 . Using the triangle inequalit y , this implies that Appro xima te- g is p oin twise 48 η (12 R/r ) -close to some linear function with probability at least 1 − ε/ 4 . Moreov er, from Observ ation 5.9 , we get: Appro xima te- g is in fact p oin twise 750 Rn 1 . 5 η -close to f , with probability at least 1 − ε/ 2 . With R = 2 √ n , following the guarantee of Claim 5.12 , which we ensure by our assumption on η : O ( n 2 η ) ≤ 1 100 k 2 min k ∈ [ n ]: a k  =0 | a k | , or equiv alently η ≤ O  min i ∈ [ n ]: f ( e i )  =0 | f ( e i ) | ( nk ) 2  , 20 w e get that FindInfBuck ets ( Appr oxima te- g , B , [ r ]) will return at most k -influen tial v ariables of f with probabilit y at least 1 − 128 ⌈ log k ⌉ 2 / 5 k . Thus with probability at least 1 − ε/ 2 − 128 ⌈ log k ⌉ 2 / 5 k , Test- k -Linear will Accept . Soundness: Let us consider the case when f is ε -far from k -linearit y . W e will pro ve the con trap ositiv e. W e will show that if Test- k -Linear do es not reject f with probabilit y at least 1 − δ ( ≥ 2 / 3) , then f is p oin twise close to some k -linear function, with non-zero probabilit y . Note that if Appr oxima te Additivity Tester rejects f with probability ≥ 1 − δ , w e are done. So, let us consider the case when Appro xima te Additivity Tester accepts with probability ≥ δ . As in the completeness pro of, from Lemma 5.5 , and Lemma 5.7 , we know that g is p oin t wise 42 η (12 R/r ) -close to linearit y in B ( 0 , R ) with probabilit y at least 1 − 12 / 125 , and from Claim 5.8 , w e ha ve that g and Appro xima te- g are p oin t wise 6 η (12 R/r ) -close in B ( 0 , R ) with probability at least 1 − ε/ 4 . This implies that Appr oxima te- g is p oin twise 48 η (12 R/r ) -close to some linear function with probability at least 1 − ε/ 4 − 12 / 125 . Again, from Observ ation 5.9 , we get: Appr o xima te- g is p oin t wise 750 Rn 1 . 5 η -close to f , with probabilit y at least 1 − ε/ 4 , implying no w f m ust b e p oin t wise (48(12 R/r ) + 750 Rn 1 . 5 ) η -close to linearity , with probability at least 1 − ε/ 2 − 12 / 125 . Note, here R = 2 √ n , and r = 1 / 50 . With our assumption on η again ensuring the conditions for Claim 5.12 are met, i.e., (48(12 R/r ) + 750 Rn 1 . 5 ) η ≤ 1 100 k 2 min k ∈ [ n ]: a k  =0 | a k | , w e get FindInfBuck ets ( Appr oxima te- g , B , [ r ]) will return at most 8 k -influen tial v ariables of f with probabilit y at least 1 − 128 ⌈ log k ⌉ 2 / 5 k . Since Test- k -Linear accepts f with probability ≥ δ , this implies that the total num b er of influen tial v ariables returned by FindInfBuck ets is at most k , with probability ≥ δ . Combining the ab o v e, w e can conclude that f is p oin twise (48(12 R/r ) + 750 Rn 1 . 5 ) η -close to a k -linear function, with probability at least δ − 12 / 125 − ε/ 2 − 128 ⌈ log k ⌉ 2 / 5 k . This concludes the soundness argument. Query complexity: F rom Theorem 5.4 , we know that Appro xima te Additivity Tester p erforms O ( 1 ε ) queries. F ollo wing Claim 5.12 , we also know that FindInfBuck ets p erforms e O ( k log k ) queries. Combining them, we hav e: Test- k -Linear p erforms e O ( k log k + 1 / ε ) queries in total. 6 k -Sparse Low Degree T esting In this section, w e present and analyze our sparse low degree tester ( Algorithm 4 ). Theorem 6.1 (Generalization of Theorem 2.6 ) . L et η < min { ε, 1 / 2 2 n } . Given η -appr oximate query ac c ess to f : R n → R that is b ounde d in B ( 0 , 2 d √ n ) , ther e exists a tester Test- k -Sp arse ( A lgorithm 4 ) that p erforms e O ( d 5 + d 2 / ε + dk 3 ) queries and guar ante es: • Completeness: If f is a k -sp arse, de gr e e- d p olynomial, then A lgorithm 4 A ccepts with pr ob ability at le ast 1 − ε/ 4 . • Soundness: If f is ε -far fr om al l k -sp arse de gr e e- d p olynomials, then Algorithm 4 Rejects with pr ob ability at le ast 2 / 3 . W e will prov e Theorem 6.1 in Section 6.3 , after developing the necessary machinery . Algorithm 4 first in vok es Appr oxLo wDegreeTester ( Algorithm 9 ) to reject functions that are far from an y low-degree p olynomial. W e record some useful claims ab out Algorithm 9 , and its subroutines ( Algorithm 10 ) from [ ABF + 23 ] and (its impro vemen t in) [ AKM25 ]: 21 Algorithm 4: Test- k -Sp arse 1 Inputs: η -approximate query oracle e f for f : R n → R , that is b ounded in B( 0 , R ) , sparsity parameter k ∈ N , pro ximity parameter ε ∈ (0 , 1) , degree parameter d ∈ N . 2 Output: Accept iff f is k -sparse function. 3 Run Appr oxLo wDegreeTester ( f , d, N ( 0 , I n ) , η , ε, R, R ) ( Algorithm 9 ). 4 if Algorithm 9 r eje cts then 5 return Reject . 6 Call Appr ox-Pol y-Sp arsity-Test ( Appro xQuer y- g ) ( Algorithm 5 ). 7 if Appro x-Pol y-Sp arsity-Test ( Appro xQuer y- g ) ac c epts then 8 return A ccept . ▷ Appr oxQuer y- g in Algorithm 10 9 else 10 return Reject . Theorem 6.2 ([ AKM25 , Theorem 3.6]) . L et d ∈ N , for L > 0 , f : R n → R b e b ounde d in the b al l B ( 0 , L ) , and for ε ∈ (0 , 1) , R > 0 , let D b e an ( ε/ 4 , R ) -c onc entr ate d distribution. F or α > 0 , β ≥ 2 (2 n ) O ( d ) ( R/L ) d α , given α -appr oximate query ac c ess to f , and sampling ac c ess to D , ther e is an one-side d err or, O ( d 5 + d 2 ε ) -query Appro xLo wDegreeTester ( Algorithm 9 ) which, distinguishes b etwe en the c ase when f is p ointwise α -close to some de gr e e- d p olynomial and the c ase when, for every de gr e e- d p olynomial h : R n → R , Pr p ∼D [ | f ( p ) − h ( p ) | > β ] > ε . Lemma 6.3 ([ ABF + 23 , Lemma 4.4]) . L et r = (4 d ) − 6 , δ = 2 d +1 α , as set in Algorithm 10 , and R > r . If Appr oxChara cteriza tionTest fails with pr ob ability at most 2 / 3 , then g is p ointwise 2 (2 n ) 45 d ( R/L ) d δ -close to a de gr e e- d p olynomial in B ( 0 , 2 dR √ n/L ) . F urthermor e, for every p oint p ∈ B( 0 , 2 dR √ n/L ) , Appr oxQuer y- g ( p ) wel l appr oximates g ( p ) with high pr ob ability, that is, Pr p ∼D h | g ( p ) − Appr oxQuer y- g ( p ) | ≤  24 dR √ n Lr  d 2 d +4 δ i ≥ 1 − ε 4 . T o in vok e these results, we assume: (i) f is b ounded in B ( 0 , R ) , i.e., we set L = R , and (ii) α = η . Additionally , since we work ov er standard Gaussians, w e set R = 2 d √ n . 