Estimation of Riemannian Quantities from Noisy Data via Density Derivatives

Estimation of Riemannian Quan tities from Noisy Data via Densit y Deriv ativ es Junhao Chen ∗ Ruo wei Li † Zhigang Y ao ‡ Abstract W e study the reco very of geometric structure from data generated b y con volving the uniform measure on a smo oth compact submanifold M ⊂ R D with am bient Gaussian noise. Our main result is that several fundamental Riemannian quantities of M , including tangent spaces, the in trinsic dimension, and the second fundamental form, are identiﬁable from deriv ativ es of the noisy densit y . W e ﬁrst derive uniform small-noise expansions of the data densit y and its deriv a- tiv es in a tubular neigh b orho o d of M . These expansions show that, at the p opulation lev el, tangen t spaces can b e recov ered from the densit y Hessian with O ( σ 2 ) error, while the intrinsic dimension can b e estimated consisten tly . W e further construct estimators for the second funda- men tal form from densit y deriv ativ es, obtaining O ( d ( y , M ) + σ ) and O ( d ( y , M ) + σ 2 ) errors for h yp ersurfaces and submanifolds with arbitrary co dimension. A t the sample level, w e estimate the density and its deriv ativ es by kernel methods in the am bient space and plug them into the p opulation constructions, yielding uniform nonparametric rates in the ambien t dimension. Fi- nally , w e sho w that these densit y-based constructions admit a geometric interpretation through densit y-induced ambien t metrics, linking the geometry of M to ambien t geo desic structure. Keyw ords: geometric inference, submanifold geometry , tangent space estimation, second funda- men tal form estimation, densit y-induced metric 1 In tro duction A common mo deling paradigm for high-dimensional data in R D p osits an unknown d -dimensional smo oth submanifold M ⊂ R D with d ≪ D . In m uc h of the literature, ho w ever, the geometry of M is imposed primarily as an a priori condition, rather than treated as a collection of geometric ob jects to b e inferred from data. This viewp oint leads to a question of statistical identiﬁabilit y: what geometric structure of M is reco v erable from noisy observ ations, under what noise conditions, and at what accuracy? One line of researc h, often referred to as manifold estimation or manifold ﬁtting, studies recov ery of the underlying set M from discrete observ ations. The aim is to construct a d -dimensional sub- manifold c M that is close to M , while satisfying essential regularity conditions, such as smo othness, b ounded curv ature, and a p ositive lo wer bound on the reach; see, e.g., Genov ese et al. (2012, 2014); F eﬀerman et al. (2018); Y ao and Xia (2025); Sober and Levin (2020); F eﬀerman et al. (2025); Y ao et al. (2023); F eﬀerman et al. (2023). Bey ond set estimation, one may also seek to infer regularity quan tities such as the reac h (Aamari et al., 2019). ∗ jhochenn@gmail.com , Independent Researc her † ruoweili@nus.edu.sg , Department of Statistics and Data Science, National Universit y of Singap ore ‡ zhigang.yao@nus.edu.sg ; zhigang.yao@cmsa.fas.harvard.edu . Departmen t of Statistics and Data Science, National Universit y of Singap ore; Center of Mathematical Sciences and Applications, Harv ard Universit y 1 In this pap er, w e study a complementary problem of recov ering the intrinsic Riemannian struc- ture of M . F or an em b edded submanifold M ⊂ R D , t wo fundamental extrinsic primitives are the tangen t spaces { T x M } x ∈ M and the second fundamen tal form { Π x } x ∈ M . The tangen t space de- termines the induced metric, while Π x describ es the normal comp onent of the ambien t deriv ativ e; together, via the Gauss formula and the Gauss equation, they determine the induced Levi–Civita connection and curv ature. Accordingly , these t wo ob jects pro vide natural statistical targets for geometric inference. In this pap er, we study their p oin twise reco very: giv en an observ ation y near M , with pro jection π ( y ), our goal is to construct a linear subspace and a symmetric bilinear map that appro ximate, resp ectiv ely , the tangen t space T π ( y ) M and the second fundamental form Π π ( y ) . A represen tative approach to tangent space estimation is local principal comp onent analysis (LPCA). In the noiseless setting, Singer and W u (2012) analyzes the LPCA co v ariance matrix and derives asymptotic accuracy guarantees for tangent-space estimation. In contrast, Ty agi et al. (2012) provides suﬃcient conditions on the neigh b orho o d width and local sample size to guarantee a small principal angle betw een the PCA subspace and the true tangen t space. Under additiv e noise, Kaslovsky and Meyer (2014) establishes nonasymptotic p erturbation b ounds for LPCA in an am bien t Gaussian noise model, highligh ting a trade-oﬀ betw een noise and curv ature. Both T yagi et al. (2012) and Kaslo vsky and Mey er (2014) adopt a co ordinate-based lo cal mo del in whic h tangent co ordinates are sampled uniformly and then mapp ed to the manifold via a lo cal graph parametrization. Beyond linear ﬁtting, Cheng and Chiu (2016) shows that incorp orating a quadratic term in the local approximation can impro v e the angular error from ﬁrst-order to second- order, while Aamari and Levrard (2019) studies joint estimation of the tangent space and higher- order lo cal p olynomial terms. In parallel with LPCA, sev eral recen t approac hes exploit Laplacian structure to infer tangen t directions. Kohli et al. (2025) estimate tangen t spaces b y orthogonalizing gradien ts of lo w-frequency graph-Laplacian eigenv ectors. Jones (2025) ﬁrst estimates the manifold Laplacian and then uses the carr´ e du champ to recov er the induced metric and tangent spaces. F or estimating the second fundamen tal form, a natural starting p oin t is that under a suit- able lo cal parametrization, the quadratic term encodes the second fundamen tal form. Along this line, Aamari and Levrard (2019) ﬁts lo cal p olynomials to a lo cal graph parametrization, where the quadratic term pro vides an appro ximation of the second fundamen tal form. They further es- tablish minimax low er b ounds for estimating b oth the tangent space and the second fundamental form under a b ounded tubular-noise model. In the noiseless case, the accuracy is go verned b y the T a ylor remainder of the lo cal parametrization; with additional noise, noise deviations enter the local p olynomial ﬁt as a con trolled perturbation, whose eﬀect is ampliﬁed only through the lo cal scale. F rom a dual viewp oin t, since the shap e op erator (W eingarten map) is equiv alent to the second fundamen tal form via the ambien t metric, one ma y estimate curv ature through normal v ariations. Cao et al. (2021) prop oses a tw o-step estimator that ﬁrst estimates tangent and normal spaces b y LPCA and then ﬁts the W eingarten map by least squares in the estimated lo cal frame. Their asymptotic rate analysis is carried out under the assumption of exact tangent spaces and a giv en normal v ector ﬁeld. Bey ond lo cal regression, Buet et al. (2022) adopts a v arifold p ersp ectiv e and introduces a w eak second fundamental form deﬁned via suitable v ariation op erators, together with regularized approximate curv ature tensors that are computable for general v arifolds, includ- ing p oin t cloud v arifolds; con v ergence guarantees are then expressed in terms of the discrepancy b et ween the input v arifold and the target submanifold and the regularization scale. Jones (2025) exploits the diﬀusion-geometric framew ork for second-order geometry , combining tangent and nor- mal information with Hessian-level carr ´ e du champ iden tities to reco v er the second fundamen tal form. In contrast to approaches based on lo cal co ordinates, b ounded tubular noise, or manifold esti- mation, w e study Gaussian-corrupted observ ations generated by conv olving the uniform measure 2 ˜ T y M T π ( y ) M y ˜ Π y ( u , u ) Π π ( y ) ( u , u ) π ( y ) u M Figure 1: Illustration of Riemannian geometry estimation from noisy data. The underlying sub- manifold M (blac k) is observ ed through noisy samples (gra y). F or a p oint y near M , let π ( y ) denote the pro jection of y on to M . F or a tangent vector u , the true tangen t space and second fundamen tal form at π ( y ) are T π ( y ) M and Π π ( y ) ( u, u ), resp ectively , while the estimated tangent space and second fundamental form at y are denoted by e T y M and e Π y ( u, u ). on a smo oth compact submanifold M ⊂ R D with ambien t Gaussian noise. Our analysis shows that fundamen tal Riemannian quan tities of M , including tangent spaces, the in trinsic dimension, and the second fundamental form, are identiﬁable from deriv atives of the noisy density in a tubular neigh b orho o d of M . This is ac hieved by deriving uniform small-noise expansions for the density and its deriv ativ es, which then lead to p opulation- and sample-level estimators of these geometric ob jects. W e also dev elop a geometric interpretation of these constructions through density-induced am bient metrics. Our approac h is built on the s mall-noise structure of the noisy density . In this regime, the log-densit y b eha ves like a scaled squared distance to M , and its ﬁrst three deriv ativ es encode, resp ectiv ely , a normal ﬁeld, a sp ectral splitting betw een tangent and normal directions, and the v ariation of the tangential pro jection. This leads to plug-in geom etric estimators based on deriv a- tiv es of the log-densit y up to order three. At the sample level, we estimate these deriv ativ es by k ernel metho ds in R D and deriv e conv ergence rates for the resulting geometric estimators. The pap er is organized as follo ws. Section 2 introduces the geometric setup and preliminaries. Section 3 develops uniform small-noise expansions for the densit y and its deriv ativ es in a tubular neigh b orho o d. Section 4 uses these expansions to construct p opulation-level estimators of tangent spaces, the in trinsic dimension, and the second fundamen tal form. Section 5 presen ts the density- induced metric in terpretation. Section 6 establishes sample-level guaran tees via kernel estimation of densit y deriv atives. Section 7 rep orts n umerical studies. 1.1 Main Results W e now summarize the main con tributions of the pap er. Throughout, M ⊂ R D is a compact C k ( k ≥ 6), d -dimensional submanifold with positive reach τ > 0, and the clean data X are sampled 3 uniformly from M . W e observe Y = X + ξ , X ∼ P M , ξ ∼ N (0 , σ 2 I D ) , and write P σ = P M ∗ N (0 , σ 2 I D ) for the resulting density . F or y in a ﬁxed tubular neighborho o d T ( τ − ε ), let π ( y ) b e its nearest-point pro jection on to M and let v y = y − π ( y ) b e the normal displacemen t. Our main observ ation is that tangent spaces, the intrinsic dimension, and the second funda- men tal form of M are identiﬁable from the deriv ativ es of the log-density G σ ( y ) = ∇ log P σ ( y ) , H σ ( y ) = ∇ 2 log P σ ( y ) , T σ ( y ) = ∇ 3 log P σ ( y ) . Based on this, we establish b oth p opulation-level iden tiﬁcation results and sample-level statistical rates. W e use c ( · ) for population-level constructions and f ( · ) for their sample-level coun terparts. Figure 1 provides a schematic illustration of the prop osed geometric estimators. A t the p opulation lev el, w e ﬁrst derive uniform small-noise expansions for P σ and the deriv ativ es of log P σ on T ( τ − ε ). These expansions show that the Hessian H σ ( y ) has d eigenv alues corresp onding to tangent directions and D − d eigen v alues corresp onding to normal directions, separated by a gap of order σ − 2 . As a consequence, the tangent space can b e recov ered from the eigenspace asso ciated with the largest d eigenv alues of H σ ( y ), for y ∈ T ( τ − ε ) with error sin { Θ( b T y M , T π ( y ) M ) } = O ( σ 2 ) . The same sp ectral splitting also yields a consisten t estimator of the in trinsic dimension d . Next, w e construct estimators of the second fundamen tal form from densit y deriv atives, with diﬀeren t formulas depending on the geometric setting. • F or totally um bilical submanifolds, the gradien t G σ restricted to M recov ers the mean curv a- ture v ector and thereb y yields an O ( σ ) estimator of the second fundamen tal form. • F or h yp ersurfaces, combining the gradien t G σ ( y ) and Hessian H σ ( y ) yields an estimator sat- isfying ∥ b Π y − Π π ( y ) ∥ op = O ( ∥ v y ∥ + σ ) , o ver subsets U σ ⊂ T ( τ − ε ) on whic h ∥G σ ( y ) ∥ is b ounded b elow. • In arbitrary codimension, diﬀeren tiating the Hessian-based tangential pro jector gives an es- timator on T ( τ − ε ) based on ( H σ , T σ ), with error ∥ b Π y − Π π ( y ) ∥ op = O ( ∥ v y ∥ + σ 2 ) , y ∈ T ( τ − ε ) . Th us, tangen t spaces, the in trinsic dimension, and the second fundamen tal form are all iden tiﬁable from deriv ativ es of the noisy log-density . A t the sample level, given i.i.d. samples y 1 , . . . , y N ∼ P σ , w e estimate P σ and its deriv atives b y kernel metho ds in R D and plug them into the p opulation constructions. This yields uniform sample-lev el estimators for the tangent space and second fundamental form. Moreo ver, the same eigengap argument also yields consistency of the sample-level in trinsic dimension estimator. Under suitable bandwidth choices, the tangent-space estimator satisﬁes sin { Θ( e T y M , T π ( y ) M ) } = O ( σ 2 ) + O P  log N N  2 D +8  , 4 while the second fundamental form estimator satisﬁes ∥ e Π y − Π π ( y ) ∥ op =      O ( ∥ v y ∥ + σ ) + O P  log N N  2 D +8  , for h yp ersurfaces , O ( ∥ v y ∥ + σ 2 ) + O P  log N N  2 D +10  , in arbitrary co dimension . These are am bien t-dimensional nonparametric rates, reﬂecting the fact that our metho d estimates deriv ativ es of the am bient densit y directly , rather than ﬁrst ﬁtting an in trinsic manifold represen- tation. Finally , we sho w that the ab ov e density-based constructions admit a geometric interpretation through ambien t metrics. In particular, suitable densit y-induced metrics on R D mak e the intrinsic geo desics of M asymptotically compatible with am bien t tra jectories as the noise level v anishes. F or totally um bilical submanifolds and h yp ersurfaces, we construct, resp ectively , a Riemannian metric and a degenerate metric: g P σ = P 4 /d σ g E , g P σ = d (log P σ ) ⊗ d (log P σ ) . 