The Limit of Convexity Based Isoperimetry: Sampling Harmonic-Concave Functions

Sampling Harmonic Conca v e F unctions: The Limit of Con v exit y Based Isop erimetry Karthek ey an Chandrasek aran ∗ Amit Deshpande † San tosh V empala ∗ Abstract Logconcave functions represent the current frontier of efficien t algo rithms for sampling, opti- mization and in teg ration in R n [L V06a]. Efficient sampling algo rithms to sample acco rding to a probability density (to whic h the other tw o pr oblems can b e r educed) re lie s on go o d isop erimetr y which is known to hold for arbitr ary log c o ncav e densities. In this pap er, we extend this frontier in tw o wa ys: first, we characterize conv exity-lik e conditions that imply go o d iso p er imetry , i.e., what condition o n function v alues a lo ng every line g uarantees go o d isop er imetry? The answer turns out to be the set of (1 / ( n − 1)) -harmonic c onc ave funct ions in R n ; we also prov e that this is the b est po ssible characterization a long every line, of functions having go o d isop erimetry . Next, we give the first efficie nt algo rithm for s ampling according to such functions with complexity depe nding on a smo othnes s pa rameter. F ur ther, noting that the multiv ariate Cauch y densit y is an imp orta nt distribution in this c la ss, we ex ploit certa in prop erties of the Cauch y density to give a n e fficient sa mpling algor ithm based on random walks with a mixing time that matches the current b est b o unds known for sampling logconcave functions. 1 In tro du ction Ov er the past t wo decades, logconca ve functions ha ve emerged as the common fron tier for the complexit y of sampling, optimizatio n a n d int egration. More p r ecisely , give n a f u nction f : R n → R + , accessible b y querying the function v alue at an y p oint x ∈ R n , and an error p arameter ǫ > 0, three fund amental problems are: (i) Integratio n: estimate R f to w ith in 1 ± ǫ , (ii) Maximization: find x that appro ximately maximizes f , i.e., f ( x ) ≥ (1 − ǫ ) max f , and (iii) Samplin g: generate x from d ensit y π with d tv ( π , π f ) ≤ ǫ wh ere d tv is the tota l v ariation distance and π f is the densit y prop ortional to f . (F or eac h of these, exact solutions are intracta ble.) The complexit y of an algorithm is measured by the n umber of queries for the function v alues and the n umb er of a r ithmetic o p erations. The most general class of fun ctions for whic h these problems are kn o wn to ha ve p olynomial complexit y in the d imension n , is the class of logco n ca v e functions. A function f : R n → R + is logconca v e if its logarithm is conca v e on its supp ort, i.e., for an y tw o p oints x, y ∈ R n and an y λ ∈ (0 , 1), f ( λx + (1 − λ ) y ) ≥ f ( x ) λ f ( y ) 1 − λ . (1) This p o werful class generalizes indicator functions of con vex b o dies (and hence th e problems sub- sume con v ex optimization and volume computation) as w ell as Gaussians. F ollo wing the celebrated result of Dy er, F rieze and Kannan [DFK91] giving a p olynomial algorithm for estimating the v olume of a con ve x b o dy , a long lin e of w ork [AK91, Lo v90 , DF91, LS92, LS93, KL S97, L V07, L V06c, L V06b] ∗ School of Computer Science, Georgia T ech. Email: kart he@gatech.edu, vempala@cc.gatec h.edu † Microsof t R esearc h. Email : amitdesh@m icrosoft.com 1 culminated in the results that b oth s ampling and int egration h a v e p olynomial complexit y for any logconca ve d ensit y . In tegration is d one by a reduction to sampling an d sampling also pro vides an alternativ e to the Ellipsoid metho d for optimization [BV04, KV06, L V06a]. Sampling itself is ac hiev ed b y a random walk w hose stationary distribution has densit y prop ortional to the given function. The k ey qu estion is thus the rate of con v ergence of the w alk, wh ic h dep end s (among other thin gs) on the isoperimetry of the target function. Roughly s p eaking, isop erimetry is the minim u m r atio of the measure of the b ound ary of a partition of sp ace in to tw o sets to the measure of the smaller of the tw o sets. Logconca v e f unctions satisfy the follo wing isop erimetric inequalit y: Theorem 1. [DF91, LS93] L et π f b e a distribution in R n with density pr op ortional to a lo gc onc ave function f . L et K b e the supp ort of f , D the diameter of K and S 1 , S 2 , S 3 b e any p artition of K . Then, π f ( S 3 ) ≥ 2 d ( S 1 , S 2 ) D min { π f ( S 1 ) , π f ( S 2 ) } . wher e d ( S 1 , S 2 ) r efers to the minimum distanc e b etwe en any two p oints in S 1 and S 2 . While th ese results are fairly general, they do n ot captur e the complete class of fu n ctions whic h ha ve goo d isoperimetry . Logconca ve f u nctions in R n are defined by a conv ex-combinatio n based condition for every t wo p oints in the supp ort of the fu nction, sayi n g that the function is logconca v e along ev ery line. Th is is a generalisation of the case of conv ex b o dies w here we hav e that the lin e segmen t connecting any tw o