6.1 T esting sparsit y of p olynomials given exact query access In this section, w e design an algorithm for testing sparsity of p olynomial functions f : R n → R , b y extending the machinery developed in [ GJR10 ] and [ BOT88 ] to the real num b ers. Recall the notion of Hank el Matrices asso ciated with p olynomials ( Definition 2.5 ). Definition 2.5 (Hankel Matrix for p olynomials [ GJR10 , BOT88 ]) . Consider any u ≜ ( u 1 , . . . , u n ) ∈ R n , and define u i ≜ ( u i 1 , . . . , u i n ) ∈ R n , ∀ i ∈ N . F or a function f : R n → R and an in teger t ∈ Z > 0 , define the t -dimensional Hankel matrix associated with f at u to b e the following: H t ( f , u ) ≜      f ( u 0 ) f ( u 1 ) . . . f ( u t − 1 ) f ( u 1 ) f ( u 2 ) . . . f ( u t ) . . . . . . . . . . . . f ( u t − 1 ) f ( u t ) . . . f ( u 2 t − 2 )      ∈ R t × t . W e note the following observ ation ab out Hankel matrices. It follo ws essentially the same argument as in [ BOT88 , GJR10 ], since the decomp osition that they use o ver finite domains also works ov er the reals. F or completeness, a pro of of this observ ation is provided in Section B . 22 Observ ation 6.4 (Generalization of [ BOT88 , Section 4], and [ GJR10 , Lemma 4]) . L et f : R n → R b e an exactly k -sp arse p olynomial over the r e als, i.e., f ( x ) = P k i =1 a i M i ( x ) , wher e a 1 , . . . , a k ∈ R \ { 0 } , and M 1 , . . . , M k ar e the monomials of f . Then for al l ℓ + 1 ≤ k , det ( H ℓ +1 ( f , x )) = X S ⊆ [ k ] | S | = ℓ +1 Y i ∈ S a i Y i,j ∈ S i k , det ( H ℓ +1 ( f , x )) ≡ 0 . As a preliminary , we argue that Observ ation 6.4 can b e used to test whether a p olynomial f is ( ≤ k ) -sparse, or not (with error probability 0 , in fact!), given exact query access to f . Lemma 6.5. L et f : R n → R b e a p olynomial function (of any de gr e e). Given exact query ac c ess to f , ther e is an algorithm that makes 2 k + 1 queries to f , and exactly tests whether ∥ f ∥ 0 ≤ k (r eturns A ccept ), or ∥ f ∥ 0 > k (r eturns Reject ) with err or pr ob ability 0 . Pr o of. The algorithm is as follows: Sample a single p oin t u ∼ N ( 0 , I n ) . Construct the Hank el matrix H k +1 ( f , u ) as in Definition 2.5 using 2 k − 1 exact queries to f . T est whether det ( H k +1 ( f , u ) ) ? = 0 and return A ccept if the determinant is 0 , and Reject otherwise. Let Q ( x ) ≡ det ( H k +1 ( f , x )) ∈ R [ x ] , so that the test is Q ( u ) ? = 0 . If ∥ f ∥ 0 ≤ k , by Observ ation 6.4 , Q is the zero p olynomial and hence the algorithm will alwa ys Accept after finding that Q ( u ) = 0 . If ∥ f ∥ 0 ≥ k + 1 , then Q is a non-zero polynomial, and hence Pr u ∼N ( 0 ,I n ) [ Q ( u ) = 0] = 0 ; this is b ecause the Leb esgue measure, and hence the probability measure with resp ect to any contin uous distribution, of the zero-set of an y non-zero real p olynomial is 0 . Th us, the algorithm will Reject with probabilit y 1 ov er the randomness of u . 6.2 T esting sparsit y of p olynomials given appro ximate query access No w, instead of exact query access, we hav e η -appro ximate query access to a p olynomial f . W e mak e the follo wing observ ation ab out the Hank el matrix constructed with the queried v alues ˜ f , since the approximate-query oracle guarantees that | ˜ f ( z ) − f ( z ) | ≤ η for all p oin ts z ∈ R n . W e provide a pro of of this observ ation in Section B . Observ ation 6.6. L et f : R n → R and let ˜ f b e an η -appr oximate query or acle to f . Then for any t ≥ 1 and u ∈ R n , we c an expr ess H t ( ˜ f , u ) = H t ( f , u ) + E t ( u ) , wher e E t ( u ) is a Hankel-structur e d, noise matrix , such that ∥ E t ( u ) ∥ ∞ ≤ η , and ∥ E t ( u ) ∥ op ≤ η t . W e also need the following probabilistic upp er b ound on the eigenv alues of the Hankel matrix of a p olynomial, that was briefly introduced in Section 2.2 . W e restate it here for conv enience. Theorem 2.11 (Probabilistic Upp er Bound on σ max ) . L et f : R n → R , f ( x ) = P k i =1 a i M i ( x ) b e a k -sp arse, de gr e e- d p olynomial, wher e M i ’s ar e its non-zer o monomials, and σ max ( H t ( f , u )) denote the lar gest singular value of the t -dimensional Hankel matrix asso ciate d with f at a p oint u ∈ R n , H t ( f , u ) . Then, for any γ ∈ (0 , 1) , with a ≜ ( a 1 , · · · , a k ) ⊤ ∈ R k , Pr u ∼N ( 0 ,I n )   σ max ( H t ( f , u )) ≥ ∥ a ∥ 2 2 2 d/ 2 ⌈ d/ 2 ⌉ ! + s k γ 2 d/ 2 √ d ! ! 2 t   ≤ γ . 23 Pr o of of The or em 2.11 . Let H u denote H t ( f , u ) . Then for any z = ( z 1 , . . . , z t ) ⊤ ∈ R t , w e hav e | z ⊤ H u z | =       X i ∈ [ t ] X j ∈ [ t ] [ H u ] i,j z i z j       ≤ X i ∈ [ t ] X j ∈ [ t ]   f ( u i + j − 2 )   | z i || z j | ( ∵ [ H u ] i,j = f ( u i + j − 2 ) ) ≤ X i ∈ [ t ] X j ∈ [ t ] | z i || z j |   X p ∈ [ k ] | a p || M p ( u i + j − 2 ) |   = z ⊤ V ⊤ D V z , where z = ( | z 1 | , . . . , | z t | ) ⊤ ∈ R t ≥ 0 , V =        1 1 . . . 1 | M 1 ( u ) | | M 2 ( u ) | . . . | M k ( u ) | | M 1 ( u ) | 2 | M 2 ( u ) | 2 . . . | M k ( u ) | 2 . . . . . . . . . . . . | M 1 ( u ) | t − 1 | M 2 ( u ) | t − 1 . . . | M k ( u ) | t − 1        ⊤ , and D =      | a 1 | 0 . . . 0 0 | a 2 | . . . 0 . . . . . . . . . . . . 0 0 . . . | a k |      , wherein in the last step, we use the same decomposition as in the pro of of Observ ation 6.4 , but applied to H t ( | f | , u ) , with | f | ( u ) ≜ P p ∈ [ k ] | a p || M p ( u i + j − 2 ) | . Th us, if ∥ z ∥ 2 = ∥ z ∥ 2 ≤ 1 , we hav e σ max ( H u ) ≤ z V ⊤ ( D 1 2 ) ⊤ D 1 2 V z =    D 1 2 V z    2 2 ≤ ∥ D 1 2 V ∥ 2 op ∥ z ∥ 2 2 ≤ ∥ D 1 2 V ∥ 2 op ≤ ∥ D 1 2 V ∥ 2 F , where D 1 2 V =      | a 1 | | a 1 || M 1 ( u ) | · · · | a 1 || M 1 ( u ) | t − 1 | a 2 | | a 2 || M 2 ( u ) | · · · | a 2 || M 2 ( u ) | t − 1 . . . . . . . . . . . . | a k | | a k || M k ( u ) | · · · | a k || M k ( u ) | t − 1      . By U ≥ max  2 , max p ∈ [ k ] | M p ( u ) |  , we get ∥ D 1 2 V ∥ 2 F ≤ P t − 1 r =0 ∥ a ∥ 2 2 U 2 r = ∥ a ∥ 2 2  U 2 t − 1 U − 1  ≤ ∥ a ∥ 2 2 · U 2 t . Finally , for any α ≜ ( α 1 , . . . , α n ) ∈ N n , w e hav e E u ∼N ( 0 ,I n ) [ | u α | ] = Q i ∈ [ n ] E u i ∼N (0 , 1) [ | u α i i | ] . F rom [ Ela61 , Equations (1), (4), and (15)], for s ∈ N > 0 , w e get E u ∼N (0 , 1) [ | u s | ] =    s ! 2 s/ 2 ( s/ 2)! ≤ 2 s/ 2 ⌊ s/ 2 ⌋ ! √ π , if s is ev en (and > 0 ), and 2 ⌊ s/ 2 ⌋  ⌊ s 2 ⌋  ! q 2 π = 2 s/ 2 ⌊ s/ 2 ⌋ ! √ π , if s is o dd. (2) The inequality in the “ s is even” case follows from: (i) s ! 