Estimation of Conformal Metrics

Estimation of Confo rmal Metrics Jérôme T aupin Univ ersité P aris-Sacla y , F rance INRIA Sacla y , F rance Abstract W e study deformations of the geo desic distances on a domain of R N induced by a function called conformal factor. W e show that under a p ositive reach assumption on the domain (not necessarily a submanifold) and mild assumptions on the conformal factor, geo desics for the conformal metric hav e go od regularity prop erties in the form of a low er b ounded reach. This regularit y allows for eﬃcient estimation of the conformal metric from a random p oin t cloud with a relative error prop ortional to the Hausdorﬀ distance b et ween the p oint cloud and the original domain. W e then establish con vergence rates of order n − 1 /d that are close to sharp when the intrinsic dimension d of the domain is large, for an estimator that can b e computed in O ( n 2 ) time. Finally , this pap er includes a useful equiv alence result b etw een ball graphs and nearest-neighbors graphs when assuming Ahlfors regularit y of the sampling measure, allowing to transpose results from one setting to another. 2012 ACM Subject Classiﬁcation Theory of computation → Computational geometry Keyw o rds and phrases Geometric inference, metric estimation, conformal metric, geo desics, sets of p ositiv e reach 1 Intro duction This pap er studies metrics o ver subsets of the Euclidean space ( R N , ∥· ∥ ) obtained b y conformal deformation of the shortest-path metric via a positive function. W e are particularly in terested in the regularity of such metrics and in their estimation via i.i.d. sampling of p oin ts. ▶ Deﬁnition 1.1. L et M ⊂ R N b e a close d p ath-c onne cte d domain and f : M → R ∗ + b e a conformal factor . F or al l x, y ∈ M the conformal distance b etwe en x and y over M via f is deﬁne d as D M ,f ( x, y ) def = inf γ ∈ Γ M ( x,y ) Z I f ( γ ( t )) ∥ ˙ γ ( t ) ∥ d t (1) wher e Γ M ( x, y ) is the set of al l Lipschitz p aths γ : I → M wher e I = [ a, b ] is a nontrivial se gment, γ ( a ) = x and γ ( b ) = y . The quantity minimize d over al l p aths γ is denote d | γ | f def = Z I f ( γ ( t )) ∥ ˙ γ ( t ) ∥ d t and r eferr e d to as the c onformal length of the curve γ via f . In the c ase wher e f = 1 , we write D M ( x, y ) = D M , 1 ( x, y ) and | γ | = R I ∥ ˙ γ ∥ , r esp e ctively the distanc e b etwe en x and y induc e d by the ambient metric over M and the Euclide an length of a curve γ . In the follo wing, to ensure regularit y of the conformal metric, the domain M is assumed to ha v e p ositive reach τ M , which we recall is deﬁned as the supremum of all r > 0 suc h that an y p oin t in the oﬀset M r = { x ∈ R N : d ( x, M ) < r } has a unique pro jection onto M —where d ( x, M ) = inf y ∈ M ∥ x − y ∥ denotes the distance from point x to subset M —see [ 11 ]. Moreov er, the conformal factor f is assumed to b e κ -Lipsc hitz and lo wer b ounded by f min > 0 . These are the sole assumptions used in this pap er regarding M and f . Notice that f is deﬁned ov er M in Deﬁnition 1.1 as paths are constrained to the domain. Since any Lipschitz function deﬁned ov er a subset of R N can b e extended to the whole space preserving its Lipschitz prop ert y and low er bound (see [ 17 , Theorem 1] for instance), one 2 Estimation of Conformal Metrics ma y assume without loss of generality that f is deﬁned ov er the entire space R N , which is useful to estimate the conformal metric eﬃcien tly . The connectedness and p ositive reach of M ensure that for any endp oin ts x, y ∈ M , there exists a Lipschitz path from x to y in M , so that Γ M ( x, y ) is alwa ys nonempty and D M ,f ( x, y ) is alwa ys ﬁnite. This can b e deduced from Lemma 2.1 b elow. Most of the time, a path γ is chosen to b e parameterized with constant v elo cit y , i.e., ∥ ˙ γ ∥ is constant o v er I , with I b eing either [ 0 , 1] or [0 , | γ | ] . The latter is referred to as an ar c-length parameterization, where ∥ ˙ γ ∥ = 1 almost ev erywhere. The induc e d metric D M = D M , 1 is the metric induced b y the ambien t space R N on to M . If M is a C 1 submanifold of R N and f is C 1 , D M ,f is a Riemannian metric with tensor f ( x ) 2 · g x at p oint x ∈ M where g x is the tensor of the induced metric D M at x . The term “conformal” used in this pap er is b orro wed from the Riemannian literature. Finally , if µ is a measure ov er a submanifold M with density ρ with regard to ( w.r.t. ) the volume form of M and f is a negative p ow er of the density ρ , D M ,f has already b een studied and is sometimes referred to as the F ermat distanc e [ 14 ]. This particular case is one of the main motiv ations for this article as the F ermat distance has b een used for v arious practical applications, see for instance [ 13 ] and the references therein. Particular examples of the conformal factor f are discussed in Section 5. Conformal metrics are included in the more general class of length-spaces, for whic h it is kno wn that the inﬁm um in Deﬁnition 1.1 is in fact a minimum [ 7 , Theorem 2.5.23]. That is, for all x, y ∈ M there exists a path γ ∈ Γ M ( x, y ) suc h that D M ,f ( x, y ) = | γ | f , called a minimizing ge o desic —and shortened to geo desic in this pap er for brevit y . Denote Γ ⋆ M ,f ( x, y ) the set of suc h geo desics b etw een tw o endp oin ts x and y along with Γ ⋆ M ,f = S x  = y ∈ M Γ ⋆ M ,f ( x, y ) the set of all geodesics w.r.t. the conformal metric. W e discuss in Section 2 the regularit y of geo desics w.r.t. D M ,f and show in Prop osition 2.4 that under the aforementioned assumptions on M and f , geo desics hav e p ositiv e reach that is low er b ounded by an explicit constant dep ending on the reach of the domain and on the conformal factor. In particular, an y geo desic may b e parameterized as a C 1 , 1 curv e with an explicit upp er b ound on the Lipschitz constant of its ﬁrst deriv ative. W e then sho w in Section 3 that the conformal metric can b e approached using p olygonal paths on a weigh ted graph built from a p oin t cloud X ⊂ M , provided that the graph only con tains edges of length at most some threshold r and that X is close to M in Hausdorﬀ distance. The small edges condition ensures that paths on the graph cannot ven ture to o far outside the domain. This kind of construction is the same as the one used for the Isomap algorithm [ 5 ], although we allow more generality by adapting the weigh ts to the conformal factor f . When f = 1 and M is a geo desically conv ex C 2 submanifold, [ 5 ] pro vides a relative error b ound of order r 2 τ 2 + ρ r where τ is the minim um radius of curv ature of M and ρ denotes the Hausdorﬀ distance b etw een M and X . The term dep ending on ρ can b e made quadratic as shown in [ 2 ], where an assumption called geo desic smo othness