2 Notation and Preliminaries In this section, we collect the geometric notation and standard facts that are used throughout the pap er. Pro ofs of the lemmas are pro vided in App endix A. Throughout the pap er w e ﬁx in tegers d, D , and k ≥ 6, and consider a compact C k d -dimensional submanifold M ⊂ R D . The am bien t space R D is equipp ed with the Euclidean metric g E = ⟨· , ·⟩ . The induced Riemannian metric on M is denoted by g := g E | M . W e write V M for the volume of M under the induced v olume measure. F or a Euclidean space E , a p oint z ∈ E , and r > 0, we write B E ( z , r ) for the op en Euclidean ball of radius r centered at z . In particular, when E = R m w e also write B m ( z , r ). 2.1 Geometric Setup and Reac h F or a closed set A ⊂ R D w e recall the notion of reac h. Deﬁnition 2.1. Let A be a closed subset of R D . The reach of A is the largest τ ≥ 0 such that ev ery p oint at distance less than τ from A has a unique nearest p oint in A . Denote the reach of M by τ . W e assume throughout that M has p ositiv e reach τ > 0. F or p oin ts x ∈ M w e write T x M and T ⊥ x M for the tangen t and normal spaces at x , resp ectiv ely . The tangen t and normal bundles are denoted by T M and T ⊥ M . F or τ > 0 we deﬁne the op en τ -tubular neigh b orho o d of M in the normal bundle b y T ( τ ) := { ( x, v ) ∈ T ⊥ M : ∥ v ∥ < τ } . W e iden tify ( x, v ) ∈ T ( τ ) with the p oint x + v ∈ R D . F or 0 < ε < τ w e also introduce the closed tubular neigh b orho o d T ( τ − ε ) := { ( x, v ) ∈ T ⊥ M : ∥ v ∥ ≤ τ − ε } . Since M is compact and τ > 0, the set T ( τ − ε ) is compact. W e use T ( τ ) for the op en tubu- lar neigh b orho o d where the nearest-p oin t pro jection is deﬁned, and T ( τ − ε ) for its closed inner truncation on which all uniform es timates b elo w are stated. 5 The exp onential map at x ∈ M is denoted b y exp x : T x M → M . The injectivity radius at x ∈ M , denoted by inj( x ), is the largest radius r > 0 suc h that exp x : B T x M (0 , r ) → M is a diﬀeomorphism, and the global injectivity radius of M is inj M := inf x ∈ M inj( x ) . The reac h provides a lo wer b ound on the injectivity radius. Lemma 2.2. (Alexander and Bishop, 2006, Cor ol lary 4) L et M b e a C 2 emb e dde d submanifold with r e ach τ > 0 . Then inj M ≥ π τ . 2.2 T ubular Neighborho o ds and Pro jection F or a set A ⊂ R D and a p oint z ∈ R D , w e write d ( z , A ) := min a ∈ A ∥ z − a ∥ for the Euclidean distance from z to A . When the nearest point is unique w e denote the corresp onding pro jection b y π A ( z ) := arg min a ∈ A ∥ z − a ∥ . By default, for eac h y ∈ T ( τ ), we write π ( y ) for the nearest-p oint pro jection onto M and let v y := y − π ( y ) denote the corresp onding normal displacement. The p ositiv e reach assumption ensures that the pro jection π : T ( τ ) → M is well-deﬁned. Since M is C k with k ≥ 6, the tubular neighborho o d theorem implies that π and v y are C k − 1 on T ( τ ). Moreo ver, we write P T ( y ) : R D → T π ( y ) M , P N ( y ) : R D → T ⊥ π ( y ) M for the orthogonal pro jections on to the tangen t and normal spaces at π ( y ), resp ectively . Since P T ( x ) and P N ( x ) are C k − 1 in x ∈ M and π is C k − 1 on T ( τ − ε ), P T ( y ) and P N ( y ) are C k − 1 on T ( τ − ε ). 2.3 Exp onen tial Co ordinates and Curv ature F or r < inj( x ), the exp onen tial map in tro duces an exponential co ordinate chart. In particular, the map ( x, u ) 7→ exp x ( u ), its lo cal in verse ( x, z ) 7→ exp − 1 x ( z ), and the metric coeﬃcients in exponential co ordinates ( x, u ) 7→ g ij ( x, u ) are C k − 2 . W e record the following standard expansions under the standing C k assumption. Lemma 2.3. Ther e exist c onstants r 0 ∈ (0 , inj M ) and C such that, for every x ∈ M and every u ∈ T x M with ∥ u ∥ ≤ r 0 , exp x ( u ) = x + u + 1 2 Π x ( u, u ) + Q e ( x, u ) , ∥ Q e ( x, u ) ∥ ≤ C ∥ u ∥ 3 . Lemma 2.4. Ther e exist c onstants r 0 ∈ (0 , inj M ) and C such that, for every x ∈ M and every u ∈ T x M with ∥ u ∥ ≤ r 0 , q det g ij ( x, u ) = 1 + Q g ( x, u ) , | Q g ( x, u ) | ≤ C ∥ u ∥ 2 . W e denote b y ∇ the am bient deriv ative in R D (i.e., the Levi–Civita connection of g E ), and b y ∇ M the Levi–Civita connection on ( M , g ). The second fundamental form at x ∈ M is the symmetric bilinear map Π x : T x M × T x M → T ⊥ x M , Π x ( X , Y ) = ( ∇ X Y ) ⊥ , where ( · ) ⊥ denotes orthogonal pro jection on to T ⊥ x M . In particular, x 7→ Π x has C k − 2 regularit y . F or a normal v ector n ∈ T ⊥ x M , the associated shape op erator S n : T x M → T x M is giv en by S n ( X ) = − ( ∇ X n ) ⊤ , where ( · ) ⊤ denotes the pro jection on to T x M . The mean curv ature vector at 6 x is the trace H x = 1 d T r(Π x ) ∈ T ⊥ x M . W e denote the op erator norm b y ∥ · ∥ op . F or multilinear maps, ∥ · ∥ op is the usual supremum ov er unit inputs. P ositive reach implies a uniform b ound on the second fundamen tal form: Lemma 2.5. (Niyo gi et al., 2008, Pr op osition 6.1) L et M b e an emb e dde d submanifold with r e ach τ > 0 . Then sup x ∈ M ∥ Π x ∥ op ≤ τ − 1 . 2.4 Generic Notation F or tw o linear subspaces E , F ⊂ R D , w e write Θ( E , F ) for their largest principal angle, so that sin { Θ( E , F ) } denotes the corresp onding subspace discrepancy . Unless stated otherwise, generic constan ts C, C 1 , C 2 , . . . and O ( · ) b ounds in Sections 3–5 are uniform ov er y ∈ T ( τ − ε ) and 0 < σ ≤ σ 0 for suﬃciently small σ 0 , and may dep end on M , d , D , k , τ , and ε . In Section 6, where σ is ﬁxed, uniformity is only with resp ect to y ∈ T ( τ − ε ), and the implied constan ts may dep end on σ . 3 Asymptotic Expansions of the Noisy Density and Its Deriv atives In this section, we study the small-noise structure of the manifold-generated density P σ = P M ∗ N (0 , σ 2 I D ) on the tubular neigh b orho o d T ( τ − ε ). Our goal is to obtain uniform expansions of P σ and of the deriv atives of log P σ that we later use to reco ver tangen t spaces and the second fundamen tal forms. Throughout, M is a compact C k submanifold with k ≥ 6 and p ositive reach τ > 0, and 0 < ε < τ is ﬁxed. F or eac h ﬁxed ε , we work in the small-noise regime 0 < σ ≤ σ 0 , where σ 0 > 0 is suﬃciently small. Pro ofs of the theorems and corollaries are giv en in Section 8.1; the pro of of Lemma 3.3 is given in App endix B. 3.1 Manifold-Generated Noisy Densit y Let P M denote the uniform probability measure on M , namely dP M ( x ) = V − 1 M dµ ( x ), where µ is the induced volume measure on M . Let ξ ∼ N (0 , σ 2 I D ) and let Φ σ b e its density . Then the noisy observ ation Y = X + ξ has density P σ ( y ) = Z M Φ σ ( y − x ) dP M ( x ) = 1 V M (2 π σ 2 ) D 2 Z M exp  − ∥ y − x ∥ 2 2 σ 2  dµ ( x ) . Recall that π ( y ) denotes the pro jection onto M and v y the corresponding normal displacemen t. F or Π π ( y ) the second fundamen tal form at π ( y ) ∈ M , we deﬁne a self-adjoint map A y : T π ( y ) M → T π ( y ) M b y A y = I T π ( y ) M − ⟨ v y , Π π ( y ) ⟩ , ⟨ A y u, v ⟩ = ⟨ u, v ⟩ − ⟨ v y , Π π ( y ) ( u, v ) ⟩ . Since y 7→ π ( y ) is C k − 1 on T ( τ − ε ) and x 7→ Π x is C k − 2 on M , for k ≥ 6 the map y 7→ A y is at least C 3 . The follo wing theorem giv es an expansion of P σ ( y ). Theorem 3.1. Uniformly for y ∈ T ( τ − ε ) and 0 < σ ≤ σ 0 , P σ ( y ) = 1 V M (2 π σ 2 ) D − d 2 exp  − ∥ v y ∥ 2 2 σ 2  1 p det A y  1 + O ( σ ∥ v y ∥ + σ 2 )  . 7 The proof is based on T aylor expansions in exponential co ordinates around π ( y ). The lead- ing factor separates the Gaussian decay in the normal direction from the curv ature correction (det A y ) − 1 / 2 . F or the analysis of deriv ativ es, it is conv enient to rewrite Theorem 3.1 at the lev el of the log- densit y and to record uniform b ounds on the remainder and its deriv ativ es. Corollary 3.2. Uniformly for y ∈ T ( τ − ε ) and 0 < σ ≤ σ 0 , log P σ ( y ) = log  V M (2 π σ 2 ) D − d 2  − 1 − ∥ v y ∥ 2 2 σ 2 − 1 2 log det A y + R ( y , σ ) , (1) wher e R ( y , σ ) is C 3 on T ( τ − ε ) and its ambient derivatives satisfy | R ( y , σ ) | ≤ C 0  σ ∥ v y ∥ + σ 2  , ∥∇ m y R ( y , σ ) ∥ op ≤ C m σ, m ∈ { 1 , 2 , 3 } . 3.2 Gradien t Expansion W e no w derive an expansion for the gradient G σ ( y ) = ∇ log P σ ( y ) , y ∈ T ( τ − ε ) . T o diﬀerentiate the expansion in Corollary 3.2 with resp ect to the am bient co ordinate y , we ﬁrst record the deriv ativ es of π ( y ) and v y . Recall that ∇ ω ( · ) denotes the Euclidean directional deriv ativ e in the direction ω ∈ R D , and ω ⊤ denotes the orthogonal pro jection of ω onto T π ( y ) M . Lemma 3.3. F or every y ∈ T ( τ ) and every ω ∈ R D , ∇ ω π ( y ) = A − 1 y ω ⊤ , ∇ ω v y = ω − A − 1 y ω ⊤ . The pro of is based on the c haracterization of A y in terms of the second fundamental form. Com bining Lemma 3.3 with the log-densit y expansion in Corollary 3.2, w e obtain the follo wing expression for the gradient. Theorem 3.4. Uniformly for y ∈ T ( τ − ε ) and 0 < σ ≤ σ 0 , G σ ( y ) = − v y σ 2 + 1 2 dH π ( y ) + O ( ∥ v y ∥ + σ ) , (2) wher e the O ( · ) term is taken in the Euclide an norm. The leading term − v y /σ 2 is purely normal and comes from the Gaussian conv olution, while the ﬁrst curv ature correction is giv en by the mean curv ature vector. In particular, for x ∈ M , G σ ( x ) = d 2 H x + O ( σ ) . Aw ay from M , the gradient is asymptotically aligned with the normal direction. 8 3.3 Hessian Expansion W e next study the Hessian of the log-densit y: H σ ( y ) = ∇ 2 log P σ ( y ) = ∇G σ ( y ) . Using the gradient expansion in Theorem 3.4 together with Lemma 3.3, we obtain the follo wing description of H σ ( y ) in T ( τ − ε ). Theorem 3.5. Uniformly for y ∈ T ( τ − ε ) and 0 < σ ≤ σ 0 , H σ ( y ) = − 1 σ 2 P N ( y ) − 1 σ 2 ( I T π ( y ) M − A − 1 y ) P T ( y ) + O (1) , wher e the O (1) term is taken in the op er ator norm. Here P T ( y ) and P N ( y ) are orthogonal pro jections as in Section 2. Deﬁne the leading-order op erator H 0 ( y ) = − 1 σ 2 P N ( y ) − 1 σ 2 ( I T π ( y ) M − A − 1 y ) P T ( y ). W e next record a corollary on the sp ectral gap of H 0 and H σ . Corollary 3.6. F or e ach ﬁxe d ε ∈ (0 , τ ) , ther e exist c onstants c ε > 0 and σ 0 > 0 such that the fol lowing holds. L et λ 1 ( y , σ ) ≥ · · · ≥ λ D ( y , σ ) b e the eigenvalues of H σ ( y ) . Then, for every y ∈ T ( τ − ε ) and every 0 < σ ≤ σ 0 , λ d ( y , σ ) − λ d +1 ( y , σ ) ≥ c ε σ − 2 . The same sp e ctr al gap holds for H 0 ( y ) . 4 P opulation Geometric Estimators from Densit y Deriv ativ es In this section, we develop p opulation-level identiﬁcation formulas and estimators for geometric quan tities of the em b edded manifold M from deriv ativ es of the noisy log-densit y log P σ . A t this stage we assume access to the gradient G σ = ∇ log P σ , the Hessian H σ = ∇ 2 log P σ , and the third- order deriv ativ e T σ = ∇ 3 log P σ . Fix ε ∈ (0 , τ ) and let σ 0 > 0 b e as in Section 3; throughout this section w e consider y ∈ T ( τ − ε ) and 0 < σ ≤ σ 0 , so that the expansions of Theorems 3.1, 3.4, 3.5, Corollary 3.2, and Corollary 3.6 apply . Our results reveal a simple hierarch y in the geometric information enco ded b y density deriv a- tiv es. Second-order deriv atives identify tangen t spaces and the in trinsic dimension by a spectral gap. F or totally umbilical submanifolds, the second fundamental form collapses to the mean cur- v ature v ector, so ﬁrst-order deriv ativ es already suﬃce. F or h yp ersurfaces, ﬁrst- and second-order deriv ativ es reco ver the second fundamental form through an estimated normal direction. In arbi- trary co dimension, recov ering the full second fundamen tal form requires diﬀeren tiating the tangent pro jector and therefore inv olv es third-order deriv ativ es. W e presen t the constructions in this order of increasing geometric complexity . The second fundamen tal form estimators in this section are p opulation-lev el constructions. They are deﬁned as bilinear forms on the true tangent space T π ( y ) M , and are analyzed relative to the corresp onding p opulation target. The sample-level counterparts are obtained in Section 6 b y replacing G σ , H σ , and T σ with plug-in estimators based on noisy observ ations, together with estimated tangen t spaces when needed. Pro ofs of theorems and corollaries are giv en in Section 8.2; pro ofs of lemmas are giv en in App endix C. 9 4.1 T angen t Spaces and the In trinsic Dimension Estimation By Theorem 3.5 and Corollary 3.6, the d tangential eigenv alues of the Hessian H σ ( y ) are separated from the remaining D − d normal eigenv alues by a sp ectral gap of order σ − 2 . It is therefore natural to estimate the tangent space at π ( y ) by the eigenv ectors asso ciated with the d largest eigen v alues, and to estimate the intrinsic dimension d b y the lo cation of the largest sp ectral gap. F ormally , let λ 1 ≥ λ 2 ≥ · · · ≥ λ D b e the eigen v alues of H σ ( y ) and let e 1 , . . . , e D b e the asso ciated orthonormal eigenv ectors. If the intrinsic dimension d is known, w e deﬁne the tangen t space estimator by b T y M = span { e 1 , . . . , e d } . If d is unknown, we estimate it b y the lo cation of the largest consecutive sp ectral gap, b d ∈ arg max k =1 ,...,D − 1 | λ k − λ k +1 | . The following theorem shows that the tangent space estimator is consistent when d is known, and that the sp ectral-gap estimator recov ers the correct intrinsic dimension in the small-noise regime. Theorem 4.1. L et b T y M b e the eigensp ac e asso ciate d with the d lar gest eigenvalues of H σ ( y ) . Then ther e exist c onstants C and σ 0 > 0 such that, for every y ∈ T ( τ − ε ) and every 0 < σ ≤ σ 0 , sin { Θ( b T y M , T π ( y ) M ) } ≤ C σ 2 . Mor e over, for any se quenc e with ∥ v y ∥ → 0 and σ → 0 , the lar gest c onse cutive sp e ctr al gap of H σ ( y ) is attaine d at k = d . Conse quently, b d = d along the se quenc e. W e note that Stanczuk et al. (2024) exploits the small-noise geometry of the gradient ﬁeld ∇ log P σ to estimate tangen t spaces and the in trinsic dimension. Relying on the prop erty that ∇ log P σ is asymptotically normal, they prop ose to sample enough gradient v ector in a small neigh- b orho o d of a p oint so as to span the normal space. Then the sp ectrum of this stac k ed gradient matrix tells tangen t spaces and the in trinsic dimension. In con trast, our estimators are based on the local sp ectral structure of the Hessian, whic h enables reco very from p oint wise deriv ative information. 