2 s ( s/ 2)! = Γ ( s 2 + 1 2 ) √ π [ W ol24 ], and (ii) the monotonicit y of the Gamma function in ( 1 2 , ∞ ) which implies Γ  s 2 + 1 2  ≤ Γ  s 2 + 1  = s 2 ! for even s . Since E u ∼N (0 , 1) [ | u 0 | ] = 1 , for an y α = ( α 1 , . . . , α n ) ∈ N n with ∥ α ∥ 1 = P i ∈ [ n ] α i ≤ d , we hav e E u ∼N ( 0 ,I n ) [ | u α | ] ≤ Y i : α i  =0 2 α i / 2 ⌊ α i / 2 ⌋ ! √ π ! ≤ 2 d/ 2 ⌈ d/ 2 ⌉ ! . By a similar argumen t, V ar[ | u α | ] ≤ E u [ | u 2 α | ] ≤ 2 d · d ! . Th us, by Chebyshev’s inequality , for any d ≥ 0 , k ≥ 1 , and α ∈ N n with ∥ α ∥ 1 ≤ d , Pr u ∼N ( 0 ,I n ) " | u α | ≥ 2 d/ 2 ⌈ d/ 2 ⌉ ! + s k γ 2 d/ 2 √ d ! # ≤ γ k . Th us, from the ab o ve discussion, M p ( u ) ≤  2 d/ 2 ⌈ d/ 2 ⌉ ! + q k γ 2 d/ 2 √ d !  for every p ∈ [ k ] , w.p ≥ 1 − γ (using the union b ound), and hence σ max ( H t ( f , u )) ≤ ∥ a ∥ 2 2  2 d/ 2 ⌈ d/ 2 ⌉ ! + q k γ 2 d/ 2 √ d !  2 t . 24 W e now present and analyze the correctness of Algorithm 5 : Algorithm 5: Appr ox-Pol y-Sp arsity-Test : Approx-query sparsit y test for p olynomials 1 Input: η -approximate query oracle ˜ f to a p olynomial f : R n → R of total degree ≤ d , sparsit y parameter k ∈ N . 2 Output: Accept iff f is a k -sparse p olynomial. 3 Let T ← 4 min { C , 1 } · d ( k + 1) 2 , where C is the absolute constant in Theorem 4.16 . 4 for t ∈ [ T ] do 5 Sample u t = ( u t, 1 , . . . , u t,n ) ∼ N ( 0 , I n ) . 6 for i = 0 , . . . , 2 k do 7 Compute u i t ≜ ( u i t, 1 , . . . , u i t,n ) , and ˜ f ( u i t ) . 8 Compute the Hank el matrix H k +1 ( ˜ f , u t ) as in Definition 2.5 . 9 Compute the smallest singular v alue σ ( t ) min of H k +1 ( ˜ f , u t ) . 10 if σ ( t ) min ≤ η ( k + 1) for al l t ∈ [ T ] then 11 return A ccept (( ≤ k )-sparse). 12 else 13 return Reject (( > k )-sparse). Theorem 6.7 (Sparsity testing of p olynomials with approximate queries) . Given η -appr oximate query ac c ess e f to a p olynomial f : R n → R of total de gr e e ≤ d , assuming η ≤  coeff d Q ( Q )  1 2 (2( k +1))2 Θ( k 3 d ) ( σ max ( H u )) k , wher ein H u = H k +1 ( f , u ) , Q u = ( det ( H u )) 2 , d Q = deg ( Q ) , and co eff 2 d Q ( Q ) is as in The or em 4.16 , Appr ox-Pol y-Sp arsity-Test ( Algorithm 5 ) p erforms at most O ( dk 3 ) queries and guar ante es: (i) If f has sp arsity at most k , the algorithm always A ccepts , and (ii) If f has sp arsity > k , the algorithm Rejects with pr ob ability at le ast 2 3 . Pr o of. The query complexity follows directly from Algorithm 5 , since the tester p erforms at most (2 k + 1) T queries for T ≤ O ( k 2 d ) . It remains to argue the completeness and soundness guarantees. F or any u ∈ R n , let us denote H k +1 ( ˜ f , u ) by ˜ H u and H k +1 ( f , u ) by H u . Observe, since ˜ H u and H u are symmetric matrices, their singular v alues are just the absolute v alues of their eigenv alues, i.e., σ min ( ˜ H u ) = min i ∈ [ k +1] | λ i ( ˜ H u ) | and σ min ( H u ) = min j ∈ [ k +1] | λ j ( H u ) | . Completeness: W e hav e ∥ f ∥ 0 ≤ k . By Observ ation 6.4 , we get det ( H k +1 ( f , x )) ≡ 0 , giving us: for an y u ∈ R n , det ( H u ) = 0 , and hence, λ i ∗ ( H u ) = 0 for some i ∗ ∈ [ k + 1] . F rom Observ ation 6.6 , w e can write ˜ H u = H u + E , for a symmetric matrix E with ∥ E ∥ op ≤ η ( k + 1) . Then, b y W eyl’s inequalit y ( Theorem 4.14 ), we hav e | λ i ∗ ( ˜ H u ) − λ i ∗ ( H u ) | ≤ η ( k + 1) , whic h implies σ min ( ˜ H u ) ≤ | λ i ∗ ( ˜ H u ) | ≤ η ( k + 1) . So, Algorithm 5 will alwa ys Accept (Lines 10–11). Observ e: F or any non-singular symmetric matrix A ∈ R ( k +1) × ( k +1) , w e hav e ( σ min ( A )) 2( k +1) ≤ (det( A )) 2 = k +1 Y i =1 ( σ i ( A )) 2 ≤ ( σ min ( A )) 2 ( σ max ( A )) 2 k , where σ max ( A ) is the largest magnitude eigen v alue of A and σ min ( A ) is the smallest magnitude eigen v alue of A . Thus, σ min ( A ) ≤ Ξ implies det( A ) 2 ≤ (Ξ) 2 σ max ( A ) 2 k . 25 Soundness: W e hav e ∥ f ∥ 0 > k . Let Q ( x ) ≜ (det ( H k +1 ( f , x ))) 2 . By Observ ation 6.4 , Q is a non-zero p olynomial with total degree ≤ 2( k + 1) 2 d , and (det( H u )) 2 = Q ( u ) for all u ∈ R n . Let u ∼ N ( 0 , I n ) , and σ min ( ˜ H u ) ≤ η ( k + 1) . Then, as in completeness, b y W eyl’s Inequalit y ( Theorem 4.14 ), and Observ ation 6.6 , we hav e σ min ( H u ) ≤ 2 η ( k + 1) . Thus, we must hav e Q ( u ) = (det( H u )) 2 ≤ (2 η ( k + 1)) 2 σ max ( H u ) 2 k ≜ ∆ , as long as det( H u )  = 0 (whic h will happ en with probabilit y 1 ov er the choice of u , since H u ≡ 0 ). First, supp ose that f is not a constant p olynomial. Then, d Q ≜ deg ( Q ) ≥ 2 by construction, since it is the square of a non-constant p olynomial. Now, we may inv oke an anti-concen tration result. F rom Theorem 4.16 , we hav e Pr u ∼N ( 0 ,I n ) [ Q ( u ) ≤ ∆] ≤ C d Q  ∆ coeff d Q ( Q )  1 /d Q . Observe, C d Q  ∆ co eff d Q ( Q )  1 /d Q ≤ 1 − 1 C d Q ⇐ ⇒  ∆ co eff d Q ( Q )  ≤ 1 C d Q −  1 C d Q  2 ! d Q . Assuming C ≥ 1 , so that 1 / ( C d Q ) ≤ 1 / 2 (since d Q ≥ 2 ) 4 , w e hav e ln   1 C d Q −  1 C d Q  2 ! d Q   = d Q  ln  1 − 1 C d Q  − ln( C d Q )  ≥ d Q  − 1 / ( C d Q ) 1 − 1 / ( C d Q ) − ln( C d Q )  ≥ − d Q ln( C d Q ) − 2 C d Q ≥ − ( d Q ln( C d Q ) + 1) , where w e use the inequality ln(1 + z ) ≥ z 1+ z for z > − 1 , and the fact that 1 / ( C d Q ) ≤ 1 / 2 . Then 1 C d Q −  1 C d Q  2 ! d Q ≥ exp ( − ( d Q ln( C d Q ) + 1)) ≥ exp  − 2( k + 1) 2 d ln(2 C ( k + 1) 2 d ) + 1  | {z } ≜ UB d,k ≥ 2 − Θ( k 3 d ) . Th us, if ∆ ≤ UB d,k · co eff d Q ( Q ) , we will hav e Pr u ∼N ( 0 ,I n ) [ Q ( u ) ≤ ∆] ≤ 1 − 1 C d Q . If η ≤  coeff d Q ( Q )  1 2 (2( k +1))2 Θ( k 3 d ) ( σ max ( H u )) k ≤  UB d,k · coeff d Q ( Q )  1 2 (2( k +1))( σ max ( H u )) k , then ∆ = (2 η ( k +1)) 2 σ max ( H u ) 2 k ≤ UB d,k · co eff d Q ( Q ) . With the ab o v e condition satisfied, we will hav e Pr u h σ min ( ˜ H u ) ≤ η ( k + 1) i ≤ Pr u [ σ min ( H u ) ≤ 2 η ( k + 1)] ≤ Pr u [ Q ( u ) ≤ ∆] ≤ 1 − 1 C d Q . No w, consider the op eration of Algorithm 5 . In every iteration t ∈ [ T ] for T = 4 min { C, 1 } · d ( k + 1) 2 ≥ 2 C d Q , we will hav e Pr u t h σ min ( ˜ H u t ) ≤ η ( k + 1) i ≤ 1 − 1 / ( C d Q ) b y the ab o ve argument. Then, o ver T indep enden t rounds, the probability that this even t o ccurs every time, i.e., f is accepted, is Y t ∈ [ T ] Pr u t h σ min ( ˜ H u t ) ≤ η ( k + 1) i ≤  1 − 1 C d Q  T ≤ 1 e 2 < 1 3 for our c hoice of T . Th us, the algorithm will Reject with probability at least 2 3 . 4 W e renormalize the constant from Theorem 4.16 appropriately . 