that is slightly weak er than that of a p ositive reach allo ws to obtain a b ound of order Ar + ρ 2 r 2 where A dep ends on the assumption. This assumption how ever lacks precision when comparing distances at a lo cal scale which results in the ﬁrst term b eing of order r instead of r 2 . By using a p ositiv e reac h assumption instead, the upp er b ound can b e further improv ed to r 2 τ 2 + ρ 2 r 2 according to [ 3 ] where τ is the reach of the domain, although the setup also implicitly assumes that the domain M is a smo oth manifold isometric to a conv ex domain. Going back to our setup, we adapt these prior w orks to ﬁt any conformal c hange. Under some condition on the w eigh t function used, that in particular alw ays holds for the induced metric, w e establish in Theorem 3.5 the same upper b ound on the appro ximation error as in [ 3 ], that is r 2 τ 2 + ρ 2 r 2 where τ is the explicit low er b ound on the reac h of any geo desic provided by Prop osition 2.4. This J. T aupin 3 result emphasizes the fact that the only assumption on the domain needed for this level of precision is that of a p ositive reac h. The approximation error can thus b e made prop ortional to the Hausdorﬀ distance b etw een the p oin t cloud and the domain b y choosing appropriately r ≍ √ τ ρ . When the conformal factor f is unknown, replacing it with an estimate g do es not alter the results provided that g is close enough to f , as shown in Lemma 3.6. Assuming that X = X n is the outcome of n ≥ 1 i.i.d. samples from a d -standard measure µ o v er M , we study in Section 4 the estimator of D M ,f built on X n follo wing the construction of Section 3. This estimator is sho wn in Theorem 4.2 to conv erge to D M ,f at a rate of n − 1 /d pro vided that f is known or can b e estimated with same rate. This rate follo ws from Theorem 3.5 and the fact that X n is known to conv erge to M in Hausdorﬀ distance at a rate of n − 1 /d . In particular, this allo ws for eﬃcien t estimation of the induced metric with the sole assumption that M has p ositive reac h. This conv ergence is shown using ball graphs. How ever, practical estimation is made easier by using nearest-neighbors graphs instead, in part b ecause it do es not require to know the intrinsic dimension d to obtain an optimal conv ergence sp eed. F or this reason, we show in Theorem 4.5 that it is p ossible to retriev e the same conv ergence sp eed when replacing the ball graph in the estimator with a k -nearest-neigh b ors graph with k ≍ √ n . T o do so, we establish an equiv alence with high probability b etw een nearest-neigh b ors graphs and ball graphs when their resp ectiv e parameters are prop erly scaled and the underlying measure is d -Ahlfors, see Prop osition 4.4. Finally , the nearest-neighbors estimator of the conformal distance b et ween tw o ﬁxed p oints ma y b e computed in O ( n 2 ) if the time complexity of ev aluating f is considered constant. Under the stronger assumption that M is a C k submanifold of dimension d with k ≥ 2 , the minimax optimal con v ergence rate for the induced metric is known to b e of order n − k/d [ 1 ]. When only assuming p ositive reach and no diﬀerential structure on M , the discussion of the optimal minimax rate prov es to b e harder as w e are not able to match the upp er b ound of n − 1 /d and obtain a low er b ound of order n − 1 / ( d − 1 / 2) instead, see Theorem 4.6. W e discuss with more details the comparison b etw een our setup and the C k case in Section 4.4. 2 Regula rity of the Geo desics for the Conformal Metric Recall that the term “geo desic” refers in this paper to a curve that ac hieves a global minim um of the conformal length. In this section we study the regularit y of the geo desics w.r.t. D M ,f . W e show that geo desics are C 1 , 1 curv e with reac h b ounded from b elow b y a constan t depending only on τ M , κ and f min . This regularity prop ert y is crucial to obtain a go o d approximation of the geo desics by p olygonal paths. The reach of a set can b e characterized through the lo cal distortion of the induced metric compared to the Euclidean metric. ▶ Lemma 2.1. [6, The or em 1] The r e ach of M may b e expr esse d as τ M = sup  r > 0 : ∀ x, y ∈ M , ∥ x − y ∥ < 2 r ⇒ D M ( x, y ) ≤ 2 r arcsin  ∥ x − y ∥ 2 r  . Giv en a path γ ∈ Γ M parameterized o v er an interv al I and without self-in tersection, denote τ γ the reach of the curve γ ( I ) ⊂ R N along with D γ = D γ ( I ) the induced metric ov er the curve for short. D γ ( x, y ) is nothing more than the length of the curve γ ( I ) b et ween tw o in termediate point x and y . In particular if x and y are the endp oints of γ then D γ ( x, y ) = | γ | . One consequence of Lemma 2.1 is that the length of a geo desic b etw een tw o p oin ts at most 2 τ M apart is upp er b ounded by the length of an arc of radius τ M b et ween b oth p oints. That 4 Estimation of Conformal Metrics is, if x, y ∈ M and ∥ x − y ∥ < 2 τ M then D M ( x, y ) ≤ 2 τ M arcsin  ∥ x − y ∥ 2 τ M  . (2) Moreo v er, if γ is a geo desic, it then induces a geo desic b etw een any of the p oin ts it go es through, which implies that D γ is exactly the restriction of D M to the curv e γ . Then, another consequence of the c haracterization of Lemma 2.1 is that the reac h of M is the minimal reac h of any geo desic, that is τ M = inf γ ∈ Γ ⋆ M τ γ . W e now introduce a notion of reach asso ciated with the conformal deformation of M b y f using the same p oint of view of the metric and its geo desics. ▶ Deﬁnition 2.2. The conformal reach of M via f is deﬁne d as the minimal r e ach of any ge o desic w.r.t. the c onformal metric, that is τ M ,f def = inf γ ∈ Γ ⋆ M,f τ γ . In the case of the induced metric this notion coincides with the usual notion of reach, that is τ M , 1 = τ M . The characterization given by Lemma 2.1 also holds for the conformal reach. ▶ Prop osition 2.3. The c onformal r e ach of M via f may b e expr esse d as τ M ,f = sup      r > 0 : ∀ x, y ∈ M , ∀ γ ∈ Γ ⋆ M ,f ( x, y ) , ∥ x − y ∥ < 2 r ⇒ | γ | ≤ 2 r arcsin  ∥ x − y ∥ 2 r       . (3) Bew are that Equation (3) inv olves geodesic w.r.t. the conformal metric D M ,f , but compares their Euclidean length—not conformal—to the one of an arc of radius r . Pro of. Denote A the subset of R ∗ + that app ears in the right-hand side of Equation (3) and let us show that its supremum is indeed τ M ,f . Since a geo desic induces a geo desic b etw een an y pair of p oin ts it go es through, one can write A = T γ A γ where the intersection is taken o v er all geo desics w.r.t. D M ,f and A γ =  r > 0 : ∀ x, y ∈ γ , ∥ x − y ∥ < 2 r ⇒ D γ ( x, y ) ≤ 2 r arcsin  ∥ x − y ∥ 2 r  . Lemma 2.1 states that τ γ = sup A γ and standard reasoning ov er ordered sets shows that τ M ,f = inf γ τ γ = inf γ sup A γ ≥ sup \ γ A γ = sup A . Moreo v er, for all x, y ∈ M , r 7→ 2 r arcsin  ∥ x − y ∥ / 2 r  is a non-increasing function ov er ( ∥ x − y ∥ , + ∞ ) , whic h implies that for all γ , if r ∈ A γ then s ∈ A γ for all 0 < s ≤ r . In particular, it follows that the inequality ab o ve is in fact an equalit y , hence τ M ,f = sup A whic h concludes the pro of. ◀ Using Prop osition 2.3, w e are able to low er b ound the reach of any geo desic w.r.t. D M ,f . J. T aupin 5 ▶ Prop osition 2.4. A ssume that M has p ositive r e ach τ M > 0 and that f is κ -Lipschitz and lower b ounde d by f min > 0 . Then any c onformal ge o desic γ w.r.t. D M ,f has p ositive r e ach τ γ > 0 . Pr e cisely, the c onformal r e ach of M via f is lower b ounde d as fol lows. τ M ,f ≥ T M ,f wher e T M ,f def = min  τ M 2 , f min 8 κ  . The reasoning b ehind Prop osition 2.4 is the following. If the reach of a geo desic γ w.r.t. D M ,f is small compared to τ M , then according to Prop osition 2.3 there exists a path in M signiﬁcan tly shorter than γ Euclidean-wise. If τ γ is also small compared to f min /κ , then this path is shown to also ha v e a shorter conformal length than γ due to the prop erties of f , whic h implies a contradiction with the geo desic nature of γ . This reasoning is detailed in App endix B.2. Now, consider an arc-length parameterized geo desic γ : I → M w.r.t. D M ,f , i.e., such that ∥ ˙ γ ∥ = 1 almost everywhere ov er I . Being a geo desic, γ has no self-intersection. Then, stating that γ has p ositiv e reac h is equiv alent to stating that γ is a C 1 , 1 curv e, i.e., that ˙ γ is Lipschitz w.r.t. the angular distance. Precisely , for all t, s ∈ I , ∠ ( ˙ γ ( t ) , ˙ γ ( s )) ≤ | t − s | τ γ . (4) See [ 11 , Remark 4.20] and [ 16 , Theorem 4] for references. Equation (4) allo ws to obtain a ﬁner con trol on the approximation error of the p olygonal paths on the graph by upp er b ounding eﬃcien tly the diﬀerence b etw een small successive steps γ ( t ) − γ ( t − δ ) and γ ( t + δ ) − γ ( t ) of a geo desic path. ▶ Lemma 2.5. L et γ : [0 , | γ | ] → R N b e an ar c-length p ar ameterize d curve without self- interse ction and with p ositive r e ach τ γ . Then for al l t 0 ∈ [0 , | γ | ] and δ ∈ (0 , π 2 τ γ ] such that [ t 0 − δ, t 0 + δ ] ⊂ [0 , | γ | ] , the angle b etwe en the velo city ve ctor of γ at t 0 and the dir e ction fr om γ ( t 0 ) to γ ( t 0 + δ ) is upp er b ounde d as fol lows: ∠  γ ( t 0 + δ ) − γ ( t 0 ) , ˙ γ ( t 0 )  ≤ δ 2 τ γ . (5) Mor e over, denoting v − = γ ( t 0 ) − γ ( t 0 − δ ) and v + = γ ( t 0 + δ ) − γ ( t 0 ) , the diﬀer enc e b etwe en smal l steps of the p ath on b oth side fr om t 0 is upp er b ounde d as fol lows:     v + ∥ v + ∥ − v − ∥ v − ∥     ≤ 1 τ γ min  ∥ v + ∥ , ∥ v − ∥  . (6) Lemma 2.5 is key to obtain an approximation error of the conformal metric proportional to the Hausdorﬀ distance b et w een the domain and a p oin t cloud. A pro of is provided in App endix A.1 and Equation (5) is illustrated by Figure 1. 3 P olygonal App roximation of the Conformal Metric Consider a p oin t cloud X ⊂ M of p oints sampled from the domain. If the p oin t cloud appro ximates well the domain, it is p ossible to estimate the conformal distance ov er M through weigh ted p olygonal paths built ov er X within a small margin of error. Assuming that f is known, the edges of the p olygonal path are weigh ted using f to approximate the conformal distance b et w een their endp oin ts. The degree of approximation of M b y X is measured using the Hausdorﬀ distance d H ( M , X ) def = max  sup x ∈ M d ( x, X ) , sup x ∈ X d ( x, M )  6 Estimation of Conformal Metrics Figure 1 Bounding the angular velocity of a path with p ositive reach. whic h simpliﬁes to d H ( M , X ) = sup x ∈ M d ( x, X ) assuming that X ⊂ M . By deﬁnition, any p oin t in M is at distance at most d H ( M , X ) from a point in X . The idea b ehind the p olygonal approximation is the following. Consider a geo desic γ ∈ Γ ⋆ M ,f ( x, y ) b et ween tw o p oints x, y ∈ M . If d H ( M , X ) is small, there exists a p olygonal path ov er X that follows closely the tra jectory of γ . Assuming that the edges of this path are equipp ed with appropriate weigh ts to simulate the conformal distance b etw een their endp oin ts, the total weigh t of the p olygonal path should b e close to the conformal length | γ | f . Moreo v er, since γ is a geo desic, p olygonal paths should not b e able to hav e a weigh t muc h smaller than | γ | f . This is ensured by using only short edges, which allows the weigh ts to appro ximate well the conformal length by preven ting shortcuts outside the domain. In the end, the shortest weigh ted path on the graph is exp ected to retrieve | γ | f = D M ,f ( x, y ) . 3.1 W eighted Graphs Let us now describ e formally this construction. The p olygonal approximation of D M ,f is deﬁned as the metric of an appropriate weigh ted graph on a p oint cloud X . ▶ Deﬁnition 3.1. L et X ⊂ R N b e a ﬁnite p oint cloud. F or al l r > 0 , the r -ball graph over X is denote d G r ( X ) and is the gr aph of vertex set X and with an e dge b etwe en p oints x and y if and only if ∥ x − y ∥ ≤ r . F or al l inte ger k ≥ 1 , the k -nearest-neigh b ors graph (or k -NN gr aph) over X is de- note d G k ( X ) and is the gr aph of vertex set X with an e dge b etwe en two p oints x and y if and only if x (or y ) is among the k ne ar est p oints of X to y (or x ) excluding self. In b oth c ases, the p ar ameter r or k is r eferr e d to as the threshold of the gr aph. Ball graphs are more conv enient to study whereas NN graphs are more practical, see Section 4.2. W e deﬁne the p olygonal metric in b oth contexts. The graph used is equipp ed with a weigh t function to approximate the conformal length of its edges. ▶ Deﬁnition 3.2. Given a function f : R N → R ∗ + , c onsider the weight functions w f ,q ( x, y ) def = ∥ x − y ∥ 2( q − 1) f ( x ) + 2 q − 1 X k =2 f  q − k q − 1 x + k − 1 q − 1 y  + f ( y ) ! deﬁne d for any q ≥ 2 and x, y ∈ R N , along with w f , ∞ ( x, y ) def = ∥ x − y ∥ Z 1 0 f  (1 − t ) x + ty  d t . The p ar ameter q ∈ { 2 , . . . , ∞} is r eferr e d to as the resolution of the weights. J. T aupin 7 The weigh t function w f ,q is meant to appro ximate the conformal distance b etw een the endp oin ts of an edge. Prop osition 3.3 b elo w shows that it is indeed the case when the endp oin ts are close, which is the reason w h y graphs with small edges are used. Recall that we ha v e assumed without loss of generality that f is deﬁned o v er the whole space, allowing its ev aluation outside M when q  = 2 . As q grows, the weigh t w f ,q b ecomes more accurate and con v erges to the weigh t with inﬁnite resolution w f , ∞ , the latter b eing exactly the conformal length of the straight path. How ever, the computational cost is linear with q