4.2 Second F undamental F orm for T otally Umbilical Submanifolds W e now turn to the estimation of the second fundamen tal form. W e b egin with the class of totally um bilical submanifolds, characterized b y Π( u, v ) = g ( u, v ) H , for all tangen t v ectors u, v , where H is the mean curv ature and g is the induced metric. In Euclidean space, this class consists of aﬃne subspaces and spheres of arbitrary dimension. F ollowing Theorem 3.4, for x ∈ M we hav e G σ ( x ) = d 2 H x + O ( σ ) . Therefore at noiseless p oints on the manifold the gradien t recov ers the mean curv ature vector up to the known factor d/ 2 and an O ( σ ) error. Com bining this with the deﬁnition of totally umbilical submanifolds suggests the following estimator. 10 Theorem 4.2. Assume M is total ly umbilic al. F or x ∈ M , u, v ∈ T x M , deﬁne b Π x ( u, v ) = 2 d g ( u, v ) G σ ( x ) . Then ther e exist c onstants C and σ 0 > 0 such that, for every x ∈ M and 0 < σ ≤ σ 0 , ∥ b Π x − Π x ∥ op ≤ C σ. Due to the presence of the leading normal term − v y /σ 2 in the gradien t expansion (Theorem 3.4), this estimator cannot b e directly generalized from x ∈ M to noisy p oints y ∈ T ( τ − ε ) uniformly in σ . Unless ∥ v y ∥ is of order o ( σ 2 ), the Gaussian term − v y /σ 2 dominates the curv ature contribution and prev ents a stable reco very of Π π ( y ) from the gradient alone. 4.3 Second F undamental F orm for Hyp ersurfaces W e next consider hypersurfaces, i.e., submanifolds of co dimension one ( D = d + 1). In this setting, the second fundamental form can be reconstructed from a normal vector ﬁeld. Lemma 4.3. L et M b e a hyp ersurfac e, and let N b e a C 1 ve ctor ﬁeld deﬁne d on T ( τ ) such that N ( y ) ∈ T ⊥ π ( y ) M , N ( y )  = 0 for al l y ∈ T ( τ ) . Then, for every y ∈ T ( τ ) and u, v ∈ T π ( y ) M , Π π ( y ) ( u, v ) = − 1 ∥N ( y ) ∥ 2  ∇ A y u N ( y ) , v  N ( y ) . Here A y is the tangent endomorphism introduced in Section 3. In particular, A y u = u − ( ∇ u v y ) ⊤ . According to Theorem 3.4, the tangen tial comp onent of the gradient G σ = ∇ log P σ is of lo wer order than its normal com p onen t near M , so G σ pro vides a natural approximate normal ﬁeld. This suggests replacing the true normal ﬁeld in Lemma 4.3 b y G σ . Moreov er, using the deﬁnition of the Hessian H σ ( u, v ) := ⟨∇ u ∇ log P σ , v ⟩ , w e give the follo wing theorem. Theorem 4.4. Assume M ⊂ R D is a hyp ersurfac e. L et { U σ } 0 <σ ≤ σ 0 b e a family of subsets of T ( τ − ε ) such that inf y ∈ U σ ∥G σ ( y ) ∥ ≥ c 0 > 0 for every 0 < σ ≤ σ 0 . F or y ∈ U σ , u, v ∈ T π ( y ) M , deﬁne b Π y ( u, v ) = − 1 ∥G σ ( y ) ∥ 2  H σ ( y ) u, v  G σ ( y ) . Then ther e exists a c onstant C such that, for every y ∈ U σ and 0 < σ ≤ σ 0 , ∥ b Π y − Π π ( y ) ∥ op ≤ C ( ∥ v y ∥ + σ ) . On regions where G σ ma y degenerate, one may instead use the pro jector-based estimator of Theorem 4.7, which also applies to h yp ersurfaces. The same formula, with M replaced by the level set L σ,c := { y ∈ R D : P σ ( y ) = c } , yields the second fundamen tal form of L σ,c at y without error whenev er G σ ( y )  = 0, since G σ ( y ) is strictly normal to the level set and Lemma 4.3 applies with N = G σ . With regard to the density lev el set, w e can measure ho w close it is to M following the expansion of P σ . Corollary 4.5. Ther e exist c onstants C and σ 0 > 0 such that, for every level set L σ,c with L σ,c ∩ M  = ∅ and 0 < σ ≤ σ 0 , sup y ∈ L σ,c d ( y , M ) = O ( σ 2 ) . This suggests that noisy density generates nearb y geometric surrogates of M through its lev el sets, whic h will b e further discussed in Section 5 with regard to the density-induced metrics. 11 4.4 Second F undamental F orm for Submanifolds of Arbitrary Co dimension F or submanifolds of arbitrary co dimension in R D , there is no unique normal direction, and the second fundamental form must b e recov ered in a diﬀerent w ay . W e refer to this as the general- co dimension case. Due to the embedding relation, the deriv ativ es on M can be view ed as the tangen tial pro jection of the deriv atives on R D . Hence, we ma y exp ect that the pro jection op erator em b o dies the curv atures of M . In the following lemma we in tro duce an approach of computing the second fundamen tal form b y taking the directional deriv ativ e of the tangential pro jection map. Lemma 4.6. L et P T ( y ) b e a C 1 tangential pr oje ction ﬁeld on T ( τ ) . Then, for every x ∈ M and u, v ∈ T x M , Π x ( u, v ) = ∇ u P T ( x ) v . (3) By Theorem 4.1, the tangent space at y can be estimated as b T y M = span { e 1 , . . . , e d } , where e 1 , . . . , e d are the eigen vectors of H σ ( y ) associated with the d largest eigen v alues. W e denote the corresp onding appro ximate pro jection by b P T ( y ) = E y E T y , where E y is the D × d matrix with columns e 1 , . . . , e d . By Corollary 3.6, the d tangential eigenv alues of H σ ( y ) are separated from the D − d normal eigenv alues on T ( τ − ε ) b y a gap of order σ − 2 . Therefore, since y 7→ H σ ( y ) is C 1 , the asso ciated sp ectral pro jector b P T ( y ) is w ell-deﬁned and C 1 on T ( τ − ε ). Substituting b P T ( y ) for P T ( y ) in the lemma leads to the following theorem. Theorem 4.7. L et b P T ( y ) b e the tangential sp e ctr al pr oje ctor asso ciate d with the d lar gest eigen- values of H σ ( y ) . F or y ∈ T ( τ − ε ) , u, v ∈ T π ( y ) M , deﬁne b Π y ( u, v ) = ∇ u b P T ( y ) v . (4) Then ther e exist c onstants C and σ 0 > 0 such that, for every y ∈ T ( τ − ε ) and 0 < σ ≤ σ 0 , ∥ b Π y − Π π ( y ) ∥ op ≤ C ( ∥ v y ∥ + σ 2 ) . The pro of uses p erturbation b ounds for sp ectral pro jectors. Since b P T ( y ) is the Riesz pro jector asso ciated with the tangen tial eigenspace of H σ ( y ), the diﬀerence ∇ u b P T ( y ) − ∇ u P T ( y ) can b e con trolled in terms of the p erturbation of H σ ( y ) and its deriv ative ∇ u H σ ( y ). In practice, for a ﬁxed direction u , the deriv ativ e X := ∇ u b P T ( y ) can be computed from the Sylv ester-type system H σ ( y ) X − X H σ ( y ) = b P T ( y ) ∇ u H σ ( y ) − ∇ u H σ ( y ) b P T ( y ) , b P T ( y ) X + X b P T ( y ) = X . Since the tangen tial and normal sp ectral clusters are separated b y a gap of order σ − 2 , this system has a unique solution. Moreo ver, the estimator (4) depends on third-order deriv ativ es of the log- densit y through ∇ u H σ ( y ) = ∇ 3 log P σ ( y )( u, · , · ) . The approac h of computing the second fundamen tal form via the deriv ativ e of the tangen- tial pro jection go es bac k to Hutchinson (1986); Ambrosio and Man tegazza (1998). Conceptually , b oth Theorem 4.1 (tangen t space) and Theorem 4.7 (second fundamental form) exploit the same structural prop ert y of the log-density (Corollary 3.2): in the small-noise regime, the leading term −∥ v y ∥ 2 / (2 σ 2 ) in the expansion of log P σ is a scaled “squared distance function” to the manifold. The squared distance function enco des b oth the tangent spaces and the second fundamental form, and our metho d recov ers these geometric quantities from the deriv ativ es of log P σ . 12 5 Densit y-Induced Metrics A central conclusion from the previous sections is that the tangent space and second fundamental form of M admit explicit represen tations in terms of the noisy densit y P σ and its deriv atives. This suggests a further geometric question: can these density-based formulas induce an am bient geometric structure on R D that reﬂects the intrinsic geometry of M together with its embedding? Sp eciﬁcally , in this section we in vestigate how to endow the am bient space R D with a density- induced metric g P σ asso ciated with P σ suc h that the inclusion ( M , g )  → ( R D , g P σ ) is a totally geo desic map: an y geodesic in M with respect to the induced Euclidean metric g remains a geodesic when viewed as a curve in ( R D , g P σ ). F rom the p ersp ective of geo desic equation, one may seek an am bient structure whose Christoﬀel symbols, restricted to tangent directions along M , cancel the second fundamental form of the Euclidean embedding. W e record tw o constructions. In the totally um bilical case, the densit y induces a conformal Riemannian metric on the am bient space. In the h yp ersurface case, the resulting metric is degenerate, so the corresp onding statement is understo o d through the asso ciated Christoﬀel-type co eﬃcient ﬁeld and its induced acceleration equation. The results in this section should b e view ed as geometric in terpretation of the densit y-based reco very form ulas, and are not of the statistical estimation theory dev elop ed in the rest of the pap er. Pro ofs of Theorem 5.1 and Theorem 5.2 are given in Section 8.3. 5.1 Metric Change for T otally Um bilical Submanifolds Theorem 4.2 shows that the second fundamental form of totally umbilical submanifolds can b e reco vered from the gradien t of log-density ∇ log P σ up to O ( σ ) error. This yields a conformal am bient metric for whic h the second fundamen tal form of the inclusion is O ( σ ). Theorem 5.1. Assume M is total ly umbilic al. Deﬁne the Riemannian metric g P σ = P 4 d σ g E . L et Π P σ b e the se c ond fundamental form of the inclusion ( M , g )  → ( R D , g P σ ) . Then ther e exist c onstants C and σ 0 > 0 such that, for every x ∈ M and 0 < σ ≤ σ 0 , ∥ Π P σ x ∥ op ≤ C σ. In p articular, the inclusion is asymptotic al ly total ly ge o desic as σ → 0 . F or the problem of mo difying the ambien t metric so that geo desics concen trate in regions where the data density is large, a broader line of w ork considers densit y-driven conformal metrics of the form ˜ g P σ = w ( P σ ) g E with factor w ( P σ ) > 0 that decreases as P σ increases. In many contin uum limits of graph-based distances, suc h as F ermat-type distances and related constructions, this yields metrics that can be written as ˜ g P σ ∝ P − α σ g E for some α > 0; geodesics under ˜ g P σ then tend to a void lo w-density regions and to follo w high-densit y zones near the data manifold (e.g., Groisman et al., 2022; T rillos et al., 2024). At ﬁrst sight this seems at o dds with our choice g P σ = P 4 /d σ g E , whose conformal factor gro ws with the densit y . The tw o constructions, ho w ever, enco de diﬀerent geometric quantities. Our conformal factor is determined b y the second fundamen tal form estimator and is chosen precisely so that the intrinsic geo desics on M remain geodesics in the c hanged am bien t space. In con trast, densit y-inv erse metrics suc h as the F ermat distance arise as contin uum limits of shortest-path distances on weigh ted random geometric graphs. In the noisy manifold setting considered here, b oth approac hes nev ertheless bias geo desics tow ard high-probability regions near M while serving diﬀerent purposes. 