p oint s in the b o dy lies completely within the b o d y . T his motiv ates the follo wing question: What is the condition that needs to b e satisfied along ev ery line by a function, for it to ha ve go o d isop erimetry? In this p ap er, we present the complete class of fu nctions with go o d isop erimetry that can b e describ ed by suc h co nv ex-com bination based conditions. W e a lso giv e an efficien t algorithm to sample from these functions. F ur ther, w e iden tify the Cauc hy den sit y (wh ich is not logconca v e) as a w ell-known example in this s et of f unctions and obtain an efficient algo rithm for samp lin g the m ultiv ariate Cauc hy densit y r estricted to a conv ex b o dy . Its complexit y mat ches the b est- kno wn b ound s for logconca ve f u nctions. W e note th at the density functions satisfying the con v ex- com bination based c h aracterization that w e p resen t h ere, could b e hea vy-tailed with un b oun ded momen ts as is the case with the Cauc hy den s it y . T o motiv ate and state our results, we b egin with a discus s ion of 1-dimensional cond itions. 1.1 F rom conca v e to quasi-conca ve A fu nction f : R n → R + is s aid to b e                            c onc ave if , f ( λx + (1 − λ ) y ) ≥ λf ( x ) + (1 − λ ) f ( y ) lo gc onc ave if , f ( λx + (1 − λ ) y ) ≥ f ( x ) λ f ( y ) 1 − λ s -harmonic-c onc ave if , f ( λx + (1 − λ ) y ) ≥  λ f ( x ) s + 1 − λ f ( y ) s  − 1 s quasi-c onc ave if , f ( λx + (1 − λ ) y ) ≥ min { f ( x ) , f ( y ) } for all λ ∈ [0 , 1] , ∀ x, y ∈ R n . 2 These conditions are progressively weak er, r estricting the function v alue at a conv ex combinatio n of x and y to b e at least the arithmetic av erage, geometric av erage, harmonic a v erage and min im um, resp ectiv ely . Not e that s = 1 giv es the usual harmonic a verage (and s 1 -harmonic-conca ve functions are also s 2 -harmonic-conca ve if s 1 < s 2 ). It is thus easy to verify th at: conca v e ( logconca v e ( s -harmonic-conca v e ( quasi-conca v e . Relaxing fu r ther w ould violate unimo dalit y , i.e., there could b e t wo distinct lo cal maxima, wh ic h app ears quite problematic for all of th e fundamental p roblems. Also, it is we ll-known that quasi- conca v e functions hav e p o or isop erimetry . W e note here the relation b et wee n the s -h armonic-conca ve probabilit y density fun ction and th e probabilit y measure as sh o wn b y C . Borell [Bor74, Bor75]. This giv es an equiv alence b et ween the one-dimensional con v exit y-like condition to a condition on the pr obabilit y measure of the fun ction. Lemma 2. L e t −∞ < κ ≤ 1 n . A n absolutely c ontinuous pr ob ability me asur e µ on R n is κ -c onc ave if and only if it is c onc entr ate d on an op e n c onvex set K in R n and has ther e a p ositive density p , which is κ ( n ) − concav e for κ ( n ) = κ 1 − κn . Th u s, w e ha v e that if the d ensit y function is s -harmonic-conca ve f or s ∈ [0 , 1 /n ], then the corresp ondin g probabilit y measure is κ -conca v e f or κ = − s 1 − ns . F ur ther, Bobk o v [Bob07] p ro ves the follo wing isop erimetric inequalit y for κ -conca v e probability measures for − ∞ < κ ≤ 1. Theorem 3. Given a κ -c onc ave pr ob ability me asur e µ , for any me asur able subset A ⊆ R n , µ ( δ A ) ≥ c ( κ ) m min { µ ( A ) , 1 − µ ( A ) } 1 − κ wher e m = R R n | x | dµ ( x ) , for some c onstant c ( κ ) dep ending on κ . Using the c haracterisation giv en b y C .Borell in Lemma 2, one can obtain a w eak er form of isop erimetric in equ alit y from th e ab ov e theorem for s -harmonic-conca ve functions f : R n → R + for s ≥ 1 n to sa y that for any m easur able su bset A , for some constant c ( s ) dep ending only on s , π f ( δ A ) ≥ c ( s ) m min { π f ( A ) , 1 − π f ( A ) } 1+ s 1 − ns W e note that our result giv es a s tronger inequalit y , in th e sense that, we remo v e the d ep enden ce on s from the inequ alit y completely . W e prov e such an inequalit y f or the more ge n eral class of  1 n − 1  -harmonic-conca ve functions rather than  1 n  -harmonic-conca ve functions. W e pro ceed to sho w that the limit of isop erimetry for con v ex-com bination based conditions alo n g ev ery line for the density f unction, is the set of (1 / ( n − 1))- h armonic-conca ve fu n ctions. F urther, w e addr ess th e problem of sampling fr om su c h distrib utions restricted to an y conv ex set (p ossib ly u n b ounded). 