and there is no closed-form formula for w f , ∞ in general. On the other hand, w f , 2 ( x, y ) = ∥ x − y ∥ f ( x ) + f ( y ) 2 is the simplest choice for the weigh ts and may b e more suitable if ev aluating f is p ossible only at p oin ts in X . Note that the weigh t w f ,q is the optimal appro ximation of w f , ∞ using only q samples of the Lipsc hitz function f . Under stronger assumptions on f suc h as C k regularit y , other deﬁnitions would b e b etter suited. ▶ Prop osition 3.3. F or al l x, y ∈ M such that ∥ x − y ∥ ≤ τ M ,f ,     1 − w f ,q ( x, y ) D M ,f ( x, y )     ≤ κ 4 f min ∥ x − y ∥ q − 1 + ∥ x − y ∥ 2 16 T 2 M ,f . (7) and we denote δ q ( ∥ x − y ∥ ) this upp er b ound. The ﬁrst term in Equation (7) is to b e interpr ete d as 0 if q = ∞ and r epr esents the oﬀset b etween w f ,q and w f , ∞ , wher e as the se c ond term r epr esents the oﬀset b etwe en w f , ∞ and D M ,f . Prop osition 3.3 shows that the conformal metric D M ,f ma y b e approximated lo cally by the weigh ts w f ,q . Note that for ﬁxed resolution q in the r -ball graph, the distortion b etw een the weigh ts and the conformal distances is linear in r . How ever, b y c ho osing q to b e in versely prop ortional to r , the distortion b ecomes quadratic in r . The pro of for Prop osition 3.3 is pro vided in App endix B.3. W e now introduce the p olygonal approximation of the conformal metric as the metric of the weigh ted graph built on X . ▶ Deﬁnition 3.4. Consider a p oint cloud X ⊂ R N , function f : R N → R ∗ + and p ar ameters r > 0 or k ≥ 1 and q ∈ { 2 , . . . , ∞} . The p olygonal metric asso ciate d with these p ar ameters is deﬁne d b etwe en two p oints x, y ∈ R N as b D X,f ( x, y ) def = min ( x 0 ,...,x K ) K − 1 X k =0 w f ,q ( x k , x k +1 ) (8) wher e the minimum is taken over the set of p aths ( x 0 , . . . , x K ) such that x 0 = x and x k = y in the gr aph G that is chosen either as the r -b al l gr aph G r ( X ∪ { x, y } ) or the k -NN gr aph G k ( X ∪ { x, y } ) . The choice of a ball graph or a NN graph along with the threshold r or k and the resolution q of the weigh ts are left implicit when writing b D X,f to av oid heavy notations. Note that when the endp oin ts x and y do not b elong to the p oin t cloud, the distance b D X,f ( x, y ) is computed by adding them to the graph. As a result, b D X,f induces a distance ov er X but not o v er M . Indeed, when considering endp oin ts outside X , b D X,f ma y not satisfy the triangular inequalit y due to the set of p ossible paths dep ending on the endp oints. 8 Estimation of Conformal Metrics 3.2 Metric Appro ximation T o ev aluate ho w eﬃcien t is the approximation of a metric, the error b et ween the true distances and their estimation may b e measured using either of the multiplicativ e loss functions ℓ ∞ ,M  D ′ , D  def = sup x  = y ∈ M     D ′ ( x, y ) − D ( x, y ) D ′ ( x, y ) ∨ D ( x, y )     and l ∞ ,M  D ′ | D  def = sup x  = y ∈ M     1 − D ′ ( x, y ) D ( x, y )     . In the case where D ′ tak es inﬁnite v alues—which happ ens for instance when D ′ is the metric of a non-connected graph—w e let ℓ ∞ ,M ( D ′ , D ) = 1 and l ∞ ,M ( D ′ | D ) = + ∞ . The inequality ℓ ∞ ,M  D ′ , D  ≤ l ∞ ,M  D ′ | D  ≤ ℓ ∞ ,M  D ′ , D  1 − ℓ ∞ ,M  D ′ , D  (9) holds in general, so that b oth losses are equiv alent. The loss ℓ ∞ ,M is how ever easier to manipulate in some situations as it is symmetric and upp er b ounded by 1 . Thanks to the regularit y of geo desics stated b y Prop osition 2.4 and the lo cal appro ximation of the conformal metric by the weigh ts w f ,q stated by Prop osition 3.3, w e are able to show that the conformal metric D M ,f is approximated by the p olygonal metric from Deﬁnition 3.4 using the r -ball graph. ▶ Theo rem 3.5. L et X ⊂ M b e a p oint cloud, r > 0 and q ∈ { 2 , . . . , ∞} two p ar ameters. A ssume that 4 d H ( M , X ) ≤ r ≤ T M ,f . Then the appr oximation b D X,f deﬁne d in Deﬁnition 3.4 using the r -b al l gr aph with r esolution q satisﬁes l ∞ ,M  b D X,f | D M ,f  ≤ 1 32 T M ,f r q − 1 + r 2 8 T 2 M ,f + 56 d H ( M , X ) 2 r 2 . (10) The ﬁrst term in Equation (10) is to b e interpr ete d as 0 if q = ∞ . Let us describ e the reasoning b ehind Theorem 3.5. Giv en x  = y ∈ X , paths in G r ( X ) cannot create signiﬁcant shortcuts outside M as r ≤ τ M ,f . This is illustrated by Prop osition 3.3 from which it can b e deduced that b D X,f ( x, y ) ≥  1 − δ q ( r )  D M ,f ( x, y ) (11) and, assuming that ∥ x − y ∥ ≤ r , b D X,f ( x, y ) ≤  1 + δ q ( r )  D M ,f ( x, y ) . (12) As for when ∥ x − y ∥ > r , consider a geo desic γ ∈ Γ ⋆ M ,f ( x, y ) . Decomp ose γ in to sections of length at most r − 2 d H ( M , X ) and for each intermediate p oin t select a p oint in X at distance at most d H ( M , X ) . This process draws a p olygonal path in G r ( X ∪ { x, y } ) as it uses edges of length at most r . Summing the approximation error from Prop osition 3.3 b et ween each edge of this path and the corresp onding section of γ even tually yields the upp er b ound b D X,f ( x, y ) ≤  1 + δ q ( r ) + c 1 d H ( M , X ) T M ,f + c 2 d H ( M , X ) 2 r 2  D M ,f ( x, y ) (13) when ∥ x − y ∥ > r and where c 1 and c 2 are univ ersal constan ts. The p ositiv e reach of γ and Lemma 2.5 play a crucial role in obtaining terms of order d H ( M , X ) / T M ,f and d H ( M , X ) 2 /r 2 instead of d H ( M , X ) /r . T ogether, Equations (11)–(13) even tually imply Equation (10). The J. T aupin 9 details of this reasoning are provided in App endix B.4. The upp er b ound in Equation (10) is made prop ortional to d H ( M , X ) b y setting r ≍ q T M ,f d H ( M , X ) and q ≍ s T M ,f d H ( M , X ) . In the case of the induced metric with f = 1 , the ﬁrst term in the right-hand side of Equation (10) disapp ears, and the error is of order r 2 /τ 2 M + d H ( M , X ) 2 /r 2 similarly to the one obtained in [ 3 ]. In the general conformal case, we keep this magnitude of error by setting the resolution q to b e at least of order 1 /r . If it is not p ossible to ev aluate f outside the p oin t cloud X , setting q = 2 instead yields an error of order r / T M ,f + d H ( M , X ) 2 /r 2 whic h is sligh tly worse. A similar upp er b ound w as achiev ed in [ 2 ] for the induced metric and the term linear in r w as due to the geo desically smo oth assumption used in the pap er b eing sligh tly w eak er than a p ositive reach assumption. In particular, the latter allo ws for an eﬃcien t lo cal estimation in the form of Equation (12). The main takea wa y of Theorem 3.5 is that the crucial assumption to achiev e an approximation error prop ortional to d H ( M , X ) is the p ositive reach. In particular, the domain need not b e C 2 , or even a submanifold. 