13 5.2 Metric Change for Hyp ersurfaces F or manifolds of higher curv ature complexit y , suc h as h yp ersurfaces or density lev el sets, a conformal rescaling is no longer suﬃcient to incorp orate the full second fundamental form. In this case, Theorem 4.4 shows that the second fundamental form can b e reco vered b y the gradient of log- densit y and its deriv ative. Motiv ated b y this, we consider the degenerate metric g P σ := d (log P σ ) ⊗ d (log P σ ) , g P σ ( u, v ) = ⟨ u, ∇ log P σ ⟩⟨ v , ∇ log P σ ⟩ , where d (log P σ ) is a 1-form. Wherever ∇ log P σ ( x )  = 0, g P σ is a symmetric, p ositiv e semi-deﬁnite (0,2)-tensor of rank one. Since g P σ is degenerate, it do es not deﬁne the standard Levi–Civita formalism. Nev ertheless, the tensor g P σ together with its Mo ore–Penrose pseudo-in verse, denoted as ( g P σ ) MP , determines a co eﬃcien t ﬁeld of Christoﬀel t yp e, which in turn yields a second-order system of acceleration. Theorem 5.2. Assume M is a hyp ersurfac e. Deﬁne the de gener ate metric g P σ = d (log P σ ) ⊗ d (log P σ ) . (5) L et γ ( t ) b e a ge o desic in ( M , g ) , r e gar de d as a curve in R D , and assume ∇ log P σ  = 0 on γ ( t ) . Deﬁne the c o eﬃcient ﬁeld e Γ k ij := ( g P σ ) mk MP 1 2  ∂ j ( g P σ ) im + ∂ i ( g P σ ) j m − ∂ m ( g P σ ) ij  . Then ther e exist c onstants C and σ 0 > 0 such that, for every γ , every t in its domain and 0 < σ ≤ σ 0 ,   ¨ γ k ( t ) + e Γ k ij ( γ ( t )) ˙ γ i ( t ) ˙ γ j ( t )   ≤ C σ. In this sense, the density-induced degenerate metric g P σ determines a second-order system that matc hes the geo desic equation of M up to an O ( σ ) error. Note that for a density lev el set L σ,c , the second fundamental form is represen ted exactly by the same gradien t-Hessian formula of h yp ersurfaces; see Section 4.3. Hence the abov e degenerate metric also induces, without appro ximation error, the corresponding acceleration equation for geodesics on densit y level sets. Lev eraging the properties of the gradien t ﬁeld ∇ log P σ , suc h as the fact that it p oints tow ard M under small noise and is almost normal to M (Theorem 3.4), sev eral recent w orks construct metrics based on this gradien t that encourage geo desics to sta y near the data manifold. In particular, Azeglio and Bernardo (2025) prop ose a metric that is a weigh ted com bination of the Euclidean metric and the degenerate metric d (log P σ ) ⊗ d (log P σ ). When the weigh t on the degenerate metric is large, their metric b ecomes close to (5). They empirically observe that b oundary-v alue geo desics under their metric remain close to the data manifold even for high-dimensional data. In the manifold-generated noisy setting considered here, our Theorem 5.2 and Corollary 4.5 pro vide a heuristic explanation for this behavior: when the b oundary p oints lie on the intersection of M and a density level set, the lev el set sta ys within an O ( σ 2 ) tubular neighborho o d when noise is small; since the geo desics of lev el sets matc h the acceleration equation of g P σ , the resulting curve connecting suc h b oundary p oin ts is conﬁned to the same neighborho o d. F or submanifolds of arbitrary co dimension, the form of the estimator in Theorem 4.7 mak es an explicit metric construction substan tially more diﬃcult. The Christoﬀel sym b ols m ust satisfy conditions such as the torsion-free and metric-compatibilit y , and the resulting metric tensor must remain p ositiv e deﬁnite near M . In this sense, Theorem 4.7 should b e in terpreted as pro viding constrain ts that any densit y-based am bient metric would hav e to satisfy along M in order to enco de the correct second fundamental form. 14 6 Sample-Lev el Estimators In this section we w ork at a ﬁxed noise level 0 < σ ≤ σ 0 and estimate the noisy density P σ and its deriv ativ es from i.i.d. samples y 1 , . . . , y N ∼ P σ . All sto c hastic orders are taken as N → ∞ , with constan ts allow ed to depend on σ . Since P σ is a smo oth densit y on R D for each ﬁxed σ > 0, the sample-lev el analysis in this section is based on a standard ambien t-space kernel density estimator (KDE). These b ounds will then b e combined in Section 6.3 with the p opulation-level constructions from Section 4 to obtain geometric error b ounds for the tangen t space and the second fundamental form. Since the subsequent geometric constructions are carried out on T ( τ − ε ), w e only require uniform con trol of the KDE and its deriv ativ es on this tubular neigh b orho o d. 6.1 Kernel Estimators of Noisy Densit y and Its Deriv atives Under the manifold-generated noise model of Section 3, P σ = P M ∗ N (0 , σ 2 I D ) is a bounded densit y on R D . Therefore we estimate P σ and its deriv atives b y standard am bient-space k ernel metho ds. Consider the D -dimensional kernel density estimators e P σ ( y ) = 1 N h D N X i =1 K  y i − y h  , y ∈ R D , where K : R D → R is a k ernel and h > 0 is the bandwidth. F or order m ∈ { 1 , 2 , 3 } , we denote the deriv ativ es of the kernel estimator with bandwidth h k b y G m ( y ) = ∇ m e P σ ( y ) = 1 N h D + m m N X i =1 ∇ m K  y i − y h m  . W e assume K satisﬁes the standard smoothness, moment, and en tropy conditions under which uniform m ultiv ariate KDE b ounds hold; see, e.g., Gin ´ e and Guillou (2002). Lemma 6.1. Fix ε ∈ (0 , τ ) and 0 < σ ≤ σ 0 . Under the kernel r e gularity c onditions state d ab ove, for e ach m ∈ { 0 , 1 , 2 , 3 } , sup y ∈ T ( τ − ε ) ∥∇ m e P σ ( y ) − ∇ m P σ ( y ) ∥ op = O ( h 2 m ) + O P  r log N N h D +2 m  . (6) In p articular, if h m ≍  log N N  1 / ( D +4+2 m ) , then sup y ∈ T ( τ − ε ) ∥∇ m e P σ ( y ) − ∇ m P σ ( y ) ∥ op = O P  log N N  2 D +4+2 m  . The pro of of this lemma follows from standard KDE theory . F or each ﬁxed σ > 0, the function P σ is smooth, and its deriv atives up to order m + 2 are bounded on the compact set T ( τ − ε ). The stated bias term is the standard second-order kernel bias, while the v ariance term is the usual uniform empirical-process b ound for multiv ariate KDEs and their deriv atives. Com bining these yields the claim; see Gin´ e and Guillou (2002); Einmahl and Mason (2005). F or general multiv ariate densit y deriv ativ e estimation, including asymptotic expansions and bandwidth selection, see Chac´ on et al. (2011); Also see Genov ese et al. (2014); Chen et al. (2015) for their use in geometric inference based on deriv ativ es of a KDE. 15 6.2 Plug-in Estimators for Log-densit y Deriv ativ es W e no w deﬁne the sample-level estimators of the gradien t, Hessian, and third deriv ative of the log-densit y . The gradient estimator of the log-densit y is e G σ ( y ) = ∇ log e P σ ( y ) = G 1 ( y ) e P σ ( y ) . The Hessian is estimated by e H σ ( y ) = ∇ 2 log e P σ ( y ) = G 2 ( y ) e P σ ( y ) − e G σ ( y ) ⊗ e G σ ( y ) , where ⊗ denotes the tensor product, and we view e G σ ( y ) ⊗ e G σ ( y ) as a D × D matrix. F or the third-order deriv ativ e, we hav e e T σ ( y ) = ∇ 3 log e P σ = G 3 ( y ) e P σ ( y ) − e H σ ( y ) ⊗ e G σ ( y ) − e G σ ( y ) ⊗ 3 , where ( A ⊗ b ) ij k = A ij b k + A ik b j + A j k b i for 1 ≤ i, j, k ≤ D . Com bining Lemma 6.1 with the p ositivit y of P σ on T ( τ − ε ), we obtain the follo wing rates for the log-densit y deriv ativ es. Corollary 6.2. Assume the c onditions of L emma 6.1. Then, for e ach ﬁxe d 0 < σ ≤ σ 0 , sup y ∈ T ( τ − ε ) ∥ e G σ ( y ) − G σ ( y ) ∥ = O P  log N N  2 D +6  , (7) sup y ∈ T ( τ − ε ) ∥ e H σ ( y ) − H σ ( y ) ∥ op = O P  log N N  2 D +8  , (8) sup y ∈ T ( τ − ε ) ∥ e T σ ( y ) − T σ ( y ) ∥ op = O P  log N N  2 D +10  . (9) Pr o of. Fix 0 < σ ≤ σ 0 . Since T ( τ − ε ) is compact and P σ is contin uous and strictly positive on it, for eac h ﬁxed σ there exists c σ > 0 such that inf y ∈ T ( τ − ε ) P σ ( y ) ≥ c σ . Hence, with probability tending to one, e P σ ≥ c σ / 2 uniformly on T ( τ − ε ) by Lemma 6.1. Moreov er, b y the uniform b ounds on the deriv atives of P σ and the corresp onding uniform estimation bounds for e P σ , ( p, ∇ p, ∇ 2 p, ∇ 3 p ) asso ciated with P σ and e P σ remain, with probabilit y tending to one, in a compact subset of the region { p ≥ c σ / 2 } . Since the map from ( p, ∇ p, ∇ 2 p, ∇ 3 p ) to the log-densit y deriv atives is smo oth on { p > 0 } , it is Lipsc hitz on an y compact subset of { p ≥ c σ / 2 } . It follo ws that the uniform estimation error for the log-densit y deriv ativ es is bounded by a constan t m ultiple of the corresp onding uniform estimation error for the density deriv atives. Applying Lemma 6.1 for orders k = 0 , 1 , 2 , 3 yields the stated rates. The displa yed rates are obtained b y choosing the bandwidth separately for each order. 6.3 Sample-Lev el Geometric Error Bounds W e now plug the estimators e G σ , e H σ , e T σ in to the p opulation constructions of Section 4 and quantify the resulting geometric errors. F or tangen t space, w e deﬁne the sample-lev el estimator e T y M as the span of the eigenv ectors of e H σ ( y ) asso ciated with its d largest eigenv alues. If the intrinsic dimension d is unknown, it ma y b e estimated from the eigengap of e H σ ( y ), in direct analogue with the p opulation construction in 16 Theorem 4.1. In particular, for suﬃciently small σ , the p opulation eigengap remains uniformly separated on T ( τ − ε ), and the ab ov e Hessian b ound then also yields consistency of e d , uniformly o ver y ∈ T ( τ − ε ). T o compare the true and estimated second fundamental forms as bilinear maps on a common domain, w e work with their am bient extensions. F or each y ∈ T ( τ − ε ), let Π π ( y ) ( u, v ) := Π π ( y )  P T ( y ) u, P T ( y ) v  , u, v ∈ R D , denote the am bient extension of the true second fundamental form. The extended sample-level estimator is lik ewise deﬁned using the estimated tangen tial pro jector e P T ( y ) obtained from the eigen vectors of e H σ ( y ) (see Section 4.4). Accordingly , all op erator norms for second fundamental forms b elo w are tak en ov er bilinear maps from R D × R D to R D . In the hypersurface setting ( D − d = 1), following Theorem 4.4 w e deﬁne e Π y ( u, v ) = − 1 ∥ e G σ ( y ) ∥ 2  e H σ ( y ) e P T ( y ) u, e P T ( y ) v  e G σ ( y ) , u, v ∈ R D . F or submanifolds of arbitrary co dimension, following Theorem 4.7 w e deﬁne e Π y ( u, v ) = ∇ e P T ( y ) u e P T ( y ) e P T ( y ) v , u, v ∈ R D . Corollary 6.3. Assume the c onditions of Cor ol lary 6.2. Then, for e ach ﬁxe d 0 < σ ≤ σ 0 , the fol lowing hold. First, uniformly over y ∈ T ( τ − ε ) , sin { Θ( e T y M , T π ( y ) M ) } = O ( σ 2 ) + O P  log N N  2 D +8  . (10) In the hyp ersurfac e c ase, uniformly over y ∈ U σ , ∥ e Π y − Π π ( y ) ∥ op = O ( ∥ v y ∥ + σ ) + O P  log N N  2 D +8  , In the arbitr ary c o dimension setting, uniformly over y ∈ T ( τ − ε ) , ∥ e Π y − Π π ( y ) ∥ op = O ( ∥ v y ∥ + σ 2 ) + O P  log N N  2 D +10  . In al l thr e e displays, the sto chastic c onstants may dep end on σ . Sketch. Fix 0 < σ ≤ σ 0 . F or the tangen t-space b ound, Corollary 3.6 sho ws that, uniformly on T ( τ − ε ), H σ ( y ) has a sp ectral gap of order σ − 2 separating its largest d eigen v alues from the remaining D − d eigenv alues. Therefore, for each ﬁxed σ , Davis–Kahan’s sin Θ theorem applied to e H σ ( y ) and H σ ( y ) yields sin { Θ  e T y M , b T y M  } ≤ C ∥ e H σ ( y ) − H σ ( y ) ∥ op . This Hessian error is controlled b y Corollary 6.2, and com bining this with the population estimate in Theorem 4.1 and the triangle inequality giv es (10). In the hypersurface case, by the lo w er b ound assumption on ∥G σ ( y ) ∥ o ver U σ , Corollary 6.2 implies that ∥ e G σ ( y ) ∥ remains b ounded a wa y from zero with high probability on U σ . Therefore, on the region where ∥G σ ∥ , ∥ e G σ ∥ ≥ c 0 / 2, the map ( G , H , P ) 7→ −∥G ∥ − 2 ⟨H P u, P v ⟩G 17 is lo cally Lipschitz for ﬁxed σ . It follows that the estimation error can b e controlled b y ∥ e G σ − G σ ∥ , ∥ e H σ − H σ ∥ op , ∥ e P T − P T ∥ op . The ﬁrst tw o terms are b ounded b y Corollary 6.2, and the last one is b ounded b y (10). Com bining this with the p opulation b ound from Theorem 4.4 giv es the stated result. In arbitrary co dimension, the comparison b etw een ∇ e P T ( y ) and ∇ b P T ( y ) is a direct analogue of the Riesz-pro jector comparison carried out in the pro of of Theorem 4.7. By that argumen t, the diﬀerence of the pro jector deriv atives is controlled b y ∥ e H σ − H σ ∥ op , ∥∇ e H σ − ∇H σ ∥ op , ∥∇H σ ∥ op . F or ﬁxed σ , the last quantit y is b ounded on T ( τ − ε ), while the ﬁrst tw o terms are controlled b y Corollary 6.2. Among them, the second term giv es the dominated order O P  (log N / N ) 2 D +10  . The am bient extension in (4) in tro duces additional factors of e P T ( y ) and b P T ( y ), whic h are con trolled b y (10). Combining this with Theorem 4.7 prov es the claim. A notable feature of these rates is that they dep end on the ambien t dimension D , rather than the intrinsic dimension d . This is inherent to the present estimator class. F or eac h ﬁxed σ > 0, the target P σ = P M ∗ N (0 , σ 2 I D ) is a smo oth b ounded densit y on R D , and our sample-lev el analysis is based on estimating this am bien t-space densit y and its deriv ativ es b y KDE-t ype metho ds. Th us the resulting rates reﬂect the classical nonparametric diﬃcult y of estimating a D -dimensional density; see, for example, Goldenshluger and Lepski (2014). By contrast, the manifold structure enters primarily through the p opulation-level geometric identities dev elop ed earlier. Obtaining sample- lev el b ounds that adapt to the intrinsic dimension w ould lik ely require a diﬀeren t estimator class, one that more directly exploits the manifold structure. Remark 6.4 (Alternative estimators) . The sample-level argumen ts ab ov e use the kernel estimator only through uniform con trol of the estimated log-densit y deriv atives. Accordingly , the same plug-in geometric estimators apply to any alternativ e estimators of G σ = ∇ log P σ , H σ = ∇ 2 log P σ , T σ = ∇ 3 log P σ that satisfy analogous b ounds on the rele v an t region. P ossible alternativ es include direct score estimators in nonparametric function classes, score- matc hing estimators, and smo oth parametric mo dels suc h as neural score netw orks; see, for example, Hyv¨ arinen (2005); Srip erumbudur et al. (2017); Li and T urner (2018); Song et al. (2020). 