26 6.3 Analysis of k -Sparsit y Low Degree T ester W e are now ready to analyze our sparse low degree tester ( Algorithm 4 ). Pr o of of The or em 6.1 . Let us start with completeness. Completeness: Since f is a k -sparse, degree- d p olynomial, from Theorem 6.2 , we hav e: Appro x- Lo w-Degree-Tester alwa ys accepts f , and hence, from Lemma 6.3 , w e hav e that g is p oin t wise 2 (2 n ) 45 d R d 2 d +1 η -close to a degree- d p olynomial, sa y h , in B( 0 , R ) . Moreo ver, Pr p ∼D h | g ( p ) − Appr oxQuer y- g ( p ) | ≤  24 d √ n r  d 2 2 d +5 η | p ∈ B( 0 , 2 d √ n ) i ≥ 1 − ε 2 . = ⇒ Pr p ∼D h | h ( p ) − Appro xQuer y- g ( p ) | ≤   24 d √ n r  d 2 2 d +5 + 2 (2 n ) 45 d R d 2 d +1  η i ≥ 1 − ε 4 . (3) By setting η = 1 / 2 2 n , the conditions of Theorem 6.7 are met, giving us that Appro x-Pol y- Sp arsity-Test(Appro xQuer y- g ) ( Algorithm 5 ) will Accept with probabilit y ≥ 1 − ε/ 4 . So with probabilit y at least 1 − ε/ 4 , Test- k -Sp arse ( Algorithm 4 ) will accept f . Soundness: Let f b e ε -far from all k -sparse, degree- d p olynomials. W e will show that if Algorithm 4 accepts with probabilit y at least 1 / 3 , then f m ust b e ε -close to some k -sparse, degree- d p olynomial. F rom the premise, Appr ox-Lo w-Degree-Tester ( Algorithm 9 ) also accepts f with probabilit y at least 1 / 3 . Then, as in completeness, from Lemma 6.3 , w e ha ve: g is p oin t wise 2 (2 n ) 45 d R d 2 d +1 η -close to some degree- d p olynomial, say h ( x ) = P ∥ M i ∥ 1 ≤ d a i M i ( x ) , i.e., for all p ∈ B( 0 , R ) , | g ( p ) − h ( p ) | ≤ 2 (2 n ) 45 d R d 2 d +1 η , and ( 3 ) still holds. F rom the premise, Appro x-Pol y-Sp arsity-Test ( Appr oxQuer y- g ) ( Algorithm 5 ) do es not reject with probability at least 1 / 3 . In this case, as long as the closeness of Appr oxQuer y- g and h satisfies the assumption in Theorem 6.7 , h will b e k -sparse. The assumption is: (3 · 2 (2 n ) 45 d + 1) η ≤  co eff d Q ( Q )  1 2 (2( k + 1))2 Θ( k 3 d ) ( σ max ( H u )) k , (4) wherein H u = H k +1 ( h, u ) , Q ( u ) = ( det ( H u )) 2 , d Q = deg ( Q ) , co eff 2 d Q ( Q ) ≤ ( ∥ a ∥ 2 2 n d ) ( k +1) (( k + 1)!) 2 , and σ max ( H u ) ma y b e b ounded using Theorem 2.11 (with γ = 0 . 01 , t = k + 1 ), i.e., Pr u ∼N ( 0 ,I n )   σ max ( H u ) ≥ ∥ a ∥ 2 2 2 d/ 2 ⌈ d/ 2 ⌉ ! + r k 0 . 01 2 d/ 2 √ d ! ! 2( k +1)   ≤ 0 . 01 . So, with probabilit y at least 0 . 99 , σ max ( H u ) ≤ ∥ a ∥ 2 2  2 d/ 2 ⌈ d/ 2 ⌉ ! + 10 √ k 2 d/ 2 √ d !  2( k +1) . Plugging these in to ( 4 ) , w e observ e, setting η ≤ 1 / 2 2 n satisfies it, implying that f is ε -close to some k -sparse, degree- d p olynomial. Query complexity: The query complexity of Algorithm 4 consists of tw o parts: • query complexity of Appro xLowDegreeTester ( Algorithm 9 ) which is O ( d 5 + d 2 /ε ) (from Theorem 6.2 ), and • the query complexit y of Appro x-Pol y-Sp arsity-Test ( Algorithm 5 ) whic h is at most O ( dk 3 ) (from Theorem 6.7 ). Com bining the ab o ve, w e can sa y that Algorithm 4 p erforms O ( d 5 + d 2 /ε + dk 3 ) queries in total. 27 7 k -Jun ta T esting In this section, w e present and analyze our k -jun ta tester. Theorem 7.1 (Generalization of Theorem 2.9 ) . L et ε ∈ (0 , 1) and η < min { ε/ 16 k 2 , O ( 1 / k 2 log 2 k ) } . Given η -appr oximate query ac c ess to an unknown function f : R n → R that is b ounde d in B ( 0 , 2 √ n ) , ther e exists a one-side d err or tester Test- k -Junt a ( Algorithm 6 ) that p erforms e O (( k log k ) /ε ) queries and guar ante es: • Completeness: If f is a k -junta, then A lgorithm 6 always Accepts . • Soundness: If f is ε -far fr om al l k -juntas, then Algorithm 6 Rejects with pr ob ability at le ast 2 / 3 . Algorithm 6: Test- k -Junt a 1 Inputs: η -Approximate function oracle e f : R n → R , k ∈ N , ε, η ∈ (0 , 1) . 2 Output: Output Accept iff f is a k -junta. 3 Cho ose a random partition B of [ n ] into r = O ( k 2 ) parts. 4 Initialize S ← [ r ] , I ← ∅ . 5 for i = 1 to O ( k /ε ) do 6 B ← FindInfBucket ( e f , B , S ) ( Algorithm 2 ). 7 if B  = ∅ ( = B j for some j ∈ S ) then 8 I ← I ∪ { j } . S ← S \ { j } . 9 if | I | > k then 10 return Reject . 11 return A ccept . In order to pro ve Theorem 7.1 , we first build some mac hinery , and then analyze Algorithm 6 and its subroutine ( Algorithm 2 for jun tas) in Section 7.1 . Let us start with the notion of influence. Definition 7.2 (Influence) . Let f : R n → R b e a function. F or an y set S ⊆ [ n ] , the influence of f o ver S with resp ect to a distribution D ov er R n is defined as follo ws: Infl f ( S ) = E x , y ∼D    f ( y ) − f ( x S y S )    . W e now prov e a structural result. Let J k denote the set of all k -jun tas on n v ariables. Theorem 7.3. If dist ( f , J k ) ≥ ε , then for al l S ⊆ [ n ] : | S | ≤ k , with S ≜ [ n ] \ S , Infl f ( S ) ≥ ε . Pr o of. Fix some S ⊆ [ n ] , such that | S | ≤ k . Let J S denote the set of all juntas on S . Note J S ⊊ J k . Define g S : R n → R as the jun ta on S that minimizes the distance from f : g S ≜ arg min g ∈J S { dist ( f , g ) } . Observ e that Infl g S ( S ) = 0 . W e give a metho d of constructing such a g S , coset-wise. F or every x ∈ R | S | , define a function f x : R n → R as f x ( y ) ≜ f ( x S y S ) . Observe that f x ∈ J S , and hence f x ∈ J k , for ev ery x ∈ R | S | . So, from the premise, we hav e, for every x ∈ R | S | , dist D ,ℓ 1 ( f , f x ) = E y ∼D [ | f ( y ) − f x ( y ) | ] = E y ∼D    f ( y ) − f ( x S y S )    ≥ ε. 28 Since, the role of x is determined only by the v alues in the v ariables in S , w e may as w ell extend it for all n v ariables, i.e., for all x ∈ R n , w e hav e E y ∼D    f ( y ) − f ( x S y S )    ≥ ε. No w we may sample x from any distribution D ′ supp orted on R n as w ell, i.e., E x ∼D ′ , y ∼D    f ( y ) − f ( x S y S )    ≥ ε. In particular, w e may set D ′ = D to get Infl f ( S ) = E x , y ∼D    f ( y ) − f ( x S y S )    ≥ ε. Next we pro ve a lemma that connects the influence of the union of tw o sets of v ariables, with the influences of individual sets. Lemma 7.4 (Sub-additivity of Influence) . F or every function f : R n → R , and any S, T ⊆ [ n ] , max { Infl f ( S ) , Infl f ( T ) } ≤ Infl f ( S ∪ T ) ≤ Infl f ( S ) + Infl f ( T ) . Pr o of. F rom Definition 7.2 , w e hav e Infl f ( S ) = E x , y ∼D    f ( y ) − f ( x S y S )    , Infl f ( T ) = E x , y ∼D    f ( y ) − f ( x T y T )    , and Infl f ( S ∪ T ) = E x , y ∼D    f ( y ) − f ( x S ∪ T y S ∪ T )    = E x , y ∼D    f ( y ) − f ( x S y S ) + f ( x S y S ) − f ( x S ∪ T y S ∪ T )    ≤ E x , y ∼D    f ( y ) − f ( x S y S )    + E x , y ∼D h    f ( x S y S | {z } ≜ y ′ ∼D ) − f ( x S ∪ T y S ∪ T | {z } = x T y ′ T )    i = E