1.2 The Cauch y densit y The generalized Cauch y probabilit y den s it y f : R n → R + can b e wr itten as f ( x ) ∝ det( A ) − 1  1 + k A ( x − m ) k 2  ( n +1) / 2 . where A ∈ R n × n . F or simp licit y , we assume m = ¯ 0 u s ing a translati on. It is easy to sample this distribution in full space (by an affine transf orm ation it b ecomes spherically symmetric and 3 therefore a one-dimensional p roblem) [Joh 87 ]. W e consider the problem of sampling according to the Cauch y densit y restricted to a con v ex set. T h is is r eminiscen t of the work of Kann an and Li who considered th e problem of sampling a Gaussian d istribution restricted to a conv ex s et [KL96]. The Cauc hy density f unction b elongs to th e broader class of L ´ evy ske w alpha-stable distr ib u- tions (or simp ly kno wn as stable d istributions) [Man06 , Nol09] which are us efu l in mo d eling many v ariables in ph ysics and mathematica l finance. Unlik e most stable d istributions, Cauch y densities ha ve a closed f orm expression for th eir d ensit y . Being a 1-stable d istribution, it finds useful app lica- tions in a v ariet y of p roblems including the approxi mate nearest neigh b or p roblem and d imension reduction in l 1 norm [Ind06, DI IM04, LHC 07]. 1.3 Our results Our first result establishes go o d isop erimetry for 1 / ( n − 1)-harmonic-conca ve functions in R n . Theorem 4. L et f : R n → R + b e a (1 / ( n − 1)) -harmonic-c onc ave function with a supp ort K . L et R n = S 1 ∪ S 2 ∪ S 3 b e a me asur able p artition of R n into thr e e non-empty subsets. Then π f ( S 3 ) ≥ d ( S 1 , S 2 ) D min { π f ( S 1 ) , π f ( S 2 ) } , wher e D is the diameter of K . It is worth noting th at the isop erimetric coefficient ab o ve is only smaller b y a f actor of 2 when compared to that of logconca v e fu nctions (Theorem 1). Next, we prov e that if we go slightly b eyo nd the class of (1 / ( n − 1))-harmonic-conca ve functions, then there exist fu nctions w ith exp onentia lly s m all isop erimetric co efficien t. Theorem 5. F or any ǫ > 0 , ther e exists a 1 / ( n − 1 − ǫ ) -harmonic-c onc ave function f : R n → R + with a c onvex supp ort K of finite diameter (i. e ., D K < ∞ ) and a p artition R n = S ∪ T such that π f ( ∂ S ) min { π f ( S ) , π f ( T ) } ≤ C n (1 + ǫ ) − ǫn for some c onstant C > 0 . T o su mmarize, w e hav e the follo wing table for isop erimetry: Nature of f Go o d Isop erim etry? Conca v e Y es Logconca ve Y es (1 / ( n − 1))-harmonic-conca ve Y es (1 / ( n − 1 − ǫ ))-harmonic-conca ve ( ǫ > 0) No Harmonic-Conca v e No Quasi-Conca v e No Next we pro v e that the ball w alk with a Met r op olis filte r can b e used to sample efficien tly according to the (1 / ( n − 1))-harmonic conca v e distribu tion function whic h satisfy a certain Lipschitz condition. A t a p oint x , the ball w alk picks a new p oint y uniform ly at random from a fixed r adius ball around x and mo ves to y with probability min { 1 , f ( y ) /f ( x ) } . A d istribution σ 0 is said to b e an H -warm start ( H > 0) for the distribu tion π f if for all S ⊆ R n , σ 0 ( S ) ≤ H π f ( S ). Let σ m b e the distribu tion after m steps of the ball walk with a Metrop olis filter. W e sa y that a function f : R n → R + has parameters ( α, δ ) if f or all p oints x, y in the sup p ort of f suc h th at k x − y k ≤ δ , we ha v e max { f ( x ) /f ( y ) , f ( y ) /f ( x ) } ≤ α . 4 Theorem 6. L et f : R n → R + b e pr op ortional to an s - harmonic-c onc ave function with p ar ameters ( α, δ ) , r estricte d to a c onvex b o dy K ⊆ R n of diameter D , wher e s ≤ 1 / ( n − 1) . L et K c ontain a b al l of r adius δ and σ 0 b e an H -warm start. Th en, ther e exists a r adius r for the b al l walk such that, after m ≥  C nD 2 δ 2 log 2 H ǫ  · max  nH 2 ǫ 2 , ( α s − 1) 2 s 2 δ 2  steps, we have that d tv ( σ m , π f ) ≤ ǫ, for some absolute c onstant C, wher e d tv ( · , · ) is the total v ariation distanc e. Applying the ab o v e theorem d irectly to sample according to the Cauc hy density , we get a mixing time of O  n 3 H 2 ǫ 2 log 2 H ǫ  · max n H 2 ǫ 2 , n o using p arameters D = 8 √ 2 nH ǫ (since the probability measure outside the D -ball is at most ǫ/ 2 H ), δ = 1 and α = e n +1 2 . Using a more careful analysis (comparison of 1-step distr ibutions), this b ound can b e impro ved to matc h the cur ren t b est b ound s for sampling logconca ve functions. Theorem 7. L et f b e pr op ortional to a Cauchy pr ob ability density r estricte d to a c onvex set K ⊆ R n c ontaining a b al l of r adius k A − 1 k 2 and let σ 0 b e an H - warm starting distribution. Then after m ≥ O  n 3 H 4 ǫ 4 log 2 H ǫ  steps with b al l-walk r adius r = ǫ/ 8 √ n , we have d tv ( σ m , π f ) ≤ ǫ wher e d tv ( ., . ) is the total v ariation distanc e. The pro of of this theorem departs fr om earlier coun terparts in a significan t wa y . In add ition to isop erimetry , and the closeness of one-step distrib utions of nearby p oin ts, we hav e to pr o v e that most of the measure is con tained in a ball of not-too-large radiu s. F or logconca ve densities, this large-ball pr obabilit y deca ys exp on entially w ith the radius. F or the Cauc hy d ensit y it only deca ys linearly (Prop osition 20). 2 Preliminaries Let r B x denote a b all of r adius r aroun d p oint x . One step of the ball w alk at a p oin t x defi n es a probabilit y distribution P x o v er R n as follo ws . P x ( S ) = Z S ∩ r B x min  1 , f ( y ) f ( x )  dy . F or eve r y measurable s et S ⊆ R n the ergo d ic flo w from S is defi ned as Φ( S ) = Z S P x ( R n \ S ) f ( x ) dx, and the measur e of S according to π f is defin ed as π f ( S ) = R S f ( x ) dx/ R R n f ( x ) dx . Th e s - conductance φ s of the Marko v c hain defined by ball w alk is φ s = inf s ≤ π f ( S ) ≤ 1 / 2 Φ( S ) π f ( S ) − s . 