7 Numerical Exp erimen ts In this section we empirically ev aluate the prop osed estimators for tangent spaces and curv ature on simulated datasets. Our theoretical analysis in previous sections shows that the geometry of the underlying manifold can b e reco vered from the deriv ativ es of the manifold-generated noisy density P σ , with error b ounds that dep end explicitly on the noise lev el σ and the sample size N . The n umerical exp eriments complemen t these results b y studying ho w the estimation error scales with σ and N , and by comparing our metho d with existing approac hes for manifold geometry estimation. W e fo cus on manifolds for which tangent spaces and second fundamen tal forms are a v ailable in closed form. In particular, we consider a tw o-dimensional torus em b edded in R 3 (a h yp ersurface) 18 and the t wo-dimensional Cliﬀord torus em b edded in R 4 (a submanifold of co dimension t wo). These examples exhibit nontrivial and spatially v arying curv ature while remaining simple enough to admit accurate numerical ground truth. In Section 7.1 we describ e the exp erimental setup. Sections 7.2– 7.4 rep ort tangent space and curv ature estimation results on the torus and the Cliﬀord torus, and Section 7.5 inv estigates geo desics under the gradient-based density-induced metric. 7.1 Exp erimen tal Setup W e ﬁrst describ e the data generation pro cedure, the estimator construction, the error metrics, and the baseline and hyperparameter choices. 7.1.1 Data Generation Let M ⊂ R D b e one of the manifolds described ab o ve, endo w ed with its induced Riemannian metric and volume measure. F or each c hoice of noise lev el σ and sample size N , we generate noisy data b y ﬁrst sampling i.i.d. p oints x 1 , . . . , x N uniformly on M , and then adding indep enden t Gaussian noise y i = x i + ξ i , ξ i ∼ N (0 , σ 2 I D ) . The resulting observ ations Y = { y i } N i =1 are i.i.d. samples from the noisy distribution P σ , consistent with the p opulation-level analysis of Section 3. T o isolate the eﬀect of the noise lev el σ and the sample size N from the additional error due to the normal oﬀset ∥ v y ∥ , w e ev aluate the geometric estimators at p oin ts in a thin tubular neighborho o d of M . Concretely , we generate an indep enden t set of ev aluation p oin ts W = { w i } ¯ N i =1 suc h that c 1 σ 2 log(1 /σ ) ≤ d ( M , w i ) ≤ c 2 σ 2 log(1 /σ ) , i = 1 , . . . , ¯ N , for ﬁxed constan ts 0 < c 1 < c 2 . Throughout the experiments w e tak e c 1 = 1 / 2, c 2 = 2, and ¯ N = 50. These p oints can b e viewed as pro xies for the output of a manifold reconstruction step. This choice is motiv ated b y existing manifold ﬁtting results that ac hiev e Hausdorﬀ accuracy on the order of O  σ 2 log(1 /σ )  in the small-noise regime (see, e.g., Genov ese et al., 2014; Y ao et al., 2023), so that the ∥ v y ∥ -dep enden t terms in our b ounds decay at a com parable scale as σ → 0. 7.1.2 Kernel Density and Deriv ative Estimators A t the sample level we follow Section 6 and estimate the noisy densit y P σ and its deriv atives using k ernel metho ds. In the n umerical exp eriments w e use the standard Gaussian k ernel K ( u ) = (2 π ) − D/ 2 exp( − 1 2 ∥ u ∥ 2 ) , u ∈ R D . As suggested by Lemma 6.1, for each estimator that inv olves deriv ativ es of order m ∈ { 0 , 1 , 2 , 3 } of P σ , w e choose the bandwidth according to h m = c  log N N  1 / ( D +4+2 m ) . In the experiments, w e c ho ose the constan ts c in a simple, data-dependent wa y follo wing Scott’s rule (Scott, 1992). Sp eciﬁcally , w e take c to b e the a verage standard deviation of the sample Y along the largest d principal components. F or each geometric estimator w e use the bandwidth asso ciated with the highest-order deriv ativ e that it requires: • the gradien t-based metric in Section 7.5 uses h 1 ; 19 • the tangent space estimator and the hypersurface curv ature estimator (which rely on the Hessian of the log-density) use h 2 ; • the general-co dimension curv ature estimator (which additionally uses the third deriv ativ e of the log-densit y) uses h 3 . 7.1.3 Error Metrics F or a point w ∈ W w e denote b y T π ( w ) M the true tangen t space and b y e T w M its estimate. The error in tangen t space estimation is quan tiﬁed b y the sine of the largest principal angle betw een the t wo subspaces, err tan ( w ) := sin Θ {  T π ( w ) M , e T w M  } . F or curv ature w e fo cus on the mean curv ature vector H x ∈ T ⊥ x M , which is a normal ﬁeld on M obtained by a veraging the second fundamental form ov er an orthonormal basis of T x M . Denoting b y e H w the estimator derived from the estimated second fundamental form, w e deﬁne the point wise curv ature error as err curv ( w ) := ∥ H π ( w ) − e H w ∥ 2 . F or each conﬁguration of ( σ, N ) w e summarize the distribution of these errors o ver the ev aluation set W b y b o x plots. 7.1.4 Baselines and Hyp erparameter Selection Our main comparisons are with the following baseline metho ds. • F or tangen t space estimation, we consider LPCA as a classical baseline and the manifold diﬀusion geometry estimator of JI24 (Jones, 2025). In both cases the tangent space at π ( w ) is estimated from observ ations in a neighborho o d of w . F or LPCA we use a kernel-w eighted co v ariance matrix based on the same Gaussian k ernel as ab ov e, with bandwidth h LPCA = cN − 1 / ( d +4) , where the constant c is the same as in our estimator so that the eﬀective lo cal scale is com- parable. JI24 requires little additional tuning, and we follow their standard implemen tation without additional tuning. • F or curv ature estimation, w e compare our metho d with JI24 and with the W eingarten map estimator of CY19 (Cao et al., 2021), which ﬁts the shap e op erator from lo cal cov ariance information. The optimal bandwidth prop osed in CY19 is of the order O ( N − 1 / ( d +4) ), and we therefore set h WM = cN − 1 / ( d +4) , again using the same constan t c as ab ov e. The JI24 curv ature estimator uses its default implemen tation. In eac h exp eriment w e ﬁrst carry out a preliminary comparison that includes all a v ailable baselines. Based on the median error ov er the ev aluation set W , we then fo cus on the stronger baseline in the more detailed comparisons. 20 σ = 0.5 σ = 0.1 σ = 0.01 N = 3 × 10 5 N = 3 × 10 2 N = 3 × 10 4 N = 3 × 10 3 σ = 1.0 σ = 0.2 σ = 0.05 Figure 2: T angent space estimation on a noisy torus. Left: noisy samples Y (gray), together with the true tangen t planes (orange) and the estimated tangent planes (blue) at representativ e p oints on the torus for N = 1000 and σ = 0 . 05. Middle and right: b ox plots of the tangent space error err tan o ver the ev aluation set W as functions of the noise level σ for ﬁxed N = 10 4 and of the sample size N for ﬁxed σ = 0 . 05, resp ectively . N = 300 N = 1000 N = 3000 LPCA Ours σ = 0.5 Ours LPCA σ = 0.01 σ = 0.1 LPCA JI24 Ours Figure 3: Comparison of tangen t space estimators on the torus. Left: box plots of tangen t space errors for N = 1000 and σ = 0 . 05, comparing LPCA, the diﬀusion-geometry estimator JI24, and the prop osed Hessian-based estimator. Middle and righ t: errors of LPCA and of the prop osed estimator as functions of the noise lev el σ for ﬁxed N = 3000 and of the sample size N for ﬁxed σ = 0 . 05, resp ectively . 7.1.5 P arameter Grids T o inv estigate the asymptotic b eha vior in σ and N , we consider tw o families of conﬁgurations. In the ﬁrst we ﬁx N = 10 4 and v ary σ ∈ { 0 . 01 , 0 . 025 , 0 . 05 , 0 . 075 , 0 . 1 , 0 . 5 , 1 . 0 } . In the second we ﬁx σ = 0 . 05 and v ary N ∈ { 300 , 3000 , 30000 , 300000 } . These grids are used in the scaling experiments for our prop osed metho d (Figures 2, 4, and 6). In the baseline comparisons we additionally consider smaller grids, ﬁxing N = 3000 and v arying σ ∈ { 0 . 01 , 0 . 1 , 0 . 5 } , or ﬁxing σ = 0 . 05 and v arying N ∈ { 300 , 1000 , 3000 } , as sp eciﬁed b elo w. 21 σ = 0.5 σ = 0.1 σ = 0.01 N = 3 × 10 5 N = 3 × 10 2 N = 3 × 10 4 N = 3 × 10 3 σ = 1.0 σ = 0.2 σ = 0.05 Figure 4: Mean curv ature estimation on the torus. Left: norm of the true mean curv ature v ectors (orange) and of the estimated mean curv ature vectors (blue) on the torus for N = 1000 and σ = 0 . 05; darker colors indicate larger norm. Middle and right: b ox plots of the curv ature error err curv o ver the ev aluation set W as functions of the noise lev el σ for ﬁxed N = 10 4 and of the sample size N for ﬁxed σ = 0 . 05, resp ectively . 7.2 T angen t Space Estimation on the T orus W e ﬁrst study the tangen t space estimator obtained from the Hessian of the log-densit y , as in Theorem 4.1. F or an ev aluation p oint w ∈ W w e compute the Hessian e H σ ( w ) of the estimated log-densit y and deﬁne e T w M as the span of its d largest eigenv ectors. The underlying manifold is a tw o-dimensional torus em b edded in R 3 . W e in vestigate the de- p endence of the tangen t space error on b oth the noise lev el σ and the sample size N using the parameter grids describ ed in Section 7.1. Figure 2 illustrates the exp erimen tal design and summa- rizes the quantitativ e results. F or mo derate noise levels σ ≤ 0 . 5, the principal angle error remains w ell b elow sin 45 ◦ across the torus. A t the extreme noise level σ = 1, the error b egins to exceed this threshold, indicating that the manifold is no longer clearly identiﬁable from the noisy observ ations. As N increases the error decreases rapidly , and the estimator remains eﬀective ev en at relatively small sample sizes such as N = 300. W e next compare our estimator with the LPCA and JI24 tangent space estimators. F or a represen tative conﬁguration N = 1000, σ = 0 . 05, Figure 3 (left) displays the error distributions of all three metho ds. In this setting LPCA yields noticeably smaller errors than JI24, and we therefore fo cus on LPCA in the more detailed comparisons. The middle and righ t panels of Figure 3 rep ort the errors of LPCA and the prop osed estimator as functions of σ and N , resp ectiv ely . Across these regimes our estimator consistently yields smaller errors than LPCA, esp ecially at smaller sample sizes and for larger noise levels, in line with the higher-order dep endence on σ predicted b y Theorem 4.1. 7.3 Curv ature Estimation on the T orus W e now turn to curv ature estimation on the same tw o-dimensional torus in R 3 , treating it as a h yp ersurface. Using the hypersurface second fundamental form estimator from Theorem 4.4, we obtain an estimate e Π at eac h ev aluation p oint and compute the corresponding mean curv ature v ector e H by a veraging e Π ov er an orthonormal basis of the estimated tangent space. This yields the curv ature error err curv deﬁned in Section 7.1. Figure 4 visualizes the norm of the mean curv ature vector on the torus and summarizes the dep endence of the curv ature error on σ and N . The estimator is accurate for mo derate noise levels 22 CY19 JI24 Ours N = 300 N = 1000 N = 3000 JI24 Ours σ = 0.5 Ours JI24 σ = 0.01 σ = 0.1 Figure 5: Comparison of mean curv ature estimators on the torus. Left: box plots of curv ature errors for N = 1000 and σ = 0 . 05, comparing the W eingarten map estimator of CY19, the diﬀusion- geometry estimator JI24, and the prop osed h yp ersurface estimator. Middle and right: curv ature errors of JI24 and of the prop osed estimator as functions of the noise level σ for ﬁxed N = 3000 and of the sample size N for ﬁxed σ = 0 . 05, resp ectively . and improv es with increasing sample size, but curv ature estimation is noticeably less stable than tangen t space estimation: the v ariance of err curv is larger, and a few outliers with relatively large errors are present. This is consistent with the theoretical b ounds, whic h scale as O ( σ ) with resp ect to the noise lev el for the hypersurface curv ature estimator (Theorem 4.4), compared with O ( σ 2 ) for the tangent space estimator (Theorem 4.1). W e compare the proposed metho d with the W eingarten map estimator of CY19 and the cur- v ature estimator of JI24. F or a represen tative conﬁguration N = 1000, σ = 0 . 05, Figure 5 (left) sho ws that CY19 p erforms substan tially w orse than b oth JI24 and our estimator. W e therefore tak e JI24 as the main curv ature baseline in the subsequen t plots. The middle and right panels of Figure 5 display the curv ature errors of JI24 and of the proposed estimator as functions of N and σ , resp ectiv ely . F or the small-noise lev el σ = 0 . 05 our estimator has a smaller v ariance and a smaller maximum error than JI24 but a slightly larger median error, suggesting that JI24 may con- v erge more rapidly in the asymptotically small-noise regime. On the other hand, as the noise lev el increases b ey ond σ ≈ 0 . 1 our estimator b ecomes substantially more robust than JI24, main taining relativ ely small errors while the p erformance of JI24 deteriorates. 7.4 Curv ature Estimation on the Cliﬀord T orus T o ev aluate the estimator for the second fundamental form of submanifolds with general codi- mension, w e c onsider the tw o-dimensional Cliﬀord torus em b edded in R 4 . In contrast to the h yp ersurface case, the normal space now has dimension tw o, and the second fundamental form has non trivial comp onents in multiple normal directions, making this a more challenging setting. W e use the general-co dimension estimator of Theorem 4.7 to obtain an estimate e Π at eac h ev aluation p oin t on the Cliﬀord torus, and deﬁne the estimated mean curv ature v ector e H and error err curv as in Section 7.3. Figure 6 visualizes the norm of the mean curv ature on the Cliﬀord torus and summarizes the dep endence of the curv ature error on σ and N . The prop osed estimator ac hiev es accurate mean curv ature estimates across a wide range of noise lev els and sample sizes. Although the general- co dimension estimator is more complex than the hypersurface coun terpart, it exhibits signiﬁcan tly b etter con vergence behavior, whic h is consistent with its O ( σ 2 ) con vergence rate. Figure 7 compares the prop osed estimator with CY19 and JI24. While JI24 is more accurate 23 σ = 0.5 σ = 0.1 σ = 0.01 N = 3 × 10 5 N = 3 × 10 2 N = 3 × 10 4 N = 3 × 10 3 σ = 1.0 σ = 0.2 σ = 0.05 Figure 6: Mean curv ature estimation on the Cliﬀord torus in R 4 . Left: norm of the true mean curv ature v ectors (orange) and of the estimated mean curv ature v ectors (blue) on the Cliﬀord torus for N = 1000 and σ = 0 . 