x , y ∼D    f ( y ) − f ( x S y S )    + E x , y ∼D    f ( y ) − f ( x T y T )    = Infl f ( S ) + Infl f ( T ) . Non-negativit y of Infl trivially completes the other part of the lemma. Next we sho w, if f is far from b eing a k -jun ta, the influence of the complement of S in Algorithm 6 (i.e., all buc kets not yet identified as influential) is high, unless we identify more than k parts. Lemma 7.5. L et f : R n → R and B = { B 1 , . . . , B r } b e a r andom p artition of [ n ] , for r = Θ( k 2 ) . If dist ( f , J k ) ≥ ε , with pr ob ability ≥ 99 / 100 over the r andomness of the p artition, we have Infl f ( S ) ≥ ε/ 4 for any S ⊆ [ n ] which is a union of at most k p arts of B . Before pro ving it, let us first define the notions of intersecting families, and p -biased measure. Definition 7.6 (Intersecting family) . Let ℓ ∈ N . A family of subsets C of [ n ] is ℓ -interse cting if for an y tw o sets S, T ∈ C , | S ∩ T | ≥ ℓ . Definition 7.7 ( p -biased measure) . Let p ∈ (0 , 1) . Construct a set S ⊆ [ n ] by including each index i ∈ [ n ] in S with probability p . Then the p -biase d me asur e is defined as follows: µ p ( C ) = Pr S [ S ∈ C ] . 29 W e will use the following result to b ound the p -biased measure of in tersecting families. Theorem 7.8 ([ DS05 , F ri08 ]) . L et ℓ ≥ 1 b e an inte ger and let C b e a ℓ -interse cting family of subsets of [ n ] . F or any p < 1 ℓ +1 , the p -biase d me asur e of C is b ounde d by µ p ( C ) ≤ p ℓ . No w we are ready to prov e Lemma 7.5 . W e note that the pro of is similar to that of [ BWY15 , Lemma 2]. W e are adding it here for completeness. Pr o of of L emma 7.5 . W e will prov e: with high probability o ver the random partition I , Infl f ( S ) ≥ ε 4 , when S is a union of ≤ k parts of I . Consider the family of all sets whose complements hav e influence at most tε , for some 0 ≤ t ≤ 1 2 : F t = { S ⊆ [ n ] : Infl f ( S ) < tε } . Consider any tw o sets S 1 , S 2 ∈ F 1 / 2 , i.e., max { Infl f ( S 1 ) , Infl f ( S 2 ) } < ε/ 2 . By Lemma 7.4 , w e get Infl f ( S 1 ∩ S 2 ) = Infl f ( S 1 ∪ S 2 ) ≤ Infl f ( S 1 ) + Infl f ( S 2 ) < 2 ε/ 2 = ε. (5) As dist ( f , J k ) ≥ ε , for every S ⊆ [ n ] of size | S | ≤ k , we hav e Infl f ( S ) ≥ ε . Comparing with ( 5 ) , w e may infer | S 1 ∪ S 2 | = | S 1 ∩ S 2 | > k . Since w e started with tw o arbitrary sets S 1 and S 2 from F 1 / 2 , F 1 / 2 m ust then b e a ( k + 1) -in tersecting family ( Definition 7.6 ). No w, consider the t wo cases: (i) F 1 / 2 con tains only sets of size at least 2 k . W e note that F 1 / 4 is a 2 k -in tersecting family . T o see this, let us assume that it is not the case. Then there exist T 1 , T 2 ∈ F 1 / 4 : | T 1 ∩ T 2 | < 2 k . So, by Lemma 7.4 , we get Infl f ( T 1 ∩ T 2 ) = Infl f ( T 1 ∪ T 2 ) ≤ Infl f ( T 1 ) | {z } <ε/ 4 + Infl f ( T 2 ) | {z } <ε/ 4 < 2 ε/ 4 < ε/ 2 . This means that T 1 ∩ T 2 ∈ F 1 / 2 (b y definition of F 1 / 2 ) with | T 1 ∩ T 2 | < 2 k , which contradicts our assumption, in this case, that all sets in F 1 / 2 ha ve cardinality ≥ 2 k . Let S ⊆ [ n ] be a union of k parts in I . Since I is a random r -partition of [ n ] , S is a random subset obtained b y including each element of [ n ] in S with probabilit y k r ≤ 1 2 k +1 . By Theorem 7.8 , w e can b ound the follo wing: Pr h Infl f ( S ) < ε 4 i = Pr[ S ∈ F 1 / 4 ] = µ k/r  F 1 / 4  ≤  k r  2 k . No w we apply the union b ound ov er all p ossible choices of such S (a union of k parts of I ). Then, the probabilit y that at least one such S has Infl f ( S ) < ε/ 4 is at most  r k   k r  2 k ≤  er k  k  k r  2 k ≤  ek r  k = O ( k − k ) ≪ 1 100 . (ii) F 1 / 2 con tains at least one set of size less than 2 k Let J ∈ F 1 / 2 b e such that | J | < 2 k . F or any random partition I with Θ( k 2 ) parts, all the elemen ts of J are in different parts of I , with high probability ( ≥ 99 / 100 ). F or any T ∈ F 1 / 2 , | J ∩ T | ≥ k + 1 (since, F 1 / 2 is a ( k + 1) -in tersecting family), and hence T is not cov ered by any union of k parts of I . That is, for any S ⊆ [ n ] which is cov ered by ≤ k parts of I , S ∈ F 1 / 2 whic h implies Infl f ( S ) ≥ ε/ 2 . 30 7.1 Pro of of correctness of Test- k -Junt a As Algorithm 6 in vok es Algorithm 2 , we first prov e its correctness for juntas. Definition 7.9 (Influential v ariables and buck ets) . Let f : R n → R , and k ∈ N . A v ariable x i , i ∈ [ n ] is said to b e influential with resp ect to f , if for some x ∈ R n , only changing x i c hanges the v alue of f ( x ) . A buck et B ⊆ [ n ] is said to b e influential if it cont ains the index of at least one influential v ariable, and non-influential otherwise. Let E denote the even t that for x , y ∼ N ( 0 , I n ) , x , y ∈ B ( 0 , 2 √ n ) . By F act 4.6 , w e hav e Pr x , y ∼N ( 0 ,I n ) [ E ] = Pr x , y ∼N ( 0 ,I n ) [ x , y ∈ B( 0 , 2 √ n ) ≤ 0 . 01 . Let κ ≜ min V ⊆ [ n ] V is influential ( E x , y ∼N ( 0 ,I n )  | f ( x V y V ) − f ( y ) | | E  ) . As men tioned b efore, we use FindInfBuck et ( Algorithm 2 ) for testing k -linearit y as well. How ev er, the pro of of correctness of Algorithm 2 in Section 5 , presented in Claim 5.11 , holds only for linear functions, whic h may not b e the case in general. So, w e prov e the same for juntas as well. Claim 7.10 (Correctness of Algorithm 2 for juntas) . L et f : R n → R b e given via the η -appr oximate query or acle ˜ f , wher e η ≤ min { κ/ 4 , 1 / (1000 k 2 log 2 k ) } , B = { B 1 , . . . , B r } b e a p artition of [ n ] , and ∅  = S ⊆ [ r ] . With κ ≥ max { 2 ∥ f ∥ ∞ , B( 0 , 2 √ n ) , ε/ (4 k 2 ) } , FindInfBuck et ( f , B , S ) guar ante es: 1. If none of the B i ’s ar e influential, FindInfBuck et ( f , B , S ) always r eturns ∅ and p erforms exactly 2 queries to f . 2. Otherwise, with pr ob ability at le ast 1 − 16 η ⌈ log | S |⌉ 2 /κ , FindInfBuck et ( f , B , S ) r eturns B j for some j ∈ S which is κ -influential and p erforms ≤ 8 ⌈ lg ( | S | ) ⌉ 2 queries to f . Pr o of. Let x and y b e the Gaussian random vectors sampled by Algorithm 2 , and w ≜ y V x V ∈ R n . 1. As none of the B i ’s are influen tial, V (= { i ∈ [ n ] : i ∈ B j for some j ∈ S } ) do es not con tain an y influential v ariable. So, f ≡ f | V , where f | V : R [ | n ] \ V | → R , i.e., for ev ery x ∈ R n , with x V ≜ ( x i , i ∈ V ) ∈ R [ | n ] \ V | , f ( x ) = f | V ( x V ) . So, in Line 5 of Algorithm 2 , f ( w ) = f ( y ) , and hence | e f ( w ) − e f ( y ) | = | e f ( w ) − f ( w ) − ( e f ( y ) − f ( y )) | ≤ ≤ η z }| { | e f ( w ) − f ( w ) | + ≤ η z }| { | e f ( y ) − f ( y ) | ≤ 2 η , b y the triangle inequality , implying it will alwa ys return ∅ , p erforming exactly 2 queries to f . 