5 T o compare t wo distributions Q 1 , Q 2 w e use the total v ariation distance b et we en Q 1 and Q 2 , defined b y d tv ( Q 1 , Q 2 ) = sup A | Q 1 ( A ) − Q 2 ( A ) | . Wh en we refer to the d istance b et w een tw o sets, w e mean the minim u m distance b etw een an y t wo p oints in the tw o sets. That is, for any tw o subs ets S 1 , S 2 ⊆ R n , d ( S 1 , S 2 ) := min {| u − v | : u ∈ S 1 , v ∈ S 2 } . Next we quote a lemma from [LS93] w h ic h relates the s -conductance to the m ixing time. Lemma 8. L et 0 < s ≤ 1 / 2 and H s = sup π f ( S ) ≤ s | σ 0 ( S ) − π f ( S ) | . Then for e very me asur able S ⊆ R n and every m ≥ 0 , | σ m ( S ) − π f ( S ) | ≤ H s + H s s  1 − φ 2 s 2  m . Finally , the follo wing lo calization lemma [LS93, KLS95] is a u seful tool in the pr o ofs of isop eri- metric inequalities. Lemma 9. L et g : R n → R a nd h : R n → R b e two lower semi-c ontinuous inte gr able functions such that Z R n g ( x ) dx > 0 and Z R n h ( x ) dx > 0 . Then ther e exist two p oints a, b ∈ R n and a line ar function l : [0 , 1] → R + such that Z 1 0 g ((1 − t ) a + tb ) l ( t ) n − 1 dt > 0 and Z 1 0 h ((1 − t ) a + t b ) l ( t ) n − 1 dt > 0 . 3 Isop erimetry Here we prov e an isop erimetric inequalit y for functions satisfying a certain unimo dalit y criterion. W e fu rther s h o w that (1 / ( n − 1))-harmonic-conca ve functions satisfy this u nimo dalit y criterion and hence hav e go o d isop erimetry . W e b egin with a simple lemma that will b e used in the pro of of the isop erimetric inequalit y . Lemma 10. L et p : [0 , 1] → R + b e a unimo dal function, and let 0 ≤ α < β ≤ 1 . Then Z β α p ( t ) dt ≥ | α − β | min  Z α 0 p ( t ) dt, Z 1 β p ( t ) dt  . Pr o of of L emma 10(Isop erimetry for 1-dimensional unimo dal functions). . Su pp ose the maximum of p o ccurs at t = t max . If t max ≤ α then Z β α p ( t ) dt ≥ p ( β ) · | α − β | ≥ | α − β | · p ( β ) · | β − 1 | ≥ | α − β | · Z 1 β p ( t ) dt Otherwise, if t max > α then Z β α p ( t ) dt ≥ p ( α ) · | α − β | ≥ | α − β | · p ( α ) · | α | ≥ | α − β | · Z α 0 p ( t ) dt No w w e are r eady to p ro ve an isop erimetric inequalit y for fun ctions satisfying a certa in uni- mo dalit y criterion. 6 Theorem 11. L et f : R n → R + b e a function whose su pp ort has diameter D , and f satisfies the fol lowing unimo dality criterion: F or any affine line L ⊆ R n and any line ar function l : K ∩ L → R + , h ( x ) = f ( x ) l ( x ) n − 1 is unimo dal. L et R n = S 1 ∪ S 2 ∪ S 3 b e a p artition of R n into thr e e non-empty subsets. Then π f ( S 3 ) ≥ d ( S 1 , S 2 ) D min { π f ( S 1 ) , π f ( S 2 ) } . Pr o of. Supp ose n ot. Define g : R n → R and h : R n → R as follo ws. g ( x ) =        d ( S 1 , S 2 ) D f ( x ) if x ∈ S 1 0 if x ∈ S 2 − f ( x ) if x ∈ S 3 and h ( x ) =        0 if x ∈ S 1 d ( S 1 , S 2 ) D f ( x ) if x ∈ S 2 − f ( x ) if x ∈ S 3 . Th u s Z R n g ( x ) dx > 0 an d Z R n h ( x ) dx > 0 , Lemma 9 implies that there exist tw o p oin ts a, b ∈ R n and a linear function l : [0 , 1] → R + suc h that Z 1 0 g ((1 − t ) a + tb ) l ( t ) n − 1 dt > 0 and Z 1 0 h ((1 − t ) a + t b ) l ( t ) n − 1 dt > 0 . (2) Moreo v er, w.l.o.g. w e can assum e that the p oin ts a and b are within the supp ort of f , and h en ce k a − b k ≤ D . W e ma y also assume that a ∈ S 1 and b ∈ S 2 . Consider a p artition of the inte r v al [0 , 1] = Z 1 ∪ Z 2 ∪ Z 3 , w h ere Z i = { z ∈ [0 , 1] : (1 − z ) a + z b ∈ S i } . F or z 1 ∈ Z 1 and z 2 ∈ Z 2 , we h a v e d ( S 1 , S 2 ) ≤ d (( 1 − z 1 ) a + z 1 b, (1 − z 2 ) a + z 2 b ) ≤ | z 1 − z 2 | · k a − b k ≤ | z 1 − z 2 | D , and therefore d ( S 1 , S 2 ) ≤ d ( Z 1 , Z 2 ) D . No w w e can rewr ite Equation (2 ) as Z Z 3 f ((1 − t ) a + tb ) l ( t ) n − 1 dt < d ( S 1 , S 2 ) D Z Z 1 f ((1 − t ) a + tb ) l ( t ) n − 1 dt ≤ d ( Z 1 , Z 2 ) Z Z 1 f ((1 − t ) a + tb ) l ( t ) n − 1 dt and similarly Z Z 3 f ((1 − t ) a + tb ) l ( t ) n − 1 dt ≤ d ( Z 1 , Z 2 ) Z Z 2 f ((1 − t ) a + tb ) l ( t ) n − 1 dt Define p : [0 , 1] → R + as p ( t ) = f ((1 − t ) a + tb ) l ( t ) n − 1 . F r om the un imo dalit y assump tion in ou r theorem, w e kno w that p is unimo dal. Rewriting the ab o v e equ ations, we hav e Z Z 3 p ( t ) dt < d ( Z 1 , Z 2 ) Z Z 1 p ( t ) dt a nd Z Z 3 p ( t ) dt < d ( Z 1 , Z 2 ) Z Z 2 p ( t ) dt. (3) No w supp ose Z 3 is a u n ion of disjoint int erv als, i.e., Z 3 = S i ( α i , β i ), 0 ≤ α 1 < β 1 < α 2 < β 2 < · · · ≤ 1. By Lemma 10 we h a v e Z β i α i p ( t ) dt ≥ | α i − β i | · min  Z α i 0 p ( t ) dt, Z 1 β i p ( t ) dt  . 7 Therefore, adding these up we get Z Z 3 p ( t ) dt = X i Z β i α i p ( t ) dt ≥ | α i − β i | · X i min  Z α i 0 p ( t ) dt, Z 1 β i p ( t ) dt  ≥ d ( Z 1 , Z 2 ) · min  Z Z 1 p ( t ) dt, Z Z 2 p ( t ) dt  . since there must b e some i su c h that Z 1 and Z 2 are separated by the in terv al ( α i , β i ). But then w e get a con tradiction to Equation (3). This completes the pro of of Theorem 11. 