3.3 Unkno wn Confo rmal Facto r In general, f ma y not b e known and needs to b e estimated from the data. The p olygonal metric b D X,g from Deﬁnition 3.4 can b e deﬁned for any function g : R N → R ∗ + without the need of Lipsc hitz and lo wer bound assumptions. Then, the approximation error b etw een b D X,g and D M ,f is upp er b ounded by the sum of the error w.r.t. the conformal factor and the error w.r.t. the domain. ▶ Lemma 3.6. L et f : R N → R + a function lower b ounde d by f min , X ⊂ M a p oint cloud and g : R N → R ∗ + such that ∥ g − f ∥ ∞ ≤ 1 2 f min . Then ℓ ∞ ,M  b D X,g , D M ,f  ≤ 2 f min ∥ g − f ∥ ∞ + 2 ℓ ∞ ,M  b D X,f , D M ,f  . (14) This r esult holds r e gar d less of the typ e of gr aph, thr eshold and r esol ution as long as they ar e shar e d b etwe en b D X,f and b D X,g . The pro of of Lemma 3.6 consists in straigh tforw ard computations and is deferred to App endix B.5. As a consequence, in the con text of estimation of the domain M b y a p oint cloud X n as in Section 4 and of the conformal factor f b y a function f n : R N → R + , the rate of conv ergence of b D X n ,f n to w ards D M ,f is the slow est rate of conv ergence among that of b D X n ,f to w ards D M ,f and that of f n to w ards f . Recall that f n is not required to satisfy the Lipsc hitz and low er b ounded assumptions for the p olygonal metric b D X n ,f n to b e deﬁned and for Lemma 3.6 to hold. It may also b e deﬁned only ov er X n if the resolution is set to q = 2 . 4 Estimation from Random Samples In this section we transp ose Theorem 3.5 to a probabilistic setup where the p oin t cloud X is replaced with a random p oint cloud X n consisting of n i.i.d. samples from a probability measure µ with supp ort M . The following assumptions on µ are needed to ensure that X n co v ers M eﬃcien tly as n gro ws, allowing to deduce explicit conv ergence rates for the estimator b D X n ,f from Theorem 3.5. ▶ Deﬁnition 4.1. Consider a pr ob ability me asur e µ with supp ort M ⊂ R N and d ≥ 2 . 10 Estimation of Conformal Metrics The me asur e µ is d -standar d with lower c onstant c µ > 0 if for al l x ∈ M and r > 0 , µ  B ( x, r )  ≥ c µ r d ∧ 1 wher e B ( x, r ) denotes the op en b al l c enter e d at x with r adius r . The me asur e µ is d -Ahlfors with lower and upp er c onstants c µ > 0 and C µ > 0 if for al l x ∈ M and r > 0 , c µ r d ∧ 1 ≤ µ  B ( x, r )  ≤ C µ r d . If µ is d -standard with lo w er constant c µ , w e also denote L µ = c − 1 /d µ whic h is related to the size of the supp ort. The deﬁnition ensures that µ  B ( x, L µ )  = 1 for all x ∈ M , hence L µ ≥ diam ( M ) . Moreov er, d acts as the intr insic dimension of the supp ort M of µ . F or instance, any measure with a density ρ w.r.t. the volume measure of a submanifold of R N of dimension d is d -Ahlfors if ρ is b ounded ab o ve and b elo w. The d -standard assumption ensures that the random p oint cloud X n con v erges to M in Hausdorﬀ distance. Ahlfors regularit y is a stronger assumption and is needed to sho w the equiv alence b etw een ball graphs and NN graphs in Prop osition 4.4. 4.1 Convergence of the Ball Graph Estimato r W e ﬁrst discuss the case of a ball graph estimator. It is known that if µ is d -standard, then d H ( M , X n ) is of order L µ ( log ( n ) /n ) 1 /d at most, see App endix A.2. This conv ergence com bined with Theorem 3.5 allows to derive conv ergence rates for the estimation of D M ,f . ▶ Theo rem 4.2. A ssume that X n is the r esult of n i.i.d. samples fr om a d -standar d pr ob ability me asur e µ with supp ort M . Consider the estimator b D X n ,f fr om Deﬁnition 3.4 using the r -b al l gr aph with r esolution q wher e r and q ar e sp e ciﬁe d b elow. Then ther e exists a c onstant n 0 dep ending on L µ , T M ,f and d such that for al l n ≥ n 0 the fol lowing holds. If r = 8 p L µ T M ,f  log( n ) n  1 2 d and q ≥ 1 + 4 T M,f r , then E h ℓ ∞ ,M  b D X n ,f , D M ,f  i ≤ 16 L µ T M ,f  log( n ) n  1 d . (15) If r = 8 L 2 / 3 µ T 1 / 3 M ,f  log( n ) n  2 3 d and q = 2 , then E h ℓ ∞ ,M  b D X n ,f , D M ,f  i ≤ 8  L µ T M ,f  2 3  log( n ) n  2 3 d . (16) The ﬁrst case in Theorem 4.2 includes q = ∞ . There is no use setting q to b e greater than the indicated threshold of order T M ,f /r , as the term in the upp er b ound dep ending on q b ecomes negligible past this threshold. Note that the optimal choice for r and q requires the kno wledge of L µ , T M ,f and most imp ortan tly d . Theorem 4.2 is stated with these choices of parameters to highlight the optimal theoretical dep endency in these constants. F or practical purp oses, the knowledge of L µ and T M ,f is not actually needed as, in general, any choice of r ≍ ( log ( n ) /n ) 1 / 2 d and q ≳ 1 /r yields a conv ergence rate of ( log ( n ) /n ) 1 /d . Likewise, any c hoice of r ≍ ( log ( n ) /n ) 2 / 3 d and q = 2 yields a conv ergence rate of ( log ( n ) /n ) 2 / 3 d . It is not p ossible how ever to get rid of the dep endency in d for the range parameter r without altering the conv ergence rate. T o circumv ent this issue, one may use the NN graph instead, for whic h the optimal choice of the parameter is shown in Section 4.2 to b e k = √ n whic h do es not J. T aupin 11 dep end on d . Theorem 4.2 is deriv ed directly from Theorem 3.5 and the conv ergence of X n to M in Hausdorﬀ distance. The precise computations are deferred to App endix C.1. In the case of the induced metric, the weigh t function w f ,q b ecomes the Euclidean distance regardless of the resolution and T M ,f is replaced with τ M . ▶ Co rollary 4.3. Under the same assumptions as in The or em 4.2 and when f = 1 , ther e exists a c onstant n 0 dep ending on L µ , τ M and d such that for al l n ≥ n 0 , setting r = 8 p L µ τ M (log( n ) /n ) 1 / 2 d , the estimator b D X n , 1 satisﬁes E h ℓ ∞ ,M  b D X n , 1 , D M  i ≤ 16 L µ τ M  log( n ) n  1 d . Recall that in the case of a submanifold of class C k with k ≥ 2 and dimension d , the optimal con v ergence rate for the estimation of the induced metric is n − k/d [ 1 ]. Corollary 4.3 extends the upp er b ound to any set of p ositiv e reach with the conv ergence rate n − 1 /d . 