2. W e will show: When S has some j suc h that B j con tains a κ -influen tial v ariable, then the test in lines 5-10 will fail with probability 1 − 8 η ⌈ log | S |⌉ /κ . Let Z ≜ | f ( x V y V ) − f ( y ) | . If E x , y [ Z ] ≥ κ , then, w e argue (by anticoncen tration) that Pr x , y [ Z > 4 η ] ≥ 1 − 8 η ⌈ log | S |⌉ /κ , and hence Pr x , y h | e f ( x V y V ) − e f ( y ) | > 2 η i with the same probability . T o use anticoncen tration on Z , w e first b ound its second momen t, conditioned on the even t E : E x , y ∼N ( 0 ,I n ) [ Z 2 | E ] = E x , y ∼N ( 0 ,I n ) [ f ( x V y V ) 2 + f ( y ) 2 − 2 f ( x V y V ) f ( y ) | E ] ≤ 4 ∥ f ∥ 2 ∞ , B( 0 , 2 √ n ) . Then, applying Theorem 4.17 (with θ = 4 η κ ∈ (0 , 1) ), conditioned on E , w e get Pr[ Z ≥ 4 η | E ] ≥ Pr " Z ≥ 4 η κ E [ Z ] | {z } ≥ κ | E # ≥  1 − 4 η κ  2 E 2 [ Z | E ] E [ Z 2 | E ] ≥  1 − 4 η κ  2 κ 2 4 ∥ f ∥ 2 ∞ , B( 0 , 2 √ n ) . 31 F rom the premise, we hav e κ ≥ 2 ∥ f ∥ ∞ , B( 0 , 2 √ n ) , giving us Pr[ Z ≥ 4 η | E ] ≥ (1 − 4 η /κ ) 2 . Th us, with probabilit y at least 1 − 8 η /κ , the condition in Line 5 will not hold, ensuring Algorithm 2 reaches the recursive step (Lines 8–13). By construction, S = S ( L ) ⊔ S ( R ) , with 1 ≤ | S ( L ) | ≤ ⌈| S | / 2 ⌉ and 1 ≤ | S ( R ) | ≤ ⌊| S | / 2 ⌋ . W e hav e tw o p ossibilities: (i) S ( L ) con tains no influential buc kets and there exists ∅  = S R ⊆ S ( R ) suc h that B j is influen tial for all j ∈ S R . (ii) There exists ∅  = S L ⊆ S ( L ) suc h that B j is influen tial for all j ∈ S L . In Case (i), FindInfBuck et ( f , B , S ( L ) ) will alwa ys return ∅ and p erform 2 queries (by Part 1) and hence the return v alue of FindInfBuck et ( f , B , S ) will b e FindInfBuck et ( f , B , S ( R ) ) . Using the strong induction h yp othesis we get, with probability at least 1 − 8 η ⌈ log | S |⌉ /κ , this return v alue will be { j } for some j ∈ S R and the total n umber of queries made will b e ≤ 2 + 2 + 4 ⌈ lg( ⌊| S | / 2 ⌋ ) ⌉ ≤ 4 + 4( ⌈ lg | S |⌉ − 1) = 4 ⌈ lg | S |⌉ . In Case (ii), again using the strong induction h yp othesis, with probabilit y at least 1 − 8 η ⌈ log | S |⌉ /κ , FindInfBuck et ( f , B , S ( L ) ) will return { j } for some j ∈ S L and thus the algorithm will return { j } (line 13). The n um b er of queries made will be ≤ 2 + 4 ⌈ lg( ⌈| S | / 2 ⌉ ) ⌉ ≤ 2 + 4( ⌈ lg | S |⌉ − 1 2 ) = 4 ⌈ lg | S |⌉ . Note that, if we run FindInfBuck et ( f , B , S ) for some set S of buck et-indices with | S | > 1 , each recursiv e step (irrespective of whether it falls in case (i) or case (ii) ab o v e) will succeed with probabilit y ≥ 1 − 8 η ⌈ log | S |⌉ /κ . F or the top lev el call to succeed, all the recursive calls must succeed. The num b er of recursive calls made is at most 2 ⌈ lg | S |⌉ . So, we do a union b ound ov er all the failure ev ents to b ound the failure probability of the top-level call by 16 η ⌈ log | S |⌉ 2 /κ . W e are now ready to prov e the main theorem of this section: Pr o of of The or em 7.1 . Let us start with the completeness pro of. Completeness: The argument of completeness of Algorithm 6 holds easily , as when f is a k -jun ta, at most k of the buc kets in the random partition B will b e influential. F rom Claim 7.10 , w e get that each execution of the F or lo op increments I b y: either the index of an influential buck et (if it finds one), or nothing, while also removing the buc ket from future lo op executions. Hence after all its executions, | I | is incremented at most k times, falsifying the chec k in line 10 , making f Accept . Soundness: Now we pro ve soundness. Here our goal is to pro ve that if f is ε -far from all k -jun tas, then Algorithm 6 rejects it with probability at least 2 / 3 . In particular, if f is ε -far from b eing k -jun ta, and S is the union of at most k -parts of I , then Line 7 of Algorithm 6 will b e satisfied with high probabilit y . By Lemma 7.5 , we hav e: if | I | ≤ k , Infl f ( S ) ≥ ε/ 4 ( S = [ n ] \ I in the execution of Algorithm 6 ). A t a p oin t in the execution, when exactly k influen tial buck ets hav e b een iden tified, Infl f ( S ) ≥ ε/ 4 , | S | = O ( k 2 ) − k = O ( k 2 ) , and thus, by Lemma 7.4 , there must exist a buck et B i , i ∈ S suc h that Infl f ( B i ) ≥ ε/ (4 k 2 ) . This buck et will then b e output b y FindInfBuck et ( f , B , S ) . So, un til k + 1 influen tial buck ets hav e b een identified, eac h inv o cation of FindInfBuck et ( f , B , S ) in an execution of its F or lo op, returns an influential buck et with probability at least 1 − 64 η ⌈ log k ⌉ 2 /κ . T reating the outcome of each inv o cation as a geometric random v ariable, we conclude: to recov er at least k + 1 suc h buck ets with probabilit y ≥ 2 / 3 , k /  1 − 64 η ⌈ log k ⌉ 2 /κ  = k + k ∞ X i =1 (64 η ⌈ log k ⌉ 2 /κ ) i = O ( k/ε ) 32 iterations of the F or lo op suffice. When | I | > k , in line 10 , algorithm rejects with the same probabilit y . Query Complexit y: F rom Claim 7.10 , w e know: each in vocation of FindInfBuck et ( f , B , S ) p erforms ≤ 4 log | S | queries to f . With O ( k /ε ) suc h inv o cations in total, and | S | = r = O ( k 2 ) , the o verall query complexity thus is O ( k log k /ε ) . 8 Lo w er Bounds In this section, w e will present our low er b ound results, restated here for conv enience: Theorem 2.10. Given exact query ac c ess to f : R n → R , some k , d ∈ N and a distanc e p ar ameter ε ∈ (0 , 1) , Ω (max { k , 1 / ε } ) queries ar e ne c essary for testing the fol lowing pr op erties with pr ob ability at le ast 2 / 3 : (i) k -line arity. (ii) k -junta. (iii) k -sp arse de gr e e- d p olynomials. Mor e over, for testing k -sp arse de gr e e- d p olynomial, the lower b ound is impr ove d to Ω (max { d, k , 1 / ε } ) . F or proving the lo wer b ounds, w e follo w the approac h of [ BBM12 ], i.e., we show the hardness of testing these problems by reducing the Set-Disjointness problem, a canonical hard problem in the field of communication complexity , to them. Pro ving low er b ounds in the exact query mo del directly gives us low er b ounds in the appro ximate query mo del. F or brevit y , we only presen t the lo wer b ound argument for k -linearit y . The other results follow the same approach. A low er b ound of Ω( 1 ε ) queries: F or an y ε ∈ (0 , 1) , it is a folklore result that distinguishing if a function f is linear, or ε -far from b eing linear in ℓ 1 -distance, requires Ω(1 /ε ) queries. An exp osition for this is pro vided in [ Fis24 ]. A low er b ound of Ω( d ) queries for k -sparse, degree- d p olynomial testing: As b efore, it is another folklore result that given a proximit y parameter ε ∈ (0 , 1) , Ω( d ) queries are necessary to distinguish if an unknown function f is a degree- d p olynomial, or a degree- ( d + 1) (or ( d − 1) ) p olynomial (and hence, is ε -far from all degree- d p olynomials) with probability at least 2 / 3 . T o prov e the Ω( k ) low er b ound, we next present a reduction from Set-Disjointness . 