05; dark er colors indicate larger norm. Middle and right: b o x plots of the curv ature error err curv o ver the ev aluation set W as functions of the noise level σ for ﬁxed N = 10 4 and of the sample size N for ﬁxed σ = 0 . 05, resp ectively . CY19 JI24 Ours N = 300 N = 1000 N = 3000 CY19 Ours σ = 0.5 Ours CY19 σ = 0.01 σ = 0.1 Figure 7: Comparison of mean curv ature estimators on the Cliﬀord torus. Left: b ox plots of curv ature errors for N = 1000 and σ = 0 . 05, comparing CY19, JI24, and the prop osed general- co dimension estimator. Middle and right: curv ature errors of CY19 and of the prop osed estimator as functions of the noise level σ for ﬁxed N = 3000 and of the sample size N for ﬁxed σ = 0 . 05, resp ectiv ely . than the lo cal ﬁtting metho d CY19 on tw o-dimensional hypersurfaces, it do es not p erform consis- ten tly across the parameter regimes considered in this higher-co dimension setting. Meanwhile, the prop osed estimator consistently outp erforms CY19, with smaller errors and lo wer v ariance across the parameter regimes considered. 7.5 Geo desics of the Gradient-based Metric Finally , we inv estigate the robustness of the density-induced metrics introduced in Section 5. In particular, we fo cus on the gradient-based degenerate metric. Recall that for a noisy distribution P σ w e deﬁne the degenerate metric, g P σ := d (log P σ ) ⊗ d (log P σ ) . Theorem 5.2 sho ws that, for a manifold-generated density , intrinsic geodesics on M satisfy the am bient acceleration equation asso ciated with g P σ up to an O ( σ ) error. 24 N = 300 N = 3000 N = 30000 Figure 8: Geo desics under the gradien t-based metric in the presence of shortcut perturbations. Left and middle: true geo desic (green) on the unit circle S 1 and estimated g P σ -geo desics (orange) b et ween t wo ﬁxed endp oin ts for ﬁxed σ = 0 . 05, N = 300 and N = 3000, respectively . Righ t: Hausdorﬀ distance b et ween the computed geo desics and the true geodesic on S 1 as a function of the sample size N for ﬁxed σ = 0 . 05. T o test the stabilit y of geo desics under perturbations of the data, we in tro duce an artiﬁcial short- cut p erturbation, analogous to the numerical shortcut exp erimen t of T aupin and Chazal (2026). Fix σ = 0 . 05 and let N ∈ { 300 , 3000 , 30000 } . F or each N and eac h rep etition, we ﬁrst randomly select t wo ev aluation p oin ts w 1 , w 2 ∈ W . W e then p erturb the data by inserting √ N additional p oin ts along the straigh t line segment connecting w 1 and w 2 in the ambien t space, and we further p erturb these shortcut points by independent Gaussian noise with the same v ariance σ = 0 . 05. This creates an artiﬁcial density bridge betw een the tw o endp oints. F or the true geo desic, w e pro ject w 1 and w 2 on to the manifold, setting x 1 = π ( w 1 ) and x 2 = π ( w 2 ), and compute the corresp onding intrinsic geo desic b etw een x 1 and x 2 analytically . F or the estimated geodesic, w e start from an initial disc retized path betw een w 1 and w 2 with 100 in termediate p oints and minimize its g P σ -length using gradien t-based optimization while k eeping the endp oin ts ﬁxed. W e then compare the resulting path with the true geo desic by computing their Hausdorﬀ distance in the ambien t Euclidean space. F or each N w e rep eat this pro cedure ov er 50 random choices of ( w 1 , w 2 ) ∈ W × W and summarize the resulting Hausdorﬀ distances by b ox plots. Figure 8 sho ws representativ e examples of the computed geo desics for N = 300 (left) and N = 3000 (middle), together with box plots of the Hausdorﬀ distance as a function of N (right). The distances decrease as the sample size grows and remain small even in the presence of the short- cut points, indicating that the gradien t-based metric g P σ k eeps geo desics close to the underlying manifold and is robust to this type of p erturbation. 8 Pro ofs This section collects pro ofs of the main results in Section 3–6. 8.1 Pro ofs for Section 3 W e ﬁrst prov e the results stated in Section 3. 25 8.1.1 Proof of Theorem 3.1 Pr o of. W rite x 0 := π ( y ). Recall that A y := I T π ( y ) M − ⟨ v y , Π x 0 ⟩ . Using ∥ Π x ∥ op ≤ τ − 1 (Lemma 2.5) and ∥ v y ∥ ≤ τ − ε , λ min ( A y ) ≥ 1 − ∥ v y ∥ /τ ≥ ε/τ =: c ε > 0 , so A y is uniformly p ositive deﬁnite on T ( τ − ε ). Since M is compact, we may choose a ﬁnite smo oth atlas and, on each chart, a smo oth or- thonormal frame e 1 ( x ) , . . . , e d ( x ) for T M . Using this frame, w e identify eac h tangent space T x M lo cally with R d b y u = ( u 1 , . . . , u d ) ∈ R d ← → d X i =1 u i e i ( x ) ∈ T x M . W e work in exp onen tial co ordinates at x 0 and write x 0 ( u ) := exp x 0 ( u ) for u ∈ R d . Cho ose r 0 ∈ (0 , inj M ) suﬃcien tly small so that, uniformly in x 0 ∈ M , the exp onen tial chart is w ell deﬁned on B d (0 , r 0 ) and the expansions in Lemma 2.3 and Lemma 2.4 hold there. Then in these coordinates p det g x 0 ( u ) = 1 + Q g ( y , u ) , | Q g ( y , u ) | ≤ C g ∥ u ∥ 2 , x 0 ( u ) = x 0 + u + 1 2 Π x 0 ( u, u ) + Q e ( y , u ) , ∥ Q e ( y , u ) ∥ ≤ C e ∥ u ∥ 3 . Moreo ver, by the regularit y of exp onen tial map, metric co eﬃcien ts, and pro jection π , we hav e | ∂ α y Q g ( y , u ) | ≤ C α ∥ u ∥ 2 , ∥ ∂ α y Q e ( y , u ) ∥ ≤ C α ∥ u ∥ 3 , | α | ≤ 3 . Expanding ∥ y − x 0 ( u ) ∥ 2 after inserting the expression of x 0 ( u ), and using the orthogonality u ⊥ Π x 0 ( u, u ), one obtains ∥ y − x 0 ( u ) ∥ 2 = ∥ v y ∥ 2 + u ⊤ A y u + R ( y , u ) , | R ( y , u ) | ≤ C  ∥ v y ∥∥ u ∥ 3 + ∥ u ∥ 4  . Since R ( y , u ) is obtained by com bining the cubic remainder Q e ( y , u ) with b ounded co eﬃcients that dep end C 3 -smo othly on y , diﬀerentiating in y preserves the cubic order: | ∂ α y R ( y , u ) | ≤ C α ∥ u ∥ 3 , | α | ≤ 3 . In particular, on the compact set T ( τ − ε ) × B d (0 , r ) we can take the abov e constants uniformly . Indeed, b y shrinking r 0 if necessary , we also hav e | R ( y , u ) | ≤ c ε 4 ∥ u ∥ 2 , ∥ u ∥ ≤ r 0 . Fix an y r ∈ (0 , r 0 ). Let ρ ∈ C k ( R d ) satisfy 0 ≤ ρ ≤ 1 , ρ ( u ) = 1 for ∥ u ∥ ≤ r / 2 , ρ ( u ) = 0 for ∥ u ∥ ≥ r. Deﬁne a cutoﬀ function on M b y χ y ( x ) = ρ (exp − 1 x 0 ( x )) for x ∈ exp x 0 ( B d (0 , r 0 )) , χ y ( x ) = 0 for x / ∈ exp x 0 ( B d (0 , r 0 )) . By the regularit y of the lo cal in verse exp onential map and the pro jection map, ( y, x ) 7→ ρ (exp − 1 π ( y ) ( x )) is C k − 2 on { ( y , x ) : y ∈ T ( τ − ε ) , d M ( x, π ( y )) < r 0 } . Moreo ver, since ρ is compactly supp orted on B d (0 , r ) with r < r 0 , the zero extension of χ y ( x ) deﬁnes C k − 2 function on T ( τ − ε ) × M . By compactness of T ( τ − ε ) × M , w e further hav e a uniform b ound | ∂ α y χ y ( x ) | ≤ C α , | α | ≤ 3 . 26 Using the cutoﬀ function, we decompose P σ ( y ) = 1 V M (2 π σ 2 ) D 2  I 1 ( y , σ ) + I 2 ( y , σ )  , where I 1 ( y , σ ) := Z M χ y ( x ) exp  − ∥ y − x ∥ 2 2 σ 2  dµ ( x ) , I 2 ( y , σ ) := Z M (1 − χ y ( x )) exp  − ∥ y − x ∥ 2 2 σ 2  dµ ( x ) . By construction, the supp ort of χ y is con tained in exp x 0 ( B d (0 , r 0 )), so I 1 can b e written in exp o- nen tial co ordinates at x 0 . Using the expansions in exponential co ordinates and extending the in tegrand b y zero outside B d (0 , r 0 ), w e obtain I 1 ( y , σ ) = Z R d χ y ( x 0 ( u )) exp  − 1 2 σ 2  ∥ v y ∥ 2 + u ⊤ A y u + R ( y , u )  1 + Q g ( y , u )  du. Mak e the change of v ariables z = u/σ , and we write η σ ( z ) = ρ ( σ z ). Then I 1 ( y , σ ) = e −∥ v y ∥ 2 / (2 σ 2 ) σ d Z R d η σ ( z ) e − 1 2 z ⊤ A y z e − R ( y ,σ z ) / (2 σ 2 )  1 + Q g ( y , σ z )  dz . W e decomp ose I 1 as I 1 ( y , σ ) = e −∥ v y ∥ 2 / (2 σ 2 ) σ d  E 0 ( y ) + E 1 ( y , σ ) + E 2 ( y , σ )  , where E 0 ( y ) := Z R d e − 1 2 z ⊤ A y z dz = (2 π ) d/ 2 p det A y , E 1 ( y , σ ) := Z R d η σ ( z ) e − 1 2 z ⊤ A y z  e − R ( y ,σ z ) / (2 σ 2 )  1 + Q g ( y , σ z )  − 1  dz , E 2 ( y , σ ) := Z R d ( η σ ( z ) − 1) e − 1 2 z ⊤ A y z dz . W e ﬁrst estimate E 2 . Since A y ≥ c ε I T π ( y ) M and η σ ( z ) − 1  = 0 implies ∥ z ∥ ≥ r / (2 σ ), we ha ve | E 2 ( y , σ ) | ≤ Z ∥ z ∥≥ r / (2 σ ) e − ( c ε / 2) ∥ z ∥ 2 dz ≤ C e − c/σ 2 . T o estimate deriv ativ es, let φ ( y , z ) := 1 2 z ⊤ A y z . Since ∂ α y A y is uniformly b ounded for | α | ≤ 3, | ∂ α y φ ( y , z ) | ≤ C α ∥ z ∥ 2 , | α | ≤ 3 . By rep eated application of the chain rule to e − 1 2 z ⊤ A y z ,    ∂ α y e − 1 2 z ⊤ A y z    ≤ C α (1 + ∥ z ∥ m α ) e − ( c ε / 2) ∥ z ∥ 2 . Therefore b y | η σ − 1 | ≤ 1 and the dominated conv ergence, | ∂ α y E 2 ( y , σ ) | ≤ C α Z ∥ z ∥≥ r / (2 σ ) (1 + ∥ z ∥ m α ) e − ( c ε / 2) ∥ z ∥ 2 ≤ C α e − c/σ 2 , | α | ≤ 3 . 27 W e next estimate E 1 . Set a ( y , z , σ ) := R ( y , σ z ) 2 σ 2 , then b y the b ounds of R ( y , u ) w e hav e | a ( y , z , σ ) | ≤ C ( σ ∥ v y ∥∥ z ∥ 3 + σ 2 ∥ z ∥ 4 ) , | ∂ α y a ( y , z , σ ) | ≤ C α σ ∥ z ∥ 3 , and | a ( y , z , σ ) | ≤ c ε 8 ∥ z ∥ 2 . W e write G ( y , z , σ ) := e − ϕ ( y ,z )  e − a ( y ,z ,σ )  1 + Q g ( y , σ z )  − 1  . First, b y | e − a (1 + Q g ) − 1 | = | ( e − a − 1)(1 + Q g ) + Q g | ≤ | a | e | a | | 1 + Q g | + | Q g | , w e hav e | e − a ( y ,z ,σ ) (1 + Q g ( y , σ z )) − 1 | ≤ C ( σ ∥ v y ∥ + σ 2 )(1 + ∥ z ∥ m ) e ( c ε / 8) ∥ z ∥ 2 . Com bining the b ound of φ ( y , z ), we get | G ( y , z , σ ) | ≤ C ( σ ∥ v y ∥ + σ 2 )(1 + ∥ z ∥ m ) e − (3 c ε / 8) ∥ z ∥ 2 . No w consider the deriv ativ es of G ( y , z , σ ). By Leibniz’ rule, ev ery term arising in ∂ α y G contains either e − a (1 + Q g ) − 1, a deriv ativ e of a ( y , z , σ ) or a deriv ativ e of Q g ( y , σ z ); b y the b ounds ab ov e, suc h terms are of order at least σ . Therefore, | ∂ α y G ( y , z , σ ) | ≤ C α σ (1 + ∥ z ∥ m α ) e − (3 c ε / 8) ∥ z ∥ 2 . Since the righ t-hand sides are integrable on R d and η σ ( z ) ≤ 1, by the dominated conv ergence it follo ws that | E 1 ( y , σ ) | ≤ C ( σ ∥ v y ∥ + σ 2 ) , | ∂ α y E 1 ( y , σ ) | ≤ C α σ, | α | ≤ 3 . Com bining the b ounds for E 1 and E 2 , w e conclude that I 1 ( y , σ ) = e −∥ v y ∥ 2 / (2 σ 2 ) σ d  (2 π ) d/ 2 p det A y + E ( y , σ )  , where | E ( y , σ ) | ≤ C ( σ ∥ v y ∥ + σ 2 ) , | ∂ α y E ( y , σ ) | ≤ C α σ, | α | ≤ 3 . W e ﬁnally estimate the complement term I 2 ( y , σ ) = Z M (1 − χ y ( x )) exp  − ∥ y − x ∥ 2 2 σ 2  dµ ( x ) . By construction of the cutoﬀ, (1 − χ y ( x ))  = 0 implies d M ( x, π ( y )) ≥ r / 2. Consider the compact set K := { ( y , x ) : y ∈ T ( τ − ε ) , d M ( x, π ( y )) ≥ r / 2 } . Since π ( y ) is the unique nearest p oint on M to y , the contin uous function f ( y , x ) := ∥ y − x ∥ 2 − ∥ v y ∥ 2 28 is strictly p ositive on K . Hence, by compactness, there exists δ ε,r > 0 such that ∥ y − x ∥ 2 ≥ ∥ v y ∥ 2 + δ ε,r whenev er (1 − χ y ( x ))  = 0. Therefore | I 2 ( y , σ ) | ≤ Z M (1 − χ y ( x )) exp  − ∥ y − x ∥ 2 2 σ 2  dµ ( x ) ≤ V M exp  − ∥ v y ∥ 2 + δ ε,r 2 σ 2  . T o estimate deriv ativ es, note that    ∂ α y  (1 − χ y ( x )) exp  − ∥ y − x ∥ 2 2 σ 2      ≤ C α σ − m α exp  − ∥ v y ∥ 2 + δ ε,r 2 σ 2  . Since the ab ov e b ound is integrable, we ma y diﬀerentiate under the integral sign. Then by Leibniz’ rule and ∥ y − x ∥ 2 ≥ ∥ v y ∥ 2 + δ ε,r , | ∂ α y I 2 ( y , σ ) | ≤ C α σ − m α exp  − ∥ v y ∥ 2 + δ ε,r 2 σ 2  . Th us I 2 and all its deriv ativ es up to order three are exponentially negligible compared with the main expansion, and may b e absorb ed in to the remainder term. Finally , putting I 1 and I 2 together. Since A y is uniformly positive deﬁnite, the factor (det A y ) − 1 / 2 is b ounded ab ov e and b elow. Hence the error coming from E ( y , σ ) and I 2 ( y , σ ) may b e normalized b y this principal term. Hence, we obtain P σ ( y ) = 1 V M (2 π σ 2 ) D − d 2 e −∥ v y ∥ 2 / (2 σ 2 ) 1 p det A y  1 + R P ( y , σ )  , where | R P ( y , σ ) | ≤ C ( σ ∥ v y ∥ + σ 2 ) , | ∂ α y R P ( y , σ ) | ≤ C α σ, | α | ≤ 3 . This reﬁned factorization also provides the deriv ative b ounds needed in Corollary 3.2. 8.1.2 Proof of Corollary 3.2 Pr o of. F ollowing the pro of of Theorem 3.1, taking logarithms in the expansion of P σ ( y ) yields log P σ ( y ) = log  V M (2 π σ 2 ) D − d 2  − 1 − ∥ v y ∥ 2 2 σ 2 − 1 2 log det A y + log(1 + R P ( y , σ )) , where | R P ( y , σ ) | ≤ C ( σ ∥ v y ∥ + σ 2 ) , | ∂ α y R P ( y , σ ) | ≤ C α σ, | α | ≤ 3 . Set R ( y , σ ) := log(1 + R P ( y , σ )) . Since 1 + R P sta ys a wa y from zero after decreasing σ 0 if necessary , the same b ound carries o ver to R ( y , σ ) with R ( y , σ ) = O ( σ ∥ v y ∥ + σ 2 ) . F urthermore, by rep eated diﬀerentiation of log(1 + R P ), together with the ab o ve b ounds on ∂ α y R P , it follo ws that | ∂ α y R ( y , σ ) | ≤ C α σ, | α | ≤ 3 , whic h satisﬁes the stated b ounds. 