3.1 Isop erimetry of (1/(n-1))-Harmonic-conca ve functions W e show that (1 / ( n − 1))-harmonic-conca ve functions satisfy the unimo dality criterion used in the pro of of Th eorem 11. Therefore, as a corollary , we get an isop erimetric in equalit y f or (1 / ( n − 1))- harmonic-conca v e fun ctions, which is a sub class of harmonic-conca v e functions. Prop osition 12. L et f : R n → R + b e a smo oth (1 / ( n − 1)) -harmonic-c onc ave function and l : [0 , 1] → R + b e a line ar function. Now let a, b ∈ R n and define h : [0 , 1 ] → R + as h ( t ) = f ((1 − t ) a + tb ) l ( t ) n − 1 . Then h is a unimo dal fu nction. Pr o of of Pr op osition 12 (U nimo dality of 1 / ( n − 1) -harmonic-c onc ave functions). Define g ( x ) = ( 1 /f ( x )) 1 / ( n − 1) . Since f is (1 / ( n − 1))-harmonic-co n ca v e, g is con ve x and we can rewrite h as h ( t ) = f ((1 − t ) a + tb ) l ( t ) n − 1 =  l ( t ) g ((1 − t ) a + tb )  n − 1 =  l ( t ) q ( t )  n − 1 , where q ( t ) = g ((1 − t ) a + t b ) wh ic h is also conv ex. Also w .l.o.g. w e can do a linear transformation that transforms l into the id entit y function l ( t ) = t withou t affecting the conv exity of q . Th us, it suffices to sh ow that if h ( t ) = ( t/q ( t )) 1 / ( n − 1) , wh ere q is a conv ex function, then h is un imo dal. Indeed, d dt h ( t ) n − 1 = q ( t ) − t d dt q ( t ) q ( t ) 2 d 2 dt 2 h ( t ) n − 1 = q ( t ) 2 t  d dt q ( t )  2 − 2 q ( t ) d dt q ( t ) − tq ( t ) d 2 dt 2 q ( t ) ! q ( t ) 4 . If there exists a lo cal op timum for h n − 1 at t = t 0 then d dt h ( t 0 ) n − 1 = 0 ⇒ q ( t 0 ) = t 0 d dt q ( t 0 ) ⇒ d 2 dt 2 h ( t 0 ) n − 1 = − t 0 q ( t 0 ) 2 d 2 dt 2 q ( t 0 ) q ( t 0 ) 4 ≤ 0 , b ecause t 0 ∈ (0 , 1) , and d 2 dt 2 q ( t ) ≥ 0 as q is con vex. This imp lies that ev ery lo cal optim um is a lo cal maxim um and hence there are n o conv ex regions in h ( t ) n − 1 . Hence, there are no conv ex regions in h ( t ) whic h imp lies that h ( t ) is u nimo dal. W e get T h eorem 4 as a corollary of Theorem 11 and Prop osition 12. 8 3.2 Lo w er b ound for isoper imetry In this section, w e sho w that (1 / ( n − 1))-harmonic-conca v e functions are the limit of isop erimetry b y sh owing a (1 / ( n − 1 − ǫ ))-harmonic conca ve fun ction with p o or isop erimetry for 0 < ǫ ≤ 1. Pr o of of The or em 5 . Th e pro of is based on the follo wing construction. Consider K ⊆ R n defined as follo ws. K =  x : 0 ≤ x 1 < 1 1 + δ and x 2 2 + x 2 3 + . . . + x 2 n ≤ (1 − x 1 ) 2  , where δ > 0. K is a parallel section of a cone symmetric around the X 1 -axis and is therefore con v ex. No w w e define a function f : R n → R + whose supp ort is K . f ( x ) =    C (1 − (1 + δ ) x 1 ) n − 1 − ǫ if x ∈ K , 0 if x / ∈ K , where C is the ap p ropriate constant so as to mak e π f ( K ) = 1. By d efinition, f is a 1 / ( n − 1 − ǫ )- harmonic conca v e fun ction. Define a p artition R n = S ∪ T as S = { x ∈ K : 0 ≤ x 1 ≤ t } and T = R n \ S . The theorem holds for a suitable choic e of t . [Pro of of Theorem 5 (Limit of isop erimetry)] W e wan t to show that π f ( ∂ S ) min { π f ( S ) , π f ( T ) } = O ( c − n ) , for some constan t c > 1, which means that f do es not satisfy the isop erimetric inequalit y . (By abuse of n otation, w e use π f ( ∂ S ) for the area measur e defined by f on the b ou n dary of S .) In order to ac hiev e this, it seems b etter for us to c h o ose a v alue of t that minimizes π f ( ∂ S ). π f ( ∂ S ) = V n − 1 C (1 − t ) n − 1 (1 − (1 + δ ) t ) 1+ ǫ − n , where V n denotes the v olume of a unit ball in R n . Simple calculus shows that π f ( ∂ S ) is conv ex as a function of t o ve r [0 , 1 / (1 + δ )] and attai n s the m in im um when t = t min = 1 1 + δ − n − 1 − ǫ ǫ · δ 1 + δ . Moreo v er, π f ( ∂ S ) is decreasing for 0 ≤ t ≤ t min and increasing for t min ≤ t ≤ 1 / (1 + δ ). T h u s , using t = t min to define the partition R n = S ∪ T , we get π f ( ∂ S ) = V n − 1 C (1 − t min ) n − 1 (1 − (1 + δ ) t min ) 1+ ǫ − n = V n − 1 C  δ 1 + δ  n − 1  n − 1 − ǫ ǫ  n − 1  δ ( n − 1 − ǫ ) ǫ  1+ ǫ − n . 9 and π f ( S ) = Z S f ( x ) dx = V n − 1 C Z t min 0 (1 − x 1 ) n − 1 (1 − (1 + δ ) x 1 ) 1+ ǫ − n dx 1 ≥ V n − 1 C Z t ′ 0 (1 − x 1 ) n − 1 (1 − (1 + δ ) x 1 ) 1+ ǫ − n dx 1 where 0 < t ′ = 1 2(1 + δ ) < t min ≥ V n − 1 C (1 − t ′ ) n − 1  1 − (1 + δ ) t ′  1+ ǫ − n t ′ = V n − 1 C  1 2 + δ 2(1 + δ )  n − 1  1 2  1+ ǫ − n 1 2(1 + δ ) ≥ V n − 1 C 1 2 ǫ · 1 2(1 + δ ) . and π f ( T ) = Z T f ( x ) dx = V n − 1 C Z 1 / (1+ δ ) t min (1 − x 1 ) n − 1 (1 − (1 + δ ) x 1 ) 1+ ǫ − n dx 1 ≥ V n − 1 C Z 1 / (1+ δ ) t ′′ (1 − x 1 ) n − 1 (1 − (1 + δ ) x 1 ) 1+ ǫ − n dx 1 where t min < t ′′ = 1 1 + δ − 1 n 2 · n − 1 − ǫ ǫ · δ 1 + δ < 1 1 + δ ≥ V n − 1 C (1 − t ′′ ) n − 1  1 − (1 + δ ) t ′′  1+ ǫ − n  1 1 + δ − t ′′  = V n − 1 C  δ 1 + δ  n − 1  1 + n − 1 − ǫ n 2 ǫ  n − 1  δ ( n − 1 − ǫ ) n 2 ǫ  1+ ǫ − n δ ( n − 1 − ǫ ) n 2 ǫ (1 + δ ) ≥ V n − 1 C  δ 1 + δ  n − 1  ( n − 1)( ǫ ( n + 1) + 1) n 2 ǫ  n − 1  δ ( n − 1 − ǫ ) n 2 ǫ  1+ ǫ − n δ ( n − 1 − ǫ ) n 2 ǫ (1 + δ ) Therefore, π f ( ∂ S ) π f ( S ) ≤  δ 1 + δ  n − 1  n − 1 − ǫ ǫ  n − 1  δ ( n − 1 − ǫ ) ǫ  1+ ǫ − n 2 1+ ǫ (1 + δ ) ≤ δ ǫ  1 1 + δ  n − 1 ( n − 1 − ǫ ) ǫ  1 ǫ  ǫ 2 1+ ǫ (1 + δ ) ≤ C n  (1 + ǫ ) − ǫn  (Using δ = 1 / (1 + ǫ ) n ) for some constan t C > 0 and π f ( ∂ S ) π f ( T ) ≤ ǫn 2(1+ ǫ ) (1 + δ ) n − 1 − δ !  