4.2 Convergence of the Nea rest-Neighb ors Graph Estimator W e now treat the case of a NN graph estimator, by observing that for adequate choice of r and k , the r -ball graph and k -NN graph are similar. Indeed, assuming that µ is d -Ahlfors regular, a ball of radius r is exp ected to contain b etw een c µ nr d and C µ nr d p oin ts from X n . ▶ Prop osition 4.4. L et µ b e a d -A hlfors me asur e as deﬁne d in Deﬁnition 4.1 and X n a p oint cloud sample d i.i.d. fr om µ . L et k ≥ 1 , ε ∈ (0 , 1) and deﬁne r − =  (1 − ε ) k C µ ( n − 1)  1 d and r + =  1 1 − ε k c µ ( n − 1)  1 d . Then with pr ob ability at le ast 1 − 2 ne − ε 2 k/ 2 , the k -NN gr aph G k ( X n ) is enclose d b etwe en two b al l gr aphs G r − ( X n ) and G r + ( X n ) , that is P  G r − ( X n ) ⊂ G k ( X n ) ⊂ G r + ( X n )  ≥ 1 − 2 n exp  − ε 2 2 k  wher e the inclusion ⊂ r efers to the inclusion of e dge sets. Prop osition 4.4 sho ws that if r and k are chosen such that k ≍ nr d , the k -NN graph and the r -ball graph are very similar with high probability . This result stems from a common in tuition and a detailed pro of is pro vided in App endix A.3 for completeness. Theorem 4.2 is then extended to k -NN graphs by choosing k ≍ p n log( n ) , which do es not dep end on d . ▶ Theo rem 4.5. A ssume that X n is the r esult of n i.i.d. samples fr om a d -A hlfors pr ob ability me asur e µ with supp ort M . Consider the estimator b D X n ,f fr om Deﬁnition 3.4 using the k -NN gr aph with p ar ameters k =  p n log( n )  and q = ⌈ n 1 / 4 ⌉ . Then ther e exists a c onstant n 0 dep ending on L µ , T M ,f and d such that for al l n ≥ n 0 the fol lowing holds. E h ℓ ∞ ,X n  b D X n ,f , D M ,f  i ≤ C  log( n ) n  1 d wher e C is a c onstant dep ending on L µ and T M ,f . Notice that in Theorem 4.5 the loss is ov er X n instead of M lik e in Theorem 4.2. The same result could b e stated ov er M , although the pro of would b e more tedious and is 12 Estimation of Conformal Metrics therefore omitted here. Recall that no knowledge on either d , c µ , C µ , τ M , κ or f min is necessary to compute the estimator, so that it can b e used in practice. Setting the resolution q to b e ( n/ log ( n )) 1 / 2 d w ould b e suﬃcient to achiev e the same upp er b ound, how ever this c hoice requires the knowledge of d . On the other hand, when setting the resolution t o q = 2 the optimal c hoice of k can b e shown similarly to b e k = n 1 / 3 log ( n ) 2 / 3 and to yield a con v ergence rate of order ( log ( n ) /n ) 2 / 3 d . Theorem 4.5 is a direct consequence of Theorem 4.2 and Prop osition 4.4 and the precise computations are pro vided in App endix C.2. Regarding the case of the induced metric, the same statement as in Corollary 4.3 holds for NN graphs. Namely , the estimator b D X n , 1 deﬁned ov er the k -NN graph with k =  p n log( n )  con v erges to D M at rate (log ( n ) /n ) 1 /d . Let us now discuss the algorithmic complexit y of the k -NN estimator with resolution q . In general, building the k -NN graph ov er a p oin t cloud X of size n is done in O ( n 2 N ) time when the ambien t dimension N is large. Now, consider tw o p oin ts x, y ∈ R N and assume that f can b e ev aluated at any p oin t with cost c f . Computing the edges of the graph G k ( X ∪ { x, y } ) takes O ( nk q c f ) time as the amount of edges in the graph is O ( nk ) and the w eigh t of each edge uses q ev aluations of f . Finally , computing the inﬁmum that deﬁnes D X,f ( x, y ) in Deﬁnition 3.4 using Dijkstra’s algorithm takes O ( n log ( n ) + nk ) time. Overall, if k =  p n log( n )  and q = ⌈ n 1 / 4 ⌉ , the time complexity is O ( n 2 N + n 7 / 4 log( n ) 1 / 2 c f ) . 4.3 Minimax low er b ound W e now study the worst case p erformance of an y estimator of the induced metric D M . ▶ Theo rem 4.6. Denote D n the set of al l estimators b D of the induc e d metric b ase d on n samples, that given any p oint cloud X of n p oints in R N pr ovides a function b D X : R N × R N → R + . L et 2 ≤ d ≤ N , L, τ > 0 and M ( d, L, τ ) b e the set of al l d -standar d me asur es µ with L µ ≤ L and supp ort M µ ⊂ R N that has p ositive r e ach lower b ounde d by τ . Then ther e exists two c onstants C > 0 and n 0 dep ending on d , L and τ such that for al l n ≥ n 0 , inf ˆ D ∈ D n sup µ ∈M ( d,L,τ ) E X ∼ µ ⊗ n h ℓ ∞ ,M µ  b D X , D M µ  i ≥ C  1 n  1 d − 1 / 2 . (17) Notice that Theorem 4.6 only addresses the case of the induced metric. This is not a loss of generality and in fact highlights the fact that the conformal change do es not make the problem any harder, as we hav e established in Section 2 that geo desics hav e the same regularit y as in the case of the induced metric. Given an y other function f satisfying the assumptions for a conformal metric, the same low er b ound ma y b e obtained with a similar reasoning as what follows, alb eit with more technicalities. The rate n − 1 / ( d − 1 / 2) is faster than the rate n 1 /d obtained in Theorems 4.2 and 4.5, although the diﬀerence b ecomes negligible when d is large. The nature of the minimax conv ergence rate remains therefore op en. Under stronger assumptions on the domain, this question is already solved—see [ 1 ]—whic h we discuss in Section 4.4. Theorem 4.6 is based on Le Cam’s metho d [ 19 ], for which a statement adapted to our setup is given in Lemma C.1. The metho d consists in ﬁnding tw o measures µ 1 and µ 2 in M ( d, L, τ ) that are at most 1 /n apart in total v ariation distance and suc h that the relativ e diﬀerence b etw een D M 1 and D M 2 is of order at least n − 1 / ( d − 1 / 2) . Consider µ 1 the uniform probabilit y on the cub e M 1 = [ − αL, αL ] d × { 0 } N − d ⊂ R N where α > 0 . Let 0 < ε ≤ αL ∧ τ and consider M 2 the result of removing from M 1 its in tersection with a ball of radius τ cen tered at (0 , t, t, . . . , t ) for some t > αL suc h that the ball intersects the edge from ( − αL, αL, αL, . . . , αL ) to ( αL, αL, αL, . . . , αL ) at tw o p oin ts x and y that J. T aupin 13 x y Figure 2 Carving an edge of the cub e in R 3 . are 2 ε apart, as pictured in Figure 2. Denote µ 2 the uniform probability ov er M 2 , whic h has reach exactly τ due to the carved area. By choosing α small enough, whic h dep ends only on d , µ 1 and µ 2 are d -standard with low er constan t L − d . Then µ 1 and µ 2 b oth b elong to M ( d, L, τ ) . The volume of the carv ed area is of order ε ( ε 2 ) d − 1 as it spans a length 2 ε b et ween x and y and a length of order ε 2 for every other dimension of the cub e. This implies that the total v ariation distance b etw een µ 1 and µ 2 is of order ε 2 d − 1 . Moreov er, The distance from x to y go es from 2 ε in M 1 to 2 τ arcsin ( ε/τ ) in M 2 , following the red arc of radius τ in Figure 2. This implies that the relative diﬀerence b et w een b oth distances is of order ε 2 . Cho osing ε ≍ n − 1 / 2 d − 1 so that the total v ariation distance is at most 1 /n ev en tually yields the desired b ound. This reasoning is detailed in App endix C.3. Using this metho d, the ab o ve construction leads to a minimax low er b ound that do es not quite ﬁt the upp er b ound from Theorem 4.2. If a b etter low er b ound can b e obtained, it is likely that it would require a diﬀerent technique. Indeed, given ﬁxed endp oints x and y at distance ε from each other and b elonging to the intersection of tw o domains M 1 and M 2 with low er b ounded reach, the relative diﬀerence b etw een D M 1 ( x, y ) and D M 2 ( x, y ) is of order at most ε 2 due to the reach assumption. Then, in order to achiev e such distortion, it is reasonable to exp ect that the volume of the diﬀerence b etw een M 1 and M 2 needs to b e of order at least ε in the direction of x − y and at least ε 2 in the other directions to ensure that the uniform measures are d -standard, whic h leads to a v olume of order ε 2 d − 1 as in the previous construction. 