29 8.1.3 Proof of Theorem 3.4 Pr o of. Corollary 3.2 gives the expansion log P σ ( y ) = log  V M (2 π σ 2 ) D − d 2  − 1 − ∥ v y ∥ 2 2 σ 2 − 1 2 log det A y + R ( y , σ ) , where the remainder R satisﬁes, ∥∇ m y R ( y , σ ) ∥ op ≤ C m σ, m ≤ 3 . T aking the ambien t gradien t yields G σ ( y ) = ∇ log P σ ( y ) = − 1 2 σ 2 ∇∥ v y ∥ 2 − 1 2 ∇ log det A y + ∇ R ( y , σ ) . (11) Using Lemma 3.3, for any ω ∈ R D w e hav e ∇ ω v y = ω − A − 1 y ω ⊤ . Since v y ∈ T ⊥ π ( y ) M and A − 1 y ω ⊤ ∈ T π ( y ) M , we obtain D ω ∥ v y ∥ 2 = 2 ⟨ v y , ∇ ω v y ⟩ = 2 ⟨ v y , ω ⟩ . By the deﬁnition of the gradient, this implies ∇∥ v y ∥ 2 = 2 v y . Consequen tly , the ﬁrst term in (11) is exactly − 1 2 σ 2 ∇∥ v y ∥ 2 = − v y σ 2 . Deﬁne F ( y ) := − 1 2 log det A y , y ∈ T ( τ − ε ) . F or x ∈ M we hav e A x = I T x M , hence F | M ≡ 0. Therefore all tangential deriv atives of F v anish on M . Then it suﬃces to compute the normal deriv ativ es. Let u ∈ T ⊥ x M with ∥ u ∥ = 1 and consider the curv e γ ( t ) := x + tu, | t | ≪ 1 . F or | t | small, γ ( t ) ∈ T ( τ − ε ), π ( γ ( t )) = x and v γ ( t ) = tu . Let S u b e the shap e op erator deﬁned at x asso ciated with u . By linearity of the shap e op erator in the normal argument, w e hav e A γ ( t ) = I T x M − ⟨ tu, Π x ⟩ = I T x M − tS u , Therefore F ( γ ( t )) = − 1 2 log det  I T x M − tS u  . Diﬀeren tiating at t = 0 gives d dt F ( γ ( t ))    t =0 = − 1 2 T r( − S u ) = 1 2 T r( S u ) . 30 F or mean curv ature H x = 1 d T r(Π x ), w e hav e T r( S u ) = d ⟨ H x , u ⟩ . Hence d dt F ( γ ( t ))    t =0 = d 2 ⟨ H x , u ⟩ , whic h shows that ∇ F ( x ) = d 2 H x ∈ T ⊥ x M . No w let y ∈ T ( τ − ε ). By the regularit y of A y , F is C 3 on T ( τ − ε ) and ∇ F is Lipsc hitz on T ( τ − ε ). Therefore ∥∇ F ( y ) − ∇ F ( π ( y )) ∥ ≤ C ∥ y − π ( y ) ∥ = C ∥ v y ∥ . Then − 1 2 ∇ log det A y = ∇ F ( y ) = d 2 H π ( y ) + O ( ∥ v y ∥ ) . F or ∇ R , by Corollary 3.2 with m = 1, ∥∇ R ( y , σ ) ∥ ≤ C 1 σ. Substituting the ab ov e gradien ts into (11), w e obtain G σ ( y ) = − v y σ 2 + d 2 H π ( y ) + O ( ∥ v y ∥ + σ ) . 8.1.4 Proof of Theorem 3.5 Pr o of. By Corollary 3.2, log P σ ( y ) = log  V M (2 π σ 2 ) D − d 2  − 1 − ∥ v y ∥ 2 2 σ 2 − 1 2 log det A y + R ( y , σ ) . T aking t wo am bien t deriv atives in y w e obtain H σ ( y ) := ∇ 2 log P σ ( y ) = − 1 2 σ 2 ∇ 2 ∥ v y ∥ 2 − 1 2 ∇ 2 log det A y + ∇ 2 R ( y , σ ) . W e treat the three terms separately . As in the pro of of Theorem 3.4, we ha v e ∇∥ v y ∥ 2 = 2 v y , F or an y ω ∈ R D , Lemma 3.3 yields ∇ ω v y = ω − A − 1 y ω ⊤ . Then, the Jacobian of v y is ∇ v y = I D − A − 1 y P T ( y ) , Therefore − 1 2 σ 2 ∇ 2 ∥ v y ∥ 2 = − 1 σ 2  I D − A − 1 y P T ( y )  = − 1 σ 2 P N ( y ) − 1 σ 2 ( I T π ( y ) M − A − 1 y ) P T ( y ) Let F ( y ) := − 1 2 log det A y . Since F ( y ) is C 3 on T ( τ − ε ), its Hessian is b ounded: ∥∇ 2 F ( y ) ∥ op = O (1) . 31 F or ∇ 2 R , by Corollary 3.2 with m = 2, ∥∇ 2 R ( y , σ ) ∥ op ≤ C ε, 2 σ. Com bining the ab ov e three terms, we can write H σ ( y ) = − 1 σ 2 P N ( y ) − 1 σ 2 ( I T π ( y ) M − A − 1 y ) P T ( y ) + O (1) . 8.1.5 Proof of Corollary 3.6 Pr o of. By Theorem 3.5, we ma y write H σ ( y ) = H 0 ( y ) + E ( y , σ ) , where H 0 ( y ) := − 1 σ 2 P N ( y ) − 1 σ 2 ( I T π ( y ) M − A − 1 y ) P T ( y ) and ∥ E ( y , σ ) ∥ op ≤ C . Then, H 0 ( y ) is blo ck diagonal: H 0 ( y )   T ⊥ π ( y ) M = − σ − 2 I , H 0 ( y )   T π ( y ) M = − σ − 2 ( I T π ( y ) M − A − 1 y ) . Hence the D − d normal eigenv alues of H 0 ( y ) are all equal to − σ − 2 , while its d tangen tial eigen v alues are those of − σ − 2 ( I T π ( y ) M − A − 1 y ). Now recall that A y = I T π ( y ) M − ⟨ v y , Π π ( y ) ⟩ . By y ∈ T ( τ − ε ) and the reac h b ound, ∥⟨ v y , Π π ( y ) ⟩∥ op ≤ ∥ v y ∥ τ ≤ τ − ε τ = 1 − ε τ < 1 . Therefore A y is uniformly inv ertible on T ( τ − ε ). Moreov er, every eigenv alue λ of A y satisﬁes λ ≤ 1 + ∥⟨ v y , Π π ( y ) ⟩∥ op ≤ 2 − ε τ , so ev ery eigenv alue of A − 1 y is b ounded b elow by λ min ( A − 1 y ) ≥ 1 2 − ε/τ . If µ is a tangential eigen v alue of H 0 ( y ), then µ = − σ − 2 (1 − λ − 1 ) , and th us   µ + σ − 2   = σ − 2 λ − 1 ≥ 1 2 − ε/τ σ − 2 . 32 It follo ws that the tangen tial and normal sp ectral clusters of H 0 ( y ) are separated by at least c ε, 0 σ − 2 , c ε, 0 := 1 2 − ε/τ . In particular, every tangen tial eigenv alue is strictly larger than ev ery normal eigenv alue, so the tangen tial cluster coincides with the d largest eigenv alues of H 0 ( y ). Finally , each eigen v alue of H σ ( y ) diﬀers from the corresp onding eigen v alue of H 0 ( y ) by at most ∥ E ( y , σ ) ∥ op ≤ C . Hence the gap b et ween the d largest eigen v alues of H σ ( y ) and the remaining D − d eigenv alues is b ounded b elo w by c ε, 0 σ − 2 − 2 C. After decreasing σ 0 if necessary , this low er b ound is at least c ε σ − 2 for some constant c ε > 0 dep ending only on ε and M . This prov es the claim. 8.2 Pro ofs for Section 4 W e next prov e the results in Section 4. 8.2.1 Proof of Theorem 4.1 Pr o of. W rite x := π ( y ). Recall that H 0 ( y ) := − 1 σ 2 P N ( y ) − 1 σ 2 ( I T x M − A − 1 y ) P T ( y ) , where H σ ( y ) = H 0 ( y ) + R σ ( y ) , ∥ R σ ( y ) ∥ op ≤ C . By Corollary 3.6, b oth H 0 ( y ) and H σ ( y ) ha ve a sp ectral gap of size at least c ε σ − 2 b et ween the largest d eigenv alues and the remaining D − d eigenv alues. The op erator H 0 ( y ) is block diagonal with resp ect to R D = T x M ⊕ T ⊥ x M . Its sp ectral pro jector asso ciated with the d largest eigenv alues is therefore exactly P T ( y ). Thus, applying the Da vis–Kahan theorem giv es sin { Θ( b T y M , T x M ) } ≤ C ∥ R σ ( y ) ∥ op c ε σ − 2 ≤ C ′ σ 2 . Next consider the consistency of b d . Let η 1 ≥ · · · ≥ η d > η d +1 = · · · = η D b e the eigen v alues of H 0 ( y ), and λ 1 ≥ · · · ≥ λ d > λ d +1 = · · · = λ D b e the eigen v alues of H σ ( y ). Since A y = I T x M − ⟨ v y , Π x ⟩ and ∥ Π x ∥ op ≤ τ − 1 , we ha ve ∥ I T x M − A − 1 y ∥ op = O ( ∥ v y ∥ ) uniformly on T ( τ − ε ). Th us sup 1 ≤ i ≤ d | η i | ≤ C ∥ v y ∥ σ 2 . Therefore, sup k  = d | η k − η k +1 | ≤ C ∥ v y ∥ σ 2 , | η d − η d +1 | ≥ c ε σ − 2 . By W eyl’s inequality , | λ k − η k | ≤ C, 1 ≤ k ≤ D , 33 so sup k  = d | λ k − λ k +1 | ≤ C 1 ∥ v y ∥ σ 2 + C 2 , whereas | λ d − λ d +1 | ≥ c ε σ − 2 − C 3 . Hence, along an y sequence such that σ → 0 and ∥ v y ∥ → 0, the gap at k = d is even tually strictly larger than all other consecutive gaps, i.e., b d = arg max k | λ k − λ k +1 | = d. 8.2.2 Proof of Theorem 4.2 Pr o of. F or x ∈ M , the estimator in the theorem is b Π x ( u, v ) = 2 d g ( u, v ) G σ ( x ) , while the true second fundamental form is Π x ( u, v ) = g ( u, v ) H x , b ecause M is totally umbilical. By Theorem 3.4, for p oints x ∈ M w e ha ve G σ ( x ) = d 2 H x + R σ ( x ) , ∥ R σ ( x ) ∥ ≤ C σ. Therefore b Π x ( u, v ) − Π x ( u, v ) = g ( u, v )  2 d G σ ( x ) − H x  = g ( u, v ) 2 d R σ ( x ) . T aking the op erator norm ov er the unit tangent vectors yields ∥ b Π x − Π x ∥ op ≤ 2 d ∥ R σ ( x ) ∥ ≤ C σ. 8.2.3 Proof of Theorem 4.4 Pr o of. W rite x := π ( y ) and r := ∥ v y ∥ . W e deﬁne tw o vector ﬁelds on U σ b y N ( y ) := P N ( y ) G σ ( y ) , T ( y ) := P T ( y ) G σ ( y ) , so that G σ ( y ) = N ( y ) + T ( y ) . Recall from the pro of of Theorem 3.4 that G σ ( y ) = − v y σ 2 + ∇ F ( y ) + ∇ R ( y , σ ) , F ( y ) := − 1 2 log det A y , where ∇ F ( y ) = d 2 H π ( y ) + O ( ∥ v y ∥ ) , ∥∇ R ( y , σ ) ∥ = O ( σ ) . 34 Since v y , H π ( y ) ∈ T ⊥ π ( y ) M , we kno w ∥T ( y ) ∥ = O ( r + σ ) , (12) and there exists C > 0 such that ∥N ( y ) ∥ ≥ r σ 2 − C for all y ∈ U σ , 0 < σ ≤ σ 0 . (13) W e show that ∥N ( y ) ∥ is low er bounded on U σ . Let C 0 ≥ C + c 0 2 . Since ∥G σ ( y ) ∥ ≥ c 0 uniformly on U σ , when r ≤ C 0 σ 2 w e ha v e ∥N ( y ) ∥ ≥ ∥G σ ( y ) ∥ − ∥T ( y ) ∥ ≥ c 0 / 2 for suﬃcien tly small σ 0 ; F or r > C 0 σ 2 , ∥N ( y ) ∥ ≥ c 0 / 2 b y (13). Therefore we ha ve ∥N ( y ) ∥ ≥ c 0 / 2 uniformly on U σ . F or an y C 1 v ector ﬁeld V on U σ with V ( y )  = 0, deﬁne Q y ( V )( u, v ) := − ⟨∇ A y u V ( y ) , v ⟩ ∥ V ( y ) ∥ 2 V ( y ) . Since M is a hypersurface and N ( y ) ∈ T ⊥ x M \ { 0 } , Lemma 4.3 yields Π x = Q y ( N ) . (14) W e will prov e ∥ b Π y − Q y ( G σ ) ∥ op = O ( r + σ ) , ∥ Q y ( G σ ) − Q y ( N ) ∥ op = O ( r + σ ) . (15) T ogether with (14), this implies the theorem. W e ﬁrst compare Q y ( G σ ) and Q y ( N ). Consider the deriv ative of T along tangent directions. W rite W ( y ) := P T ( y ) ∇ F ( y ) , S ( y ) := P T ( y ) ∇ R ( y , σ ) , so that T = W + S . W e ﬁrst treat W . F or every x ′ ∈ M , ∇ F ( x ′ ) = d 2 H x ′ ∈ T ⊥ x ′ M , hence W ( x ′ ) = P T ( x ′ ) ∇ F ( x ′ ) = 0 , x ′ ∈ M . Th us W | M ≡ 0, which implies ⟨∇ u W ( x ) , v ⟩ = 0 . Using ⟨∇ A y u W ( y ) , v ⟩ = ⟨ ( ∇ W ( y ) − ∇ W ( x )) A y u, v ⟩ + ⟨∇ W ( x )( A y u − u ) , v ⟩ + ⟨∇ u W ( x ) , v ⟩ , then the last term v anishes. Since W is C 2 on the compact set under consideration, ∇ W is Lipsc hitz, and th us ∥∇ W ( y ) − ∇ W ( x ) ∥ op = O ( r ) , and A y is uniformly b ounded on T x M , the ﬁrst term is O ( r ). Moreov er, since A y u − u = − ( ∇ u v y ) ⊤ and Π is uniformly b ounded on M , ∥ A y u − u ∥ = O ( r ) ∥ u ∥ . As ∇ W is uniformly b ounded on U σ , the second term is also O ( r ). Hence   ⟨∇ A y u W ( y ) , v ⟩   = O ( r ) . (16) Since S = P T ∇ R , the pro duct rule giv es ∇ S = ( ∇ P T ) ∇ R + P T ∇ 2 R. 35 Because P T and ∇ P T are uniformly b ounded on U σ , while b y Corollary 3.2 ∥∇ R ( y , σ ) ∥ = O ( σ ) and ∥∇ 2 R ( y , σ ) ∥ op = O ( σ ), we hav e   ⟨∇ A y u S ( y ) , v ⟩   = O ( σ ) . (17) Com bining (16) and (17), we obtain   ⟨∇ A y u T ( y ) , v ⟩   = O ( r + σ ) . (18) Set β N ( u, v ) := ⟨∇ A y u N ( y ) , v ⟩ , β T ( u, v ) := ⟨∇ A y u T ( y ) , v ⟩ . Then Q y ( G σ )( u, v ) − Q y ( N )( u, v ) = − β T ( u, v ) ∥G σ ( y ) ∥ 2 G σ ( y ) − β N ( u, v )  G σ ( y ) ∥G σ ( y ) ∥ 2 − N ( y ) ∥N ( y ) ∥ 2  . By (18) and the assumption ∥G σ ( y ) ∥ ≥ c 0 on U σ ,    β T ( u, v ) ∥G σ ( y ) ∥ 2 G σ ( y )    ≤ | β T ( u, v ) | ∥G σ ( y ) ∥ = O ( r + σ ) . F or the second term, note that (14) implies Q y ( N )( u, v ) = − β N ( u, v ) ∥N ( y ) ∥ 2 N ( y ) = Π x ( u, v ) , hence | β N ( u, v ) | ∥N ( y ) ∥ ≤ ∥ Π x ∥ op . Since ∥G σ ∥ ≥ c 0 and ∥N ( y ) ∥ ≥ c 0 / 2 uniformly on U σ , the map a 7− → a ∥ a ∥ 2 is uniformly Lipschitz on the set { a ∈ R D : ∥ a ∥ ≥ c 0 / 2 } . Thus    G σ ( y ) ∥G σ ( y ) ∥ 2 − N ( y ) ∥N ( y ) ∥ 2    = O ( ∥T ( y ) ∥ ) = O ( r + σ ) . Hence the second term is also O ( r + σ ). W e conclude that ∥ Q y ( G σ ) − Q y ( N ) ∥ op = O ( r + σ ) . (19) Next, w e compare b Π y with Q y ( G σ ). W e hav e b Π y ( u, v ) − Q y ( G σ )( u, v ) = − ⟨H σ ( y )( u − A y u ) , v ⟩ ∥G σ ( y ) ∥ 2 G σ ( y ) . Hence ∥ b Π y − Q y ( G σ ) ∥ op ≤ |⟨H σ ( y )( u − A y u ) , v ⟩| ∥G σ ( y ) ∥ . (20) Let w := u − A y u . Then w ∈ T x M and, as ab ov e, ∥ w ∥ = O ( r ) ∥ u ∥ . By Theorem 3.5, w e can write H σ ( y ) = − 1 σ 2 P N ( y ) + O  r σ 2  P T ( y ) + O (1) . 36 Th us |⟨H σ ( y ) w , v ⟩| ≤ C r σ 2 ∥ w ∥ + C ∥ w ∥ . Using ∥ w ∥ = O ( r ) ∥ u ∥ , we obtain |⟨H σ ( y )( u − A y u ) , v ⟩| ≤ C  r 2 σ 2 + r  . (21) By Theorem 3.4, there exists C 1 > 0 such that ∥G σ ( y ) ∥ ≥ r σ 2 − C 1 for all y ∈ U σ , 0 < σ ≤ σ 0 . (22) W e distinguish tw o cases. If r ≤ 2 C 1 σ 2 , then r = O ( σ 2 ), and by (21) together with the assumption ∥G σ ( y ) ∥ ≥ c 0 on U σ , ∥ b Π y − Q y ( G σ ) ∥ op ≤ C  r 2 σ 2 + r  = O ( r ) = O ( r + σ ) . If r > 2 C 1 σ 2 , then (22) yields ∥G σ ( y ) ∥ ≥ 1 2 r σ 2 , and therefore, by (20) and (21), ∥ b Π y − Q y ( G σ ) ∥ op ≤ C r 2 /σ 2 + r r /σ 2 = O ( r + σ 2 ) = O ( r + σ ) . In either case, ∥ b Π y − Q y ( G σ ) ∥ op = O ( r + σ ) . (23) Com bining (14), (19), and (23) yields ∥ b Π y − Π x ∥ op ≤ ∥ b Π y − Q y ( G σ ) ∥ op + ∥ Q y ( G σ ) − Q y ( N ) ∥ op = O ( r + σ ) = O ( ∥ v y ∥ + σ ) . This pro ves the claim. 8.2.4 Proof of Corollary 4.5 Pr o of. Let L σ,c b e a level set of P σ suc h that L σ,c ∩ M  = ∅ , and c ho ose x ∈ L σ,c ∩ M . W e ﬁrst sho w that L σ,c ⊂ T ( τ − ε ) for all suﬃciently small σ . By Theorem 3.1, uniformly for x ∈ M , P σ ( x ) = 1 V M (2 π σ 2 ) D − d 2  1 + O ( σ 2 )  . On the other hand, if y / ∈ T ( τ − ε ), then ∥ y − z ∥ ≥ τ − ε for ev ery z ∈ M , so P σ ( y ) = 1 V M (2 π σ 2 ) D/ 2 Z M exp  − ∥ y − z ∥ 2 2 σ 2  dµ ( z ) ≤ 1 (2 π σ 2 ) D/ 2 exp  − ( τ − ε ) 2 2 σ 2  . Hence sup y / ∈ T ( τ − ε ) P σ ( y ) P σ ( x ) ≤ C σ − d exp  − ( τ − ε ) 2 2 σ 2  → 0 . Therefore P σ ( y ) < P σ ( x ) for all suc h y when σ is small enough, which implies ev ery y ∈ L σ,c b elongs to T ( τ − ε ) and L σ,c ⊂ T ( τ − ε ). 37 No w ﬁx y ∈ L σ,c and write r := d ( y , M ) = ∥ v y ∥ . Since P σ ( y ) = P σ ( x ), Corollary 3.2 gives 0 = log P σ ( y ) − log P σ ( x ) = − r 2 2 σ 2 − 1 2 log det A y + R ( y , σ ) − R ( x, σ ) , where A y = I T π ( y ) M − ⟨ v y , Π π ( y ) ⟩ , | R ( y , σ ) | ≤ C ( σ r + σ 2 ) , | R ( x, σ ) | ≤ C σ 2 . Moreo ver, by Lemma 2.5, ∥⟨ v y , Π π ( y ) ⟩∥ op ≤ ∥ v y ∥∥ Π π ( y ) ∥ op ≤ r /τ ≤ ( τ − ε ) /τ < 1 . Th us A y is uniformly p ositive deﬁnite on T ( τ − ε ), and since log det is smo oth on this set, | log det A y | ≤ C ∥ A y − I T π ( y ) M ∥ op ≤ C r. Therefore r 2 2 σ 2 ≤ C r + C σ r + C σ 2 ≤ C ( r + σ 2 ) for all suﬃciently small σ . Multiplying by σ 2 yields r 2 ≤ C σ 2 r + C σ 4 . This implies r ≤ C σ 2 . Since all constants abov e are uniform ov er y ∈ L σ,c and ov er all level sets with L σ,c ∩ M  = ∅ , taking the suprem um ov er y ∈ L σ,c pro ves the claim. 