1 δ ( ǫ ( n + 1) + 1) n − 1  ≤ C n 6 1 (1 + ǫ 1+ ǫ n ) n ! (Using δ = 1 / (1 + ǫ ) n ) ≤ C 2 − n 10 for some constan t C > 0. Putting these together we ha v e π f ( ∂ S ) min { π f ( S ) , π f ( T ) } ≤ C n ((1 + ǫ ) − ǫn ) , for some constan t C > 0. 4 Sampling s-Harmonic-conca v e functions Throughout this section, let f : R n → R + b e an s -harmon ic-conca ve f unction giv en by an oracle with p arameters ( α, δ ) such that s ≤ 1 / ( n − 1). Let K b e the con v ex set ov er wh ic h w e wan t to sample p oints according to f . W e a lso assume that K con tains a ball of radius δ . W e state a tec hnical lemma r elated to the parameters and the harm onic-conca vity of the function. Lemma 13. Supp ose f : R n → R is a s -harmonic-c onc ave function with p ar ameters ( α, δ ) as define d e arlier. F or any c onstant c such that 1 < c < α , if k x − z k ≤ csδ α s − 1 , then f ( x ) f ( z ) ≤ c . Pr o of of 13 (Mo difying p ar ameters). Let x, y ∈ R n suc h that k x − y k ≤ δ and let z = (1 − t ) x + ty for some t ∈ (0 , 1). Then , by the s -harmon ic-conca vit y , we hav e that f ( z ) ≥  1 − t f ( x ) s + t f ( y ) s  − 1 s ≥  1 − t f ( x ) s + tα s f ( x ) s  − 1 s (Since k x − y k ≤ δ implies f ( x ) /f ( y ) < α ) = f ( x ) (1 + t ( α s − 1)) 1 s Since, k x − z k ≤ csδ α s − 1 , we get the d esired conclusion. Hence, for the s -harmonic-conca ve functions with parameters ( α, δ ), the ab o ve lemma states that they also hav e p arameters ( c, csδ ( α s − 1) ) for any constant c s uc h that 1 < c < α . In particular, if α > 2, this su ggests that w e ma y use (2 , 2 sδ ( α s − 1) ) as the parameters and if α ≤ 2, then we ma y use (2 , δ ) as the p arameters. Th us, the parameters are (2 , min { δ, 2 sδ ( α s − 1) } ). In order to sample, w e need to sho w that K conta in s p oints of goo d local condu ctance. F or this, defin e K r =  x ∈ K : v ol ( r B x ∩ K ) v ol ( r B x ) ≥ 3 4  . The id ea is that, for app ropriately c h osen r , the log-lipsc hitz-lik e co n strain t will enf orce that the p oints in K r ha ve goo d lo cal condu ctance. F urther, w e ha ve that the measure in K r is close to th e measure of f in K based on the radius r . Lemma 14. F or any r > 0 , the set K r is c onvex and π f ( K r ) ≥ 1 − 4 r √ n δ . Pr o of of 14 (Points of go o d lo c al c onductanc e). No w, the con v exity of K r is a direct consequence of the Bru nn-Minko wski inequalit y: for compact sets A, B and th eir Minko wski sum A + B , v ol ( A + B ) 1 /n ≥ v ol ( A ) 1 /n + vol ( B ) 1 /n . T o prov e the second part, we n eed the follo wing lemma paraphrased from [KLS97]. 11 Lemma 15. L et K b e a c onvex set c ontaining a b al l of r adius t . Then R x ∈ K R y ∈ ( x + r B ) \ K dy dx ≤ r √ n 2 t v ol ( K ) v ol ( r B ) . The target densit y f (with su pp ort K ) can b e view ed as follo ws: fir s t we pic k a lev el set L ( t ) of f , with the app ropriate marginal d istribution on t , then we p ic k x u n iformly fr om L ( t ). No w co n sider an y lev el set L ( t ) where t is p ic k ed according to the app ropriate marginal dis- tribution. If L ( t ) do es not con tain a ball of radius δ / 2, th en the probabilit y of s tepping out of K from an y p oin t in L ( t ) is 0. If L ( t ) contai n s such a b all, the pr obabilit y of steppin g ou t is b oun ded ab o ve u sing Lemma 15 by r √ n/δ . That is, v ol ( r B x \ K ) v ol ( r B ) ≤ r √ n δ where x is a rand om p oin t in L ( t ). Consider a random v ariable g ( x ) = v ol( r B x \ K ) v ol( r B ) when x is a random p oint in L ( t ). S in ce by th e ab o ve inequ alit y E ( g ( x )) ≤ r √ n/δ , we h a v e that Pr  g ( x ) > 4 r √ n δ  ≤ 1 4 This implies that at most 1 / 4-th of th e fraction of p oint s in L ( t ) step out of K with prob ab ility greater than 4 r √ n/δ . Hence, at least 3 / 4-th of the fr action of p oint s in L ( t ) step out of K w ith probabilit y at most 4 r √ n/δ . That is, 3 / 4-th of the fractio n of p oin ts in L ( t ) remain within K with probabilit y at least 1 − (4 r √ n/δ ). Hence π f ( K r ) ≥ 1 − (4 r √ n/δ ). 