8.1 Comm unication complexity setting W e consider the tw o-pla yer communication game setting, featuring Alice and Bob. Alice has a function f , and Bob has a function g , and they join tly w ant to ev aluate another function h (whic h is a function of f and g ). W e assume that b oth the play ers ha ve unbounded computational p ow er, but to ev aluate h , they need to communicate among themselves, and the ob jective is to keep this comm unication ov erhead as small as p ossible. It is known that it takes linear in the size of the bits of the largest set to solve Set-Disjointness . Theorem 8.1 ([ HW07 ], Set-Disjointness Lo wer Bound) . L et A lic e and Bob have two sets A and B , r esp e ctively, e ach of size at most k fr om a universe of size n . In or der to distinguish if | A ∩ B | = 1 , or | A ∩ B | = 0 , Ω( k ) bits of c ommunic ation b etwe en Alic e and Bob ar e r e quir e d. 33 8.2 Connection b et w een Comm unication and Query complexit y Consider tw o functions f and g , a prop ert y P , h ≜ f − g , and a communication problem C h, P : Alice and Bob receiv e f and g , resp ectively , and they wan t to decide if h ∈ P , or f is ε -far from P . [ BBM12 ] pro ved that R ( C h, P ) , the randomized communication complexity of C h, P , is at most twice Q ( P ) , the query complexity of deciding the prop erty P . Lemma 8.2 ([ BBM12 , Lemma 2.2]) . L et P b e a pr op erty of functions, and h b e a function. Then R ( C h, P ) ≤ 2 Q ( P ) . No w, given tw o sets A and B , we construct tw o functions f and g suitably as follows. 8.3 Construction of the hard instances for the low er b ound Giv en A ⊆ [ n ] : | A | = k , Alice constructs a p olynomial f = P i ∈ A x i . Similarly , Bob constructs a p olynomial g = P i ∈ B x i . Note that f , g : R n → R . Let h = f − g . Consider the tw o cases: (i) | A ∩ B | = 1 : In this case h w ould b e a (2 k + 2) linear function. (ii) | A ∩ B | = 0 : In this case, h w ould b e a 2 k -linear function. No w we show that under the standard Gaussian distribution N ( 0 , I n ) , h ’s corresp onding to cases (i), and (ii) ab ov e, are sufficiently far in ℓ 1 -distance with probabilit y at least 2 / 3 . Lemma 8.3. L et f 1 b e a (2 k + 2) -line ar function, and f 2 b e a 2 k -line ar function with c o efficients fr om { 0 , 1 } . Under the standar d Gaussian distribution N ( 0 , I n ) , f 1 is Ω(1) -far fr om f 2 in ℓ 1 -distanc e with pr ob ability at le ast 2 / 3 . Pr o of. Consider the function g = f 1 − f 2 . Observe that g is at least a 2 -linear function, with at least the tw o terms from f 1 not present in f 2 . Without loss of generality , let g ( x ) = x 1 + x 2 . W e are in terested in the even t where | g ( x ) | > O (1) . Using Theorem 4.16 (with d = 1 , t = 0 , g ( x ) = x 1 + x 2 , and co eff 1 ( g ) = √ 2 ), w e get: ∀ ε > 0 , Pr x ∼N ( 0 ,I n ) h | g ( x ) | ≤ ε i ≤ C  ε √ 2  . Setting ε = 1 2 C , w e hav e the following: Pr x ∼N ( 0 ,I n )  | g ( x ) | ≤ 1 2 C  ≤  1 2 √ 2  ≤ 1 / 3 . Th us, f 1 is at least 1 2 C -far from f 2 with probabilit y at least 2 / 3 . Sim ulation Let C h b e the comm unication problem where Alice and Bob hav e t wo sets A and B, resp ectiv ely , eac h of size k . As discussed b efore, they ha ve constructed the t wo functions f and g , resp ectiv ely , from A and B . Equiv alen tly , w e can also sa y that Alice and Bob are given tw o functions f and g resp ectiv ely , with the promise that the function h = f − g is either a 2 k -linear, or a (2 k + 2) -linear function, and they should accept if and only if h is a 2 k -linear function. F rom the ab ov e reduction, it is clear that R ( C h ) ≥ R ( Set Disjointness ) = Ω( k ) . By Lemma 8.2 , w e can say that an y testing algorithm that can distinguish b etw een 2 k -linear and 34 (2 k + 2) -linear functions with probability at least 2 / 3 , requires Ω( k ) queries. Moreov er, from Lemma 8.3 , we kno w that any 2 k -linear function is Ω(1) -far from any (2 k + 2) -linear function. Th us, an y 2 k -linearit y tester (with ε = o (1) ) that uses o ( k ) queries can not distinguish 2 k -linear functions (whic h should b e accepted) and (2 k + 2) -linear functions (whic h should b e rejected) correctly , with probabilit y ≥ 2 / 3 . This completes the pro of of the low er b ound of k -linearit y testing. 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In Pr o c e e dings of the thirty- ninth annual ACM symp osium on The ory of c omputing , pages 506–515, 2007. 2 [W ol24] W olframAlpha. Analysis of expression (alternative forms). 2024. 24 A Omitted Algorithms from the Main Bo dy Here w e state the algorithms from [ FY20 ], [ ABF + 23 ] and [ AKM25 ] for completeness. W e b egin with Algorithm 7 , a query-optimal appro ximate additivit y tester, with its subroutines pro vided in Algorithm 8 . The prop erties of this tester were recorded earlier in Theorem 5.4 . This is follo wed by Algorithm 9 , a query-optimal (w.r.t. the distance parameter ε ) approximate low-degree tester, with its subroutines in Algorithm 10 , and its prop erties recorded in Theorem 6.2 . B Omitted Pro ofs from Section 6 Observ ation 6.4 (Generalization of [ BOT88 , Section 4], and [ GJR10 , Lemma 4]) . L et f : R n → R b e an exactly k -sp arse p olynomial over the r e als, i.e., f ( x ) = P k i =1 a i M i ( x ) , wher e a 1 , . . . , a k ∈ R \ { 0 } , and M 1 , . . . , M k ar e the monomials of f . Then for al l ℓ + 1 ≤ k , det ( H ℓ +1 ( f , x )) = X S ⊆ [ k ] | S | = ℓ +1 Y i ∈ S a i Y i,j ∈ S i k , det ( H ℓ +1 ( f , x )) ≡ 0 . 