4.4 Case of a Smo oth Manifold Recall that the minimax con vergence rate for the induced metric of a C k manifold of dimension d is n − k/d [ 1 ]. The metho ds that ac hiev e suc h rates are how ever not computationally feasible in practice as they rely on manifold reconstruction via non-discrete sets. F or instance, the optimal conv ergence rate in the C 2 case was achiev ed by [ 3 ] using the tangential Delauna y complex. Precisely , these metho ds build an approximation of the manifold that is at Hausdorﬀ distance of order n − k/d , then state that the induced metric ov er this reconstruction is an estimation of the original metric with an error prop ortional to the Hausdorﬀ distance, that is of order n − k/d . In order to get a concrete estimator, one may use such manifold reconstruction, then sample a ﬁne n − k/d -net ov er it which approximates the original domain with the same Hausdorﬀ error. Our work then implies that the p olygonal metric ov er this net w ould b e an estimation of the original metric with minimax optimal error of order n − k/d . Suc h 14 Estimation of Conformal Metrics construction w ould ho w ever b e very costly as the p oint cloud size would grow from n to n k . 5 Practical Examples of Confo rmal F actors W e discuss t wo examples of conformal factors asso ciated with a measure from the literature. 5.1 Densit y as a conformal facto r If µ is a measure with density ρ w.r.t. the v olume measure o v er a submanifold M , the conformal c hange via f = ρ − β for some parameter β > 0 is sometimes referred to as the F ermat distanc e due to the parallel with the F ermat principle in optics—that may how ever b e observed with an y conformal c hange. This metric has b een applied to v arious top ological data analysis and learning problems, e.g., in [ 12 , 13 ], as it tends to accentuate features and disparities in the data. Moreo ver, the metric can b e estimated in a simple fashion by using the same ov erall reasoning as ours but using weigh ts of the form w ( x, y ) = ∥ x − y ∥ α where α > 1 is a parameter dep ending on β and d [ 14 , 15 ]. How ev er, this kind of estimation do es not feature any known conv ergence rate. Using our estimator instead provides an alternativ e metho d with quan titativ e guaran tees, gran ted that an estimator of the densit y is a v ailable. T o this extent, densit y estimation is a well-studied problem with many prop ositions in the literature. F or instance, [ 4 ] provides a kernel density estimator ρ n that conv erges in L p norm tow ards ρ at a rate of n − 1 /d +1 under our assumptions and provided that M is a C 1 submanifold and that ρ is also C 1 . Under these assumptions, setting f n = ρ − β n yields a conv ergence rate of n − 1 /d according to Lemma 3.6, at the cost of a more complex computation than the usual discrete F ermat distance studied in [14]. 5.2 Distance-to-measure as a confo rmal facto r In general, given a parameter m ∈ (0 , 1) , the distanc e-to-me asur e d µ,m : R N → R + in tro duced in [ 8 ] is deﬁned for any measure µ o v er R N . It is 1 -Lipschitz and low er b ounded by a p ositiv e v alue as long as µ has no atom, hence satisﬁes the assumption for our work. A slightly diﬀeren t setup where paths are allo w ed to leav e the domain under a sp eciﬁc constraint is studied in [ 18 ], where it is argued that this metric should b ehav e similarly to the conformal metric asso ciated with the density but with more stabilit y w.r.t. the measure. The distance- to-measure is sho wn [ 10 ] to b e estimated from n i.i.d. samples of the underlying measure with con v ergence rate n − 1 / 2 , which is faster than n − 1 /d hence do es not impact the conv ergence sp eed of the estimator of the conformal metric according to Lemma 3.6. 6 Conclusion In this work, we hav e shown that under a reach assumption on the domain and Lipschitz lo w er b ounded assumptions on the conformal change, the conformal metric has the same regularit y as the induced metric in the sense of geo desics having p ositive reach. W e hav e also sho wn that p ositiv e reach is a suﬃcient assumption to ensure metric estimation from a p olygonal metric with error prop ortional to the Hausdorﬀ distance b etw een the p oint cloud and the original domain. This leads to a conv ergence rate of order n − 1 /d for the estimation of the conformal metric using n i.i.d. samples from a d -standard measure, which in particular applies to the induced metric of any set with p ositive reach. J. T aupin 15 References 1 Eddie Aamari, Clémen t Berenfeld, and Clémen t Levrard. Optimal reac h estimation and metric learning. The A nnals of Statistics , 51(3):1086–1108, 2023-06. doi:10.1214/23- AOS2281 . 2 Catherine Aaron and Olivier Bo dart. Con vergence rates for estimators of geo desic distances and F rèchet exp ectations. Journal of Applie d Pr ob ability , 55(4):1001–1013, 2018-12. doi: 10.1017/jpr.2018.66 . 3 Ery Arias-Castro, Adel Jav anmard, and Bruno Pelletier. Perturbation Bounds for Pro crustes, Classical Scaling, and T rilateration, with Applications to Manifold Learning. Journal of Ma- chine L e arning Rese ar ch , 21(1):498–534, 2020. URL: http://jmlr.org/papers/v21/18- 720. html . 4 Clémen t Berenfeld and Marc Hoﬀmann. Density estimation on an unknown submanifold. Ele ctr onic Journal of Statistics , 15(1):2179–2223, 2021-01. doi:10.1214/21- EJS1826 . 5 Mira Bernstein, Vin Silv a, John Langford, and Joshua T enenbaum. Graph Approximations to Geo desics on Embedded Manifolds, 2000-12-20. URL: https://api.semanticscholar.org/ CorpusID:10751942 . 6 Jean-Daniel Boissonnat, André Lieutier, and Mathijs Wintraec ken. The reach, metric distortion, geo desic conv exity and the v ariation of tangent spaces. Journal of Applie d and Computational T op olo gy , 3(1):29–58, 2019-06-01. doi:10.1007/s41468- 019- 00029- 8 . 7 Dmitri Burago, Y uri Burago, and Sergei Iv ano v. A Course in Metric Ge ometry , volume 33 of Graduate Studies in Mathematics . American Mathematical Society , 2001-06-12. doi: 10.1090/gsm/033 . 8 F rédéric Chazal, David Cohen-Steiner, and Quentin Mérigot. Geometric Inference for Measures based on Distance F unctions. F oundations of Computational Mathematics , 11(6):733, 2011. doi:10.1007/s10208- 011- 9098- 0 . 9 F rédéric Chazal, Marc Glisse, Catherine Labruère, and Bertrand Mic hel. Conv ergence Rates for P ersistence Diagram Estimation in T op ological Data Analysis. Journal of Machine L e arning R ese ar ch , 16(110):3603–3635, 2015. URL: http://jmlr.org/papers/v16/chazal15a.html . 