4.1 Coupling In order to pro ve condu ctance, we need to pr o v e that wh en tw o p oin ts are geometric ally close, th en their one-step distributions o v erlap. W e will need the follo wing te chnical lemma ab out spherical caps to prov e this. Lemma 16. L et H b e a halfsp ac e in R n and B x b e a b al l whose c enter is at a dist anc e at most tr / √ n fr om H . Then e − t 2 4 > 2 v ol ( H ∩ r B ) v ol ( r B ) > 1 − t Lemma 17. F or r ≤ min { δ, 2 sδ ( α s − 1) } , if u, v ∈ K r , k u − v k < r / 16 √ n , then d tv ( P u , P v ) ≤ 1 − 7 16 Pr o of. W e ma y assume that f ( v ) ≥ f ( u ). Then, d tv ( P u , P v ) ≤ 1 − 1 v ol ( r B ) Z r B v ∩ r B u ∩ K min  1 , f ( y ) f ( v )  dy 12 Let us lo wer b ound th e second term in the righ t hand sid e. Z r B v ∩ r B u ∩ K min  1 , f ( y ) f ( v )  dy ≥ Z r B v ∩ r B u ∩ K min  1 , f ( y ) f ( v )  dy ≥  1 2  v ol ( r B v ∩ r B u ∩ K ) (Consequence of Lemma 13) ≥  1 2  (v ol ( r B v ) − vol ( r B v \ r B u ) − v ol ( r B v \ K )) ≥  1 2   v ol ( r B v ) − 1 16 v ol ( r B ) − 1 16 v ol ( r B )  ≥  7 16  v ol ( r B ) where the b oun d on v ol ( r B v \ r B u ) is deriv ed fr om Lemma 16 and vo l ( r B v \ K ) is b ounded us ing the fact that v ∈ K r . Hence, d tv ( P u , P v ) ≤ 1 − 7 16 4.2 Conductance an d mixing time Consider th e ball walk with metrop olis filter using the s -h armonic-conca ve distribution function oracle (whose parameters are ( α, δ )) with ball steps of radius r . Lemma 18. F or any ǫ 1 > 0 , let D b e the diameter of the b al l such that π f ( D B 0 ) ≥ 1 − ǫ 1 2 . L et S ⊆ R n b e such that π f ( S ) ≥ ǫ 1 and π f ( R n \ S ) ≥ ǫ 1 . Then, for b al l walk r adius r ≤ m in { ǫ 1 δ 8 √ n , 2 sδ ( α s − 1) } , we have that Φ( S ) ≥ r 2 9 √ nD min { π f ( S ) − ǫ 1 , π f ( R n \ S ) − ǫ 1 } Pr o of of L emma 18. Giv en K as the con v ex set o ver which we wan t to sample p oints according to the (1 / ( n − 1))-harmonic-conca ve distribution, defin e K r as ab o ve . F ur ther, for all subs ets A ⊆ K , w e h a v e th at π f ( A ∩ K ′ ) = π f ( A ∩ ∩ K r ) ≥ π f ( A ) − ǫ 1 (b y Lemma 14 usin g r ≤ ǫ 1 δ 8 √ n ) (4) Using S , defin e S 1 , S 2 , S 3 as follo ws. S 1 =  x ∈ S : P x ( R n \ S ) ≤ 7 32  S 2 =  x ∈ R n \ S : P x ( S ) ≤ 7 32  S 3 = R n \ ( S 1 ∪ S 2 ) . Also define S ′ i as S ′ i = S i ∩ K r , for i = 1 , 2 , 3. 13 The ergo dic flow of S can b e written as Φ( S ) = 1 2 Z S P x ( R n \ S ) f ( x ) dx + Z R n \ S P x ( S ) f ( x ) dx ! ≥ 7 64 π f ( S 3 ) ≥ 1 16 π f ( S ′ 3 ) Supp ose π f ( S ′ 1 ) ≤ π f ( S ∩ K r ) / 2, then π f ( S ′ 3 ) ≥ π f ( S ∩ K r ) / 2 b ecause S ∩ K r ⊆ S ′ 1 ∪ S ′ 3 . Th u s Φ( S ) ≥ 1 32 π f ( S ∩ K r ) ≥ 1 32 ( π f ( S ) − ǫ 1 ) (By (4)) whic h imp lies the lemma. So, we may assu m e that π f ( S ′ 1 ) ≥ π f ( S ∩ K r ) / 2, and similarly π f ( S ′ 2 ) ≥ π f (( R n \ S ) ∩ K r ) / 2. Then, using Th eorem 4, Φ( S ) ≥ 1 16 π f ( S ′ 3 ) ≥ 1 16 d ( S ′ 1 , S ′ 2 ) D min  π f ( S ′ 1 ) , π f ( S ′ 2 )  ≥ 1 32 d ( S 1 , S 2 ) D · min { π f ( S ∩ K r ) , π f (( R n \ S ) ∩ K r ) } No w, for an y u ∈ S 1 , v ∈ S 2 , d tv ( P u , P v ) ≥ 1 − 2 · max { P u ( R n \ S ) , P v ( S ) } ≥ 1 − 7 16 . Also, r ≤ min { ǫ 1 δ 8 √ n , 2 sδ ( α s − 1) } < min { δ, 2 sδ ( α s − 1) } . Hence, by Lemma 17, d ( S 1 , S 2 ) ≥ r 16 √ n . Therefore, Φ( S ) ≥ 1 32 · r 16 √ n · 1 D · min { π f ( S ) − ǫ 1 , π f ( R n \ S ) − ǫ 1 } By (4) ≥ r 2 9 √ n D min { π f ( S ) − ǫ 1 , π f ( R n \ S ) − ǫ 1 } Using the ab o ve lemma, we prov e T heorem 6. Pr o of of The or em 6 . On sett in g ǫ 1 = ǫ/ 2 H in Lemma 18, we ha v e that for ball-w alk r adius r = min { ǫδ 16 H √ n , 2 sδ ( α s − 1) } , φ ǫ 1 ≥ r 2 9 √ nD . By definition H s ≤ H · s and hence by Lemma 8, | σ m ( S ) − π f ( S ) | ≤ H · s + H · exp  − mr 2 2 19 nD 2  whic h give s us that b ey ond m ≥ 2 19 nD 2 r 2 log 2 H ǫ steps, | σ m ( S ) − π f ( S ) | ≤ ǫ . Su bstituting for r , we get the theorem. 14 4.3 Sampling the Cauc hy densit y In this section, we pr o v e certai n prop erties of the Cauc hy densit y along with th e crucial coupling lemma leading to Theorem 7. Without loss of generalit y , we ma y assu me that the distribution giv en by th e oracle is, f ( x ) ∝ ( 1 / (1 + || x || 2 ) n +1 2 if x ∈ K , 0 otherwise. (5) This is b ecause, either we are explicitly giv en the matrix A of a general Cauch y d ensit y , or we can compute it usin g the fun ction f at a small n u mb er of p oin ts and apply a linear transformation. F urther, n ote th at b y th e hyp othesis of Theorem 7, we may assume that K co ntains a unit ball. Prop osition 19. The Cauchy density function is (1 / ( n − 1)) -harmonic-c onc ave. Pr o of of Pr op osition 19 (Cauchy is 1/(n-1)-harmonic-c onc ave). T o c h ec k for (1 / ( n − 1))-harmonic- conca vit y , w e need to c h eck that g ( x ) = (1 + X i x 2 i ) n +1 2( n − 1) is conv ex. This follo ws from the follo wing t w o observ ations: 1. n +1 2( n − 1) > 1 2 2. (1 + P i x 2 i ) 1 / 2 is conv ex. Prop osition 20 sa ys that we can fin d a ball of radius O ( √ n/ǫ 1 ) outsid e wh ic h the Cauc hy densit y has at most ǫ 1 probabilit y mass. Prop osition 20. Pr k x k ≥ 2 √ 2 n ǫ 1 ! ≤ ǫ 1 . Pr o of of Pr op osition 20 (Bal l of lar ge mass). Pr ( k x k ≥ t ) = R k x k≥ t 1 (1 + k x k 2 ) ( n +1) / 2 dx R R n 1 (1 + k x k 2 ) ( n +1) / 2 dx = R S n − 1 dω · R ∞ t r n − 1 (1 + r 2 ) ( n +1) / 2 dr R S n − 1 dω · R ∞ 0 r n − 1 (1 + r 2 ) ( n +1) / 2 dr using p olar co ordinates = R ∞ t r n − 1 (1 + r 2 ) ( n +1) / 2 dr R ∞ 0 r n − 1 (1 + r 2 ) ( n +1) / 2 dr . 15 No w we will analyze the numerator, call it N , and the d enominator, call it D , s ep arately . N = Z ∞ t r n − 1 (1 + r 2 ) ( n +1) / 2 dr ≤ Z ∞ t 1 r 2 dr = 1 t , (6) and D = Z ∞ 0 r n − 1 (1 + r 2 ) ( n +1) / 2 dr = Z π / 2 0 sin n − 1 θ cos n − 1 θ 1 sec n +1 θ sec 2 θ dθ using r = tan θ = Z π / 2 0 sin n − 1 θ dθ =  − sin n − 2 θ cos θ n − 1  π / 2 0 + n − 2 n − 1 Z π / 2 0 sin n − 3 θ dθ = n − 2 n − 1 Z π / 2 0 sin n − 3 θ dθ =        ( n − 2)!! ( n − 1)!! , if n is ev en . π 4 · ( n − 2)!! ( n − 1)!! , if n is o dd . b y cont inuing the recurs ion ≥ π 4 · ( n − 2)! ( n − 1)! ≥ π 4 · √ 2 π √ n b y W allis’ inequalit y = 1 2 √ 2 n . (7) Therefore, from Equations (6) and (7 ) we get Pr ( k x k ≥ t ) ≤ N D ≤ 2 √ 2 n t . This implies Pr k x k ≥ 2 √ 2 n ǫ 1 ! ≤ ǫ 1 . Prop osition 21 shows the s m o othness prop ert y of the Cauch y density . This is th e crucial ingre- dien t u sed in th e stronger coupling lemma. Define K r as b efore. Then, Prop osition 21. F or x ∈ K r , let C x = { y ∈ r B x : | x · ( x − y ) | ≤ 4 r || x || √ n } and y ∈ C x . Th en, f ( x ) f ( y ) ≥ 1 − 4 r √ n 16 Pr o of of Pr op osition 21 (Smo othness pr op erty). W e h a v e that, || y || 2 = || x || 2 + || y − x || 2 + 2 x · ( x − y ) ≥ || x || 2 − 8 r || x || √ n Therefore f ( x ) f ( y ) =  1 + || y || 2 1 + || x || 2  n +1 2 ≥  1 − 8 r || x || √ n (1 + || x || 2 )  n +1 2 ≥  1 − 4 r √ n  n +1 2  max t 1 + t 2 = 1 2  ≥ 1 − 4 r √ n Finally , we h a v e the follo wing coupling lemma. Lemma 22. F or r ≤ 1 / √ n , if u, v ∈ K r , k u − v k < r / 16 √ n , then d tv ( P u , P v ) < 1 2 . Pr o of of L emma 22 (Coupling lemma for Cauchy). W e may assume that f ( v ) ≥ f ( u ). Then, d tv ( P u , P v ) ≤ 1 − 1 v ol ( r B ) Z r B u ∩ r B v ∩ K min  1 , f ( y ) f ( v )  dy Let us lo wer b ound low er b ound the second term in the righ t h and side. Z r B u ∩ r B v ∩ K min  1 , f ( y ) f ( v )  dy ≥ Z r B v ∩ r B u ∩ K ∩ C v min  1 , f ( y ) f ( v )  dy ≥ (1 − 4 r √ n ) v ol ( r B v ∩ r B u ∩ C v ∩ K ) (b y Lemma 21) ≥ (1 − 4 r √ n ) (v ol ( r B v ) − vol ( r B v \ r B u ) − vol ( r B v \ C v ) − vol ( r B v \ K )) ≥ (1 − 4 r √ n )  v ol ( r B v ) − 1 16 v ol ( r B ) − 1 16 v ol ( r B ) − e − 4 v ol ( r B )  ≥ (1 − 4 r √ n )  13 16  v ol ( r B ) where th e b ounds on v ol ( r B v \ r B u ) an d v ol ( r B v \ C v ) are derived from Lemm a 16 and vol ( r B v \ K ) is b ound ed usin g the fact that v ∈ K r . Since r ≤ 1 / √ n , d tv ( P u , P v ) ≤ 3 + 4 r √ n 16 ≤ 1 2 The pro of of conductance and mixing b ound follo w the pro of of mixing b ound for s -harmonic- conca v e functions closely . Comparing the ab o ve coupling lemma with th at of s -harm on ic-conca ve functions (Lemma 17), we observ e that the impro vemen t is obtained due to the constraint on the radius of the ball w alk in the coupling lemma. In the case of Cauc hy , a sligh tly relaxed radius suffices for p oints close to eac h other to h a v e a considerable o ve rlap in their one-step d istribution. 17 4.4 Discussion There are t wo asp ects of o u r algorithm and analysis that merit improv ement. Th e first is the dep end ence on the d iameter, wh ic h could p erhaps b e m ad e logarithmic by applying an appropriate affine transformation as in the case of logconca v e d ensities. The second is eliminating the dep en- dence on th e sm o othness parameter en tirely , b y allo wing for sharp change s lo cally and considering a smo other version of the original function. Both these asp ects seem to b e tied closely to pr o ving a tail b oun d on a 1-dimensional marginal of an s -harmonic conca v e fun ction. References [AK91] D. Applegate and R. 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Nolan, Stable distributions - mo dels for he avy taile d data , Birkh ¨ auser, Boston, 2009, In p rogress, Chapter 1 online at academic2.american.edu/ ∼ jpnolan. 5 App endix: pro ofs 5.1 Sampling the Cauc hy densit y The follo wing lemma giv es the p arameters for the Cauc hy d istribution fun ction. Prop osition 23. If k y − x k ≤ 1 /n , then f ( y ) f ( x ) ≥ 1 e , Pr o of. Let ∇ u denote the directional deriv ativ e of a fu nction along u . Then log f ( x ) f ( y ) = log f ( x ) − log f ( y ) ≤ sup z ∈ R n sup k u k =1 k∇ u log f ( z ) k · k x − y k = sup z ∈ R n sup k u k =1 1 f ( z ) k∇ u f ( z ) k · k x − y k = k x − y k sup z ∈ R n 1 f ( z ) sup k u k =1 k∇ u f ( z ) k . (8) 19 W e k n o w that by defin ition f ( z ) = c  1 + k z k 2  ( n +1) / 2 , for some constan t c that includes det( A ) − 1 and the normalizing f actor for Cauc h y distribu tion. Th u s sup k u k =1 k∇ u f ( z ) k = sup k u k =1      n X i =1  ∂ ∂ z i f ( z )  u i      = sup k u k =1      n X i =1 c · − ( n + 1) 2 ·  1 + k z k 2  − ( n +3) / 2 · 2 z i u i      = c ( n + 1) k z k  1 + k z k 2  ( n +3) / 2 , using u = − z k z k . Plugging this in Equation (8) w e get log f ( x ) f ( y ) ≤ k x − y k sup z ∈ R n ( n + 1) k z k 1 + k z k 2 = k x − y k ( n + 1) sup z ∈ R n k z k 1 + k z k 2 = k x − y k ( n + 1) sup r ∈ R r 1 + r 2 = k x − y k ( n + 1) 2 ≤ 1 using k x − y k ≤ 1 /n. Therefore f ( y ) f ( x ) ≥ 1 e . 20