40 Algorithm 7: [ ABF + 23 , Algorithm 7] and [ AKM25 , Algorithm 5]: (Query-)Optimal Appro ximate Additivit y T ester 1 Pro cedure Appro xima te Additivity Tester ( f , D , α, ε, R ) Giv en : Query access to f : R n → R , sampling access to an unknown ( ε/ 4 , R ) -concen trated distribution D , a noise parameter α > 0 , and a farness parameter ε > 0 ; 2 δ ← 3 α , r ← 1 / 50 ; 3 return Reject if TestAdditivity ( f , δ ) returns Reject ; 4 for N 7 ← O (1 /ε ) times do 5 Sample p ∼ D ; 6 if p ∈ B( 0 , R ) then 7 return Reject if | f ( p ) − Appr oxima te- g ( p , f , δ ) | > 5 δ n 1 . 5 κ p , or if Appr oxima te- g ( p , f , δ ) returns Reject . 8 return A ccept . Algorithm 8: [ ABF + 23 , Algorithm 8] and [ AKM25 , Algorithm 6] Additivit y Subroutines 1 Pro cedure TestAdditivity ( f , δ ) Giv en : Query access to f : R n → R , threshold parameter δ > 0 ; 2 for N 8 ← O (1) times do 3 Sample x , y , z ∼ N ( 0 , I n ) ; 4 return Reject if | f ( − x ) + f ( x ) | > δ ; 5 return Reject if | f ( x − y ) − ( f ( x ) − f ( y )) | > δ ; 6 return Reject if    f  x − y √ 2  −  f  x − z √ 2  + f  z − y √ 2     > δ ; 7 return A ccept . 8 Pro cedure Appro xima te - g ( p , f , δ ) Giv en : p ∈ R n , query access to f : R n → R , threshold parameter δ > 0 ; 9 Sample x 1 ∼ N ( 0 , I n ) ; 10 return κ p ( f ( p /κ p − x 1 ) + f ( x 1 )) . Pr o of. Since for all i ∈ [ k ] and α, β ∈ N ≥ 0 , M i ( x α ) M i ( x β ) = M i ( x α + β ) , w e get: for all u ∈ R n , H ℓ +1 ( f , u ) =      P k i =1 a i M i ( u 0 ) P k i =1 a i M i ( u 1 ) . . . P k i =1 a i M i ( u ℓ ) P k i =1 a i M i ( u 1 ) P k i =1 a i M i ( u 2 ) . . . P k i =1 a i M i ( u ℓ +1 ) . . . . . . . . . . . . P k i =1 a i M i ( u ℓ ) P k i =1 a i M i ( u ℓ +1 ) . . . P k i =1 a i M i ( u 2 ℓ )      =        M 1 ( u 0 ) M 2 ( u 0 ) . . . M k ( u 0 ) M 1 ( u 1 ) M 2 ( u 1 ) . . . M k ( u 1 ) M 1 ( u 2 ) M 2 ( u 2 ) . . . M k ( u 2 ) . . . . . . . . . . . . M 1 ( u ℓ ) M 2 ( u ℓ ) . . . M k ( u ℓ )             a 1 M 1 ( u 0 ) a 1 M 1 ( u 1 ) . . . a 1 M 1 ( u ℓ ) a 2 M 2 ( u 0 ) a 2 M 2 ( u 1 ) . . . a 2 M 2 ( u ℓ ) . . . . . . . . . . . . a k M k ( u 0 ) a k M k ( u 1 ) . . . a k M k ( u ℓ )      41 Algorithm 9: [ ABF + 23 , Algorithm 3] and [ AKM25 , Algorithm 7] Optimal Approximate Lo w Degree T ester 1 Pro cedure Appro xLo wDegreeTester ( f , d, D , α, ε, R, L ) Giv en : Query access to f : R n → R that is b ounded in B ( 0 , L ) for some L > 0 , a degree d ∈ N , sampling access to an unkno wn ( ε/ 4 , R ) -concentrated distribution D , a noise parameter α > 0 , and a farness parameter ε > 0 . 2 δ ← 2 d +1 α , r ← (4 d ) − 6 ; 3 return Reject if Appro xChara cteriza tionTest rejects ; 4 for N 9 ← O ( ε − 1 ) times do 5 Sample p ∼ 2 d √ n L D ; 6 if p ∈ B( 0 , 2 dR √ n/L ) then 7 return Reject if | f ( p ) − Appr oxQuer y- g ( p ) | > 2 · 2 (2 n ) 45 d ( R/L ) d δ , or if Appr oxQuer y- g ( p ) rejects . 8 return A ccept . =        1 1 . . . 1 M 1 ( u ) M 2 ( u ) . . . M k ( u ) ( M 1 ( u )) 2 ( M 2 ( u )) 2 . . . ( M k ( u )) 2 . . . . . . . . . . . . ( M 1 ( u )) ℓ ( M 2 ( u )) ℓ . . . ( M k ( u )) ℓ        | {z } h 1 : R k → R ℓ +1      a 1 0 . . . 0 0 a 2 . . . 0 . . . . . . . . . . . . 0 0 . . . a k      | {z } h 2 : R k → R k      1 M 1 ( u ) . . . ( M 1 ( u )) ℓ 1 M 2 ( u ) . . . ( M 2 ( u )) ℓ . . . . . . . . . . . . 1 M k ( u ) . . . ( M k ( u )) ℓ      | {z } h 3 : R ℓ +1 → R k . Th us, if ℓ + 1 > k , the mapping h 3 in the ab ov e expression is necessarily non-injective, and hence H ℓ +1 ( f , u ) is necessarily singular, for an y u . This implies that det ( H ℓ +1 ( f , x )) ≡ 0 . If k ≥ ℓ + 1 , the determinan t expansion in the observ ation follows from the following argumen t in [ BOT88 ]: 1. F or any fixed u , by standard determinant expansion, det ( H ℓ +1 ( f , u )) is a p olynomial Q ( a ) , and the total degree of eac h monomial (in the “v ariables” a 1 , . . . , a k ) in Q is exactly ℓ + 1 . 2. If rank ( H ℓ +1 ( f , u )) ≤ ∥ [ a 1 · · · a k ] ∥ 0 < ℓ + 1 , w e m ust ha ve det ( H ℓ +1 ( f , u )) ≡ 0 , irresp ective of the v alues of the non-zero { a 1 , . . . , a k } , for an y u ∈ R n . Hence each monomial (in a 1 , . . . , a k ) of Q ( a 1 , . . . , a k ) m ust ha ve at least ℓ + 1 of the v ariables { a i } . Since its total degree is exactly ℓ + 1 , each monomial must b e of the form Q i ∈ S a i for some S ⊆ [ k ] , with | S | = ℓ + 1 . 3. The co efficien t of monomial Q i ∈ S a i in Q will b e Q ( c 1 , . . . , c k ) with c i = 1 { i ∈ S } . But, b y the decomposition abov e, this will b e the square of the V andermonde determinant (see Definition 4.13 ), det ( V ℓ +1 ( { x i } i ∈ S )) = Q i,j ∈ S i δ ; 9 Sample p ∼ N ( 0 , j 2 I n ) , q ∼ N ( 0 , ( t 2 + 1) I n ) ; 10 return Reject if | P d +1 i =0 α i · f ( p + i q ) | > δ ; 11 Sample p , q ∼ N ( 0 , j 2 I n ) ; 12 return Reject if | P d +1 i =0 α i · f ( p + i q ) | > δ ; 13 return A ccept ; 14 Pro cedure Appro xQuer y- g ( p ) 15 if p ∈ B( 0 , r ) then 16 return Appr oxQuer y- g -InBall ( p ); 17 for i ∈ { 0 , 1 , . . . , d } do 18 c i ← r ∥ p ∥ 2 cos  π ( i +1 / 2) d +1  ; 19 v ( c i ) ← Appr oxQuer y- g -InBall ( c i p ) ; 20 Let p p : R → R b e the unique degree- d p olynomial such that p p ( c i ) = v ( c i ) for all i ; 21 return p p (1) ; 22 Pro cedure Appro xQuer y- g -InBall ( p ) 23 Sample q 1 ∼ N ( 0 , I n ) ; 24 return P d +1 i =1 α i · f ( p + i q 1 ) ; Pr o of of Observation 6.6 . Let u ∈ R n , and α i ≜ ˜ f ( u i ) − f ( u i ) for each i ∈ { 0 , 1 , . . . , 2 t − 2 } , so that | α i | ≤ η b y the approximation-oracle guarantee. W e can rewrite H t ( ˜ f , u ) (by Definition 2.5 ) as: H t ( ˜ f , u ) =      ˜ f ( u 0 ) ˜ f ( u 1 ) . . . ˜ f ( u t − 1 ) ˜ f ( u 1 ) ˜ f ( u 2 ) . . . ˜ f ( u t ) . . . . . . . . . . . . ˜ f ( u t − 1 ) ˜ f ( u t ) . . . ˜ f ( u 2 t − 2 )      =      f ( u 0 ) + α 0 f ( u 1 ) + α 1 . . . f ( u t − 1 ) + α t − 1 f ( u 1 ) + α 1 f ( u 2 ) + α 2 . . . f ( u t ) + α t . . . . . . . . . . . . f ( u t − 1 ) + α t − 1 f ( u t ) + α t . . . f ( u 2 t − 2 ) + α 2 t − 2      =      f ( u 0 ) f ( u 1 ) . . . f ( u t − 1 ) f ( u 1 ) f ( u 2 ) . . . f ( u t ) . . . . . . . . . . . . f ( u t − 1 ) f ( u t ) . . . f ( u 2 t − 2 )      | {z } = H t ( f , u ) +      α 0 α 1 . . . α t − 1 α 1 α 2 . . . α t . . . . . . . . . . . . α t − 1 α t . . . α 2 t − 2      | {z } ≜ E t ( u ) . The Hankel structure of E t ( u ) is eviden t. ∥ E t ( u ) ∥ ∞ ≤ η follo ws from the approximation-oracle guaran tee. This in turn sho ws ∥ E t ( u ) ∥ F ≤ p η 2 t 2 = η t , which implies the b ound on ∥ E t ( u ) ∥ op . 43