10 F rédéric Chazal, P ascal Massart, and Bertrand Michel. Rates of conv ergence for robust geometric inference. Ele ctr onic Journal of Statistics , 10(2):2243–2286, 2016-01. doi:10.1214/ 16- EJS1161 . 11 Herb ert F ederer. Curv ature measures. T r ansactions of the A meric an Mathematic al So ciety , 93(3):418–491, 1959. doi:10.1090/S0002- 9947- 1959- 0110078- 1 . 12 Ximena F ernández, Eugenio Borghini, Gabriel Mindlin, and P ablo Groisman. In trinsic p ersisten t homology via density-based metric learning. Journal of Machine L e arning R ese arch , 24(1):75:3341–75:3382, 2023-01-01. URL: https://jmlr.org/papers/v24/21- 1044.html . 13 Nicolas Garcia T rillos, Anna Little, Daniel McKenzie, and James M. Murphy . F ermat Distances: Metric Approximation, Sp ectral Con v ergence, and Clustering Algorithms. Journal of Machine L e arning Rese ar ch , 25(1):176:8331–176:8395, 2024-01-01. URL: http://jmlr.org/papers/ v25/23- 0939.html . 14 P ablo Groisman, Matthieu Jonckheere, and F acundo Sapienza. Nonhomogeneous Euclidean ﬁrst-passage percolation and distance learning. Bernoul li , 28(1):255–276, 2022-02. doi: 10.3150/21- BEJ1341 . 15 Sung Jin Hwang, Steven B. Damelin, and Alfred O. Hero Iii. Shortest path through ran- dom p oints. The Annals of Applie d Pr ob ability , 26(5):2791–2823, 2016-10. doi:10.1214/ 15- AAP1162 . 16 André Lieutier and Mathijs Wintraec ken. Manifolds of p ositive reac h, diﬀerentiabilit y , tangent v ariation, and attaining the reac h, 2024-12-06. . 17 E. J. McShane. Extension of range of functions. Bul letin of the A meric an Mathematic al So ciety , 40(12):837–842, 1934. doi:10.1090/S0002- 9904- 1934- 05978- 0 . 18 Jérôme T aupin and F rédéric Chazal. F ermat Distance-to-Measure: A robust F ermat-like metric, 2025-04-03. . 16 Estimation of Conformal Metrics 19 Bin Y u. Assouad, F ano, and Le Cam. R ese ar ch Papers in Pr ob ability and Statistics , 1997. doi:10.1007/978- 1- 4612- 1880- 7_29 . A Intermediate Results This section is devoted to the pro ofs of some intermediate results used in the main text. A.1 Rate of Change of a Curve with Positive Reach In this section we pro v e Lemma 2.5. Let v + = γ ( t 0 + δ ) − γ ( t 0 ) and v − = γ ( t 0 ) − γ ( t 0 − δ ) . The angle b etw een tw o non-zero vectors is the arccosine of the dot pro duct of the normalized v ectors, thus ∠  v + , ˙ γ ( t 0 )  = arccos  v + ∥ v + ∥ , ˙ γ ( t 0 )  . (18) Since v + = γ ( t 0 + δ ) − γ ( t 0 ) = R t 0 + δ t 0 ˙ γ ( t )d t ,  v + , ˙ γ ( t 0 )  = Z t 0 + δ t 0  ˙ γ ( t ) , ˙ γ ( t 0 )  d t = Z t 0 + δ t 0 cos  ∠  ˙ γ ( t ) , ˙ γ ( t 0 )  d t . (19) Moreo v er, ∥ v + ∥ = ∥ γ ( t 0 + δ ) − γ ( t 0 ) ∥ ≤ δ as γ is 1 -Lisp c hitz, and cos is non-increasing ov er [0 , π 2 ] . Therefore, Equations (4) and (19) imply that  v + ∥ v + ∥ , ˙ γ ( t 0 )  = 1 ∥ v + ∥ Z t 0 + δ t 0 cos  ∠  ˙ γ ( t ) , ˙ γ ( t 0 )  d t ≥ 1 δ Z t 0 + δ t 0 cos  t − t 0 τ γ  d t . (20) No w, since arccos is non-increasing and concav e ov er [0 , 1] , Equations (18) and (20) together with Jensen’s inequality yield ∠  v + , ˙ γ ( t 0 )  ≤ arccos 1 δ Z t 0 + δ t 0 cos  t − t 0 τ γ  d t ! ≤ 1 δ Z t 0 + δ t 0 t − t 0 τ γ d t = δ 2 τ γ whic h concludes Equation (5). The same result holds when comparing ˙ γ ( t 0 ) to a small v ariation b efore t 0 instead of after, therefore ∠  v + , v −  ≤ ∠  v + , ˙ γ ( t 0 )  + ∠  v − , ˙ γ ( t 0 )  ≤ δ τ γ . Moreo v er, applying Equation (2) to the induced metric on the curve γ b et ween endp oin ts γ ( t 0 ) and either γ ( t 0 + δ ) or γ ( t 0 − δ ) shows that δ ≤ min  2 τ γ arcsin  ∥ v + ∥ 2 τ γ  , 2 τ γ arcsin  ∥ v − ∥ 2 τ γ  . Finally ,     v + ∥ v + ∥ − v − ∥ v − ∥     = 2 sin  1 2 ∠  v + , v −   ≤ 2 sin  δ 2 τ γ  ≤ 1 τ γ min  ∥ v + ∥ , ∥ v − ∥  whic h concludes Equation (6). J. T aupin 17 A.2 Convergence of a P oint Cloud in Hausdorﬀ Distance Recall the conv ergence in probability of the Hausdorﬀ distance b et ween an i.i.d. p oin t cloud and the supp ort of a d -standard measure. ▶ Lemma A.1. [ 9 , The or em 2] L et X n b e sample d i.i.d. fr om a d -standar d me asur e µ with supp ort M and lower c onstant c µ . Then for al l ε > 0 , P ( d H ( M , X n ) > ε ) ≤ 4 d c µ ε d exp  − n c µ ε d 2 d  . T o establish our results in Section 4, we require the conv ergence of the Hausdorﬀ distance in exp ectation. ▶ Co rollary A.2. Under the same assumptions as in L emma A.1 and assuming that n ≥ 8 , for al l 1 ≤ p ≤ 2 d it holds that E  d H ( M , X n ) p  ≤ 2 3 p/ 2 c p/d µ  log( n ) n  p/d . Pro of. Denote Z = d H ( M , X n ) , c = 4 d /c µ and α = c µ n/ 2 d for simplicity , so that P ( Z > t ) ≤ c t d exp( − αt d ) (21) for all t > 0 according to Lemma A.1. Then, the exp ectation of Z 2 d can b e written as E  Z 2 d  = Z + ∞ 0 P ( Z 2 d > u )d u = Z + ∞ 0 2 dt 2 d − 1 P ( Z > t )d t . Giv en an y t 0 > 0 , upp er b ounding the probabilit y b y 1 when t ≤ t 0 and using Equation (21) when t > t 0 instead yields E  Z 2 d  ≤ Z t 0 0 2 dt 2 d − 1 d t + 2 c Z + ∞ t 0 dt d − 1 exp( − αt d )d t = t 2 d 0 + 2 c α exp( − αt d 0 ) . Then, setting t 0 =  log( n ) /α  1 /d and replacing c and α with their deﬁnition yields E  Z 2 d  ≤ 2 2 d  log( n ) c µ n  2 + 2 3 d +1 c 2 µ n 2 =  2 2 d + 2 3 d +1 log( n ) 2   log( n ) c µ n  2 ≤ 2 3 d  log( n ) c µ n  2 b y low er b ounding log ( n ) 2 ≥ 4 when n ≥ 8 . Finally , using Jensen’s inequality yields the desired inequality for all 1 ≤ p ≤ 2 d , that is E  Z p  ≤ E  Z 2 d  p 2 d ≤ 2 3 p 2  log( n ) c µ n  p d . ◀ A.3 Ball Graphs and Nea rest-Neighb ors Graphs In this section we pro v e Prop osition 4.4. Denote X n = ( x 1 , . . . , x n ) ∼ µ ⊗ n the p oint cloud. As a preliminary result, denote for all n ∈ N ∗ , p ∈ (0 , 1) and k ∈ R , ϕ ( n, p, k ) the probability that a random v ariable follo wing the binomial law with parameters n and p is greater than 18 Estimation of Conformal Metrics k . Denote ψ ( n, p, k ) the probability that a random v ariable with same law is smaller than k . Then for all k > np , ϕ ( n, p, k ) ≤ exp  1 + log  np k  − np k  k  , (22) and for all k < np , ψ ( n, p, k ) ≤ exp  1 + log  np k  − np k  k  . (23) Indeed, b oth quantities may b e upp er b ounded using a Chernoﬀ b ound by  np k  k  n − np n − k  n − k =  np k  k  1 + k − np n − k  n − k ≤  np k  k e k − np . No w consider k ∈ { 1 , . . . , n − 1 } , ε ∈ (0 , 1) and let r − =  (1 − ε ) k C µ ( n − 1)  1 d and r + =  1 1 − ε k c µ ( n − 1)  1 d . F or all x ∈ M , denote p − x = µ  B ( x, r − )  , which is upp er b ounded by p − = C µ ( r − ) d b y Ahlfors assumption. Denote A − i the even t that B ( x i , r − ) contains at least k p oin ts from X n \ { x i } . Then, since each of the n − 1 other sample p oin ts has a probability p − x ≤ p − of falling within B ( x, r − ) , P ( A − i ) = Z M P ( A − i | x i = x ) d µ ( x ) = Z M ϕ ( n − 1 , p − x , k ) d µ