8.2.5 Proof of Theorem 4.7 Pr o of. W rite x := π ( y ). Recall that P T ( y ) denotes the orthogonal pro jection onto T x M , while b P T ( y ) denotes the sp ectral pro jector onto the d largest eigen v alues of H σ ( y ). By Theorem 4.1, ∥ b P T ( y ) − P T ( y ) ∥ op ≤ C σ 2 . By Lemma 4.6 applied at x we ha ve, Π x ( u, v ) = ∇ u P T ( x ) v . The estimator in the theorem is b Π y ( u, v ) = ∇ u b P T ( y ) v . W e decomp ose b Π y ( u, v ) − Π x ( u, v ) =  ∇ u b P T ( y ) − ∇ u P T ( y )  v +  ∇ u P T ( y ) − ∇ u P T ( x )  v . F or the second term, since P T ( y ) is C k − 1 on T ( τ − ε ), ∇ P T ( y ) is Lipschitz on T ( τ − ε ), i.e., ∥∇ u P T ( y ) − ∇ u P T ( x ) ∥ op ≤ C ∥ v y ∥ . Next w e consider the ﬁrst term. Let P N ( y ) := I D − P T ( y ) , H 0 ( y ) := − σ − 2 P N ( y ) − σ − 2 ( I T x M − A − 1 y ) P T ( y ) . 38 Then P T ( y ) is exactly the sp ectral pro jector of H 0 ( y ) asso ciated with the largest d eigenv alues. Let e H 0 ( y ) := σ 2 H 0 ( y ) , e H σ ( y ) := σ 2 H σ ( y ) . By the uniform boundedness of A − 1 y on T ( τ − ε ) and the expansion of H σ in Theorem 3.5, the sp ectra of e H 0 ( y ) and e H σ ( y ) remain in a compact in terv al indep endent of σ . Moreov er, b y Corollary 3.6, their tangential and normal spectral clusters are separated by a uniform p ositive gap. Hence one ma y c ho ose a ﬁxed p ositiv ely orien ted con tour e Γ that encloses the tangen tial cluster and excludes the normal cluster for b oth e H 0 ( y ) and e H σ ( y ). Setting Γ := σ − 2 e Γ , w e obtain length(Γ) ≤ C 1 σ − 2 , dist  Γ , sp ec( H 0 ( y ))  ≥ C 2 σ − 2 , dist  Γ , sp ec( H σ ( y ))  ≥ C 3 σ − 2 . Therefore sup z ∈ Γ ∥ ( H 0 ( y ) − z I ) − 1 ∥ op + sup z ∈ Γ ∥ ( H σ ( y ) − z I ) − 1 ∥ op ≤ C σ 2 . F or H with sp ectrum separated b y Γ, deﬁne the corresp onding Riesz pro jector by P [ H ] = − 1 2 π i Z Γ ( H − z I ) − 1 dz . Its deriv ativ e in the direction K is D P [ H ]( K ) = 1 2 π i Z Γ ( H − z I ) − 1 K ( H − z I ) − 1 dz . Therefore, ∇ u b P T ( y ) = D P [ H σ ( y )]  ∇ u H σ ( y )  , ∇ u P T ( y ) = D P [ H 0 ( y )]  ∇ u H 0 ( y )  . Set ∆ H := H σ ( y ) − H 0 ( y ) , ∆ K := ∇ u H σ ( y ) − ∇ u H 0 ( y ) , K 0 := ∇ u H 0 ( y ) . Then ∇ u b P T ( y ) − ∇ u P T ( y ) = D P [ H σ ( y )](∆ K ) +  D P [ H σ ( y )] − D P [ H 0 ( y )]  ( K 0 ) . Let R σ ( z ) = ( H σ ( y ) − z I ) − 1 and R 0 ( z ) = ( H 0 ( y ) − z I ) − 1 . By the ab ov e choice of Γ, sup z ∈ Γ ∥ R σ ( z ) ∥ op , sup z ∈ Γ ∥ R 0 ( z ) ∥ op ≤ C σ 2 . Hence ∥ D P [ H σ ( y )](∆ K ) ∥ op ≤ C length(Γ) sup z ∈ Γ ∥ R σ ( z ) ∥ 2 op ∥ ∆ K ∥ op ≤ C σ 2 ∥ ∆ K ∥ op . Moreo ver, by the resolv ent iden tit y , R σ ( z ) − R 0 ( z ) = − R σ ( z )∆ H R 0 ( z ) , so that sup z ∈ Γ ∥ R σ ( z ) − R 0 ( z ) ∥ op ≤ C σ 4 ∥ ∆ H∥ op . 39 Therefore,    D P [ H σ ( y )] − D P [ H 0 ( y )]  ( K 0 )   op ≤ C sup z ∈ Γ ( ∥ R σ ( z ) ∥ + ∥ R 0 ( z ) ∥ ) sup z ∈ Γ ∥ R σ ( z ) − R 0 ( z ) ∥ op ∥ K 0 ∥ op ≤ C σ 4 ∥ ∆ H∥ op ∥ K 0 ∥ op . W e no w estimate the three quan tities ab ov e. First, by Theorem 3.5, ∥ ∆ H∥ op = ∥H σ ( y ) − H 0 ( y ) ∥ op = O (1) . Next, K 0 = ∇ u H 0 ( y ) = − σ − 2  ∇ u P N ( y ) + ∇ u  ( I T x M − A − 1 y ) P T ( y )   , and since ∇ P T , ∇ P N , and ∇ A − 1 y are uniformly b ounded on T ( τ − ε ), ∥ K 0 ∥ op ≤ C σ − 2 . Finally , w e claim that ∥ ∆ K ∥ op ≤ C . F rom the expansion in the pro of of Theorem 3.5, H σ ( y ) = − σ − 2 P N ( y ) − σ − 2 ( I T x M − A − 1 y ) P T ( y ) − 1 2 ∇ 2 log det A y + ∇ 2 R ( y , σ ) . Th us ∆ H = − 1 2 ∇ 2 log det A y + ∇ 2 R ( y , σ ) . Diﬀeren tiating in the tangen tial direction u ∈ T x M , the ab ov e terms satisfy ∥∇ u ∇ 2 log det A y ∥ op ≤ C , ∥∇ u ∇ 2 R ( y , σ ) ∥ op ≤ C σ, uniformly on T ( τ − ε ), by smo othness of A y and Corollary 3.2. Therefore, ∥ ∆ K ∥ op = ∥∇ u H σ ( y ) − ∇ u H 0 ( y ) ∥ op ≤ C . Substituting the b ounds for ∆ H , ∆ K , and K 0 in to the previous estimates, w e obtain ∥∇ u b P T ( y ) − ∇ u P T ( y ) ∥ op ≤ C σ 2 . Com bining this with the estimate for the second term yields ∥ b Π y − Π x ∥ op ≤ C ( ∥ v y ∥ + σ 2 ) . Since x = π ( y ), this prov es the claim. 8.3 Pro ofs for Section 5 Here w e prov e the results in Section 5. 40 8.3.1 Proof of Theorem 5.1 Pr o of. W rite P := P σ . Consider the conformal deformation g P = e 2 f ( P ) g E , where f : R + → R is to b e determined. W e write Π E and Π P for the second fundamental forms of M with resp ect to g E and g P , resp ectively , and H for the mean curv ature vector with respect to g E . It is standard that under a conformal c hange, the second fundamental form transforms as Π P x ( u, v ) = Π E x ( u, v ) − g E ( u, v )  ∇ f ( P ( x ))  ⊥ , u, v ∈ T x M , (24) where ∇ and ( · ) ⊥ are taken with respect to the ambien t Euclidean structure. Since M is totally um bilical, there is a mean curv ature v ector H x ∈ T ⊥ x M such that Π E x ( u, v ) = g E ( u, v ) H x . Substituting the umbilical form into (24) giv es Π P x ( u, v ) = g E ( u, v )  H x − ( ∇ f ( P ( x ))) ⊥  . The inclusion ( M , g )  → ( R D , g P ) is totally geo desic if and only if Π P x ≡ 0 for all x ∈ M , th us H x = ( ∇ f ( P ( x ))) ⊥ , x ∈ M . Since f dep ends on x only through P ( x ), the chain rule yields ∇ f ( P ( x )) = f ′ ( P ( x )) ∇ P ( x ) = f ′ ( P ( x )) P ( x ) ∇ log P ( x ) . By Theorem 3.4, at p oints x ∈ M the estimator satisﬁes ∇ log P ( x ) = d 2 H x + O ( σ ) . Therefore, w e obtain the scalar relation H x = f ′ ( P ( x )) P ( x ) d 2 H x + O ( σ ) . If H x = 0, the leading term matching condition is automatic. Otherwise, matching the leading term requires d 2 f ′ ( P ( x )) P ( x ) = 1 . Th us we c ho ose f ( P ) = 2 d log P + C . T aking C = 0 yields f ( P ) = 2 d log P, g P = e 2 f ( P ) g E = P 4 /d g E . With this choice, H x − ( ∇ f ( P ( x ))) ⊥ = O ( σ ) , and hence Π P x ( u, v ) = O ( σ ) g E ( u, v ) . Therefore, for every x ∈ M , ∥ Π P x ∥ op ≤ C σ, and hence the inclusion ( M , g )  → ( R D , g P ) is asymptotically totally geo desic as σ → 0. 41 8.3.2 Proof of Theorem 5.2 Pr o of. W rite P := P σ . Recall that g P = d (log P ) ⊗ d (log P ) , ( g P ) ij = ∂ i log P ∂ j log P. Since g P is rank-one, it is degenerate and do es not deﬁne a genuine Riemannian metric. W e therefore do not inv ok e the Levi–Civita connection. Instead, using the Mo ore–P enrose pseudo-inv erse ( g P ) ij MP = 1 ∥∇ log P ∥ 4 ∂ i log P ∂ j log P, w e deﬁne the asso ciated Christoﬀel-t yp e co eﬃcient ﬁeld e Γ m ij := ( g P ) mk MP e Γ ij,k , e Γ ij,k := 1 2  ∂ j ( g P ) ik + ∂ i ( g P ) j k − ∂ k ( g P ) ij  . A direct computation gives e Γ ij,k = 1 2  ∂ j ∂ i log P ∂ k log P + ∂ i log P ∂ j ∂ k log P + ∂ i ∂ j log P ∂ k log P + ∂ j log P ∂ i ∂ k log P − ∂ k ∂ i log P ∂ j log P − ∂ i log P ∂ k ∂ j log P  = ( ∂ i ∂ j log P ) ∂ k log P, and hence e Γ m ij = 1 ∥∇ log P ∥ 4 ∂ m log P ∂ k log P ( ∂ i ∂ j log P ) ∂ k log P = 1 ∥∇ log P ∥ 2 ( ∂ i ∂ j log P ) ∂ m log P. Accordingly , the induced acceleration equation of ¨ γ k ( t ) = − e Γ k ij ˙ γ i ( t ) ˙ γ j ( t ) is ¨ γ ( t ) = − 1 ∥∇ log P ( γ ( t )) ∥ 2 ∇ 2 log P ( γ ( t ))( ˙ γ ( t ) , ˙ γ ( t )) ∇ log P ( γ ( t )) . (25) No w let γ b e a geo desic on ( M , g ), viewed as a curv e in R D . Its Euclidean acceleration is ¨ γ ( t ) = Π γ ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) . By Theorem 4.4, in the hypersurface case, Π γ ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) = − 1 ∥∇ log P ( γ ( t )) ∥ 2 ∇ 2 log P ( γ ( t ))( ˙ γ ( t ) , ˙ γ ( t )) ∇ log P ( γ ( t )) + O ( σ ) , uniformly along M . Comparing w ith (25), w e conclude that γ satisﬁes the induced acceleration equation up to an O ( σ ) error. This prov es the claim. Ac kno wledgmen ts J.C. is partly supp orted by the Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS). R.L. is a researc h fello w supported b y the Singap ore Ministry of Education Tier 2 grant A- 8001562-00-00 at the National Universit y of Singap ore. Z.Y. has b een supp orted b y the Singap ore Ministry of Education Tier 2 gran t A-8001562-00-00 and the Tier 1 gran t (A-8004146-00-00 and A-8002931-00-00) at the National Universit y of Singap ore. 42 A Pro of of Lemmas in Section 2 A.1 Pro of of Lemma 2.3 Pr o of. Let x ∈ M and γ u : [0 , 1] → M b e the geodesic in M with initial condition γ u (0) = x , γ ′ u (0) = u . Then exp x ( u ) = γ u (1). Consider the T aylor expansion of exp x around the origin of T x M , it will b e written as exp x ( u ) = x + I x ( u ) + 1 2 Q x ( u ) + . . . , where I x , Q x are resp ectiv ely linear and bilinear map in T x M . Therefore γ u ( t ) = x + I x ( u ) t + 1 2 Q x ( u ) t 2 + O ( t 3 ) . Hence, I x ( u ) = γ ′ u (0) = u and Q x ( u ) = γ ′′ u (0). Since γ is a geo desic, we ha ve ∇ M γ ′ u (0) γ ′ u (0) = 0, whic h implies γ ′′ u (0) = Π x ( u, u ). Equiv alently , exp x ( u ) = x + u + 1 2 Π x ( u, u ) + O ( ∥ u ∥ 3 ) . A.2 Pro of of Lemma 2.4 Pr o of. Let x ∈ M , and work in exp onential coordinates centered at x . Since g ij | x = δ ij , D exp x is the identit y map at the origin of T x M , whic h implies ∂ ∂ u i = D (exp x ) 0 ( u i ) = u i . The geo desic parametrization in the exp onential co ordinates is giv en by c ( t ) = t ( u 1 , u 2 , . . . , u d ) . Putting this into the geo desic equation, w e get for an y k , 0 = ¨ c k ( t ) + ˙ c i ( t ) ˙ c j ( t )Γ k ij (exp( c ( t ))) = u i u j Γ k ij (exp( c ( t ))) . Let t = 0, we ha ve the quadratic form u i u j Γ k ij ( x ) = 0 for any u i u j and symmetric Γ k ij . Therefore Γ k ij (0) = 0 for all i, j, k , whic h implies ∂ i g j k + ∂ j g ik − ∂ k g ij = 0 at the origin. So ∂ k g ij | x = 0 follo ws from ∂ k g ij = 1 2 ( ∂ k g ij + ∂ j g ik − ∂ i g j k + ∂ k g ij + ∂ i g j k − ∂ j g ik ) = 0 . Hence p det g ij ( x ) = 1 and its ﬁrst deriv ativ e v anish. T aking the T aylor expansion at x w e get q det g ij ( x, u ) = 1 + O ( ∥ u ∥ 2 ) . Since M is compact and the metric is smo oth, the second deriv ativ es of g ij in these co ordinates are uniformly b ounded in x ∈ M , so the O ( ∥ u ∥ 2 ) term is uniform in x . 43 B Pro of of Lemmas in Section 3 B.1 Pro of of Lemma 3.3 Pr o of. Fix y ∈ T ( τ ) and write x := π ( y ) ∈ M and v := v y . Let S v : T x M → T x M b e the shape op erator asso ciated with the normal vector v , so that ⟨ S v ( u ) , w ⟩ = ⟨ Π x ( u, w ) , v ⟩ , S v ( u ) = − ( ∇ u v ) ⊤ , for u, w ∈ T x M . Then w e hav e A y = I T x M − ⟨ v y , Π x ⟩ = I T x M − S v on T x M . Let ω ∈ R D b e arbitrary , and choose a smo oth curve γ : ( − δ , δ ) → T ( τ ) suc h that γ (0) = y and γ ′ (0) = ω . Set π ( t ) := π ( γ ( t )) ∈ M , v ( t ) := v γ ( t ) = γ ( t ) − π ( t ) ∈ T ⊥ π ( t ) M . Then for any tangent v ector ﬁeld Y ( t ) ∈ T π ( t ) M along π ( t ) w e hav e ⟨ v ( t ) , Y ( t ) ⟩ = 0 , ∀ t. Diﬀeren tiating at t = 0 gives 0 = d dt ⟨ v ( t ) , Y ( t ) ⟩    t =0 = ⟨ v ′ (0) , Y (0) ⟩ + ⟨ v (0) , ∇ π ′ (0) Y ⟩ . Here v ′ (0) = ∇ ω v and π ′ (0) = ∇ ω π ( y ) ∈ T x M . Using the deﬁnition of the shape op erator, we hav e ⟨ v (0) , ∇ π ′ (0) Y ⟩ = ⟨ S v ( π ′ (0)) , Y (0) ⟩ . Th us ⟨ v ′ (0) , Y (0) ⟩ = ⟨ ( ∇ ω v y ) ⊤ , Y (0) ⟩ = −⟨ S v ( ∇ ω π ( y )) , Y (0) ⟩ for all Y (0) ∈ T x M , and therefore ( ∇ ω v y ) ⊤ = − S v ( ∇ ω π ( y )) . (26) On the other hand, since v y = y − π ( y ), we also ha ve ∇ ω v y = ω − ∇ ω π ( y ) . (27) T aking the tangential comp onent in (27) yields ( ∇ ω v y ) ⊤ = ω ⊤ − ∇ ω π ( y ) , b ecause ∇ ω π ( y ) ∈ T x M . Combining this with (26) we obtain ω ⊤ − ∇ ω π ( y ) = − S v ( ∇ ω π ( y )) , or equiv alen tly ( I T π ( y ) M − S v ) ∇ ω π ( y ) = ω ⊤ . By Lemma 2.5 and the reac h assumption, w e hav e ∥ S v ∥ op ≤ ∥ Π x ∥ op ∥ v ∥ ≤ τ − 1 ∥ v ∥ < 1 for y ∈ T ( τ ), hence A y = I T x M − S v is in vertible on T x M . Therefore ∇ ω π ( y ) = A − 1 y ω ⊤ . Substituting this into (27) gives ∇ ω v y = ω − A − 1 y ω ⊤ . 44 C Pro of of Lemmas in Section 4 C.1 Pro of of Lemma 4.3 Pr o of. W rite x := π ( y ). Cho ose a smo oth curve γ ( t ) ⊂ M such that γ (0) = x and ˙ γ (0) = u . By Lemma 3.3, there exists a smo oth curv e y ( t ) ⊂ T ( τ ) such that y (0) = y , π ( y ( t )) = γ ( t ) , ˙ y (0) = A y u. Let Y ( t ) be a tangent vector ﬁeld along γ ( t ) such that Y ( t ) ∈ T γ ( t ) M and Y (0) = v . Then, by the deﬁnition of the second fundamental form we ha ve Π x ( u, v ) = ( ∇ u Y (0)) ⊥ . Since N ( y ( t )) is normal to M at γ ( t ) = π ( y ( t )), we ha ve ⟨N ( y ( t )) , Y ( t ) ⟩ = 0 for all t. Diﬀeren tiating at t = 0 gives 0 = d dt    t =0 ⟨N ( y ( t )) , Y ( t ) ⟩ =  ∇ A y u N ( y ) , v  +  N ( y ) , ∇ u Y (0)  . Since N ( y ) is normal and Y ( t ) is tangent along γ ( t ), only the normal comp onent of ∇ u Y (0) con tributes, so  N ( y ) , ∇ u Y (0)  =  N ( y ) , Π x ( u, v )  . Hence  ∇ A y u N ( y ) , v  = −  N ( y ) , Π x ( u, v )  . Since T ⊥ x M is one-dimensional and spanned b y N ( y ), we m ust hav e Π x ( u, v ) = − 1 ∥N ( y ) ∥ 2  ∇ A y u N ( y ) , v  N ( y ) . This pro ves the claim. C.2 Pro of of Lemma 4.6 Pr o of. Let x ∈ M , u, v ∈ T x M and V b e a C 1 v ector ﬁeld deﬁned in a neighborho o d of x in T ( τ ) suc h that V ( x ) = v and V is tangent to M along M , i.e., V ( z ) ∈ T z M for all z ∈ M . Then the v ector ﬁeld W ( z ) := P T ( z ) V ( z ) − V ( z ) v anishes iden tically on M . In particular, W ( x ) = 0. Diﬀeren tiating along u at x and using the Levi–Civita connection in R D , w e obtain 0 = ∇ u W ( x ) = ∇ u P T ( x ) v + P T ( x ) ∇ u V ( x ) − ∇ u V ( x ) . Pro jecting this identit y on to the normal space T ⊥ x M gives ( ∇ u V ( x )) ⊥ = ( I D − P T ( x )) ∇ u V ( x ) = ( I D − P T ( x )) ∇ u P T ( x ) v , since ( I D − P T ) P T = 0. 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Estimation of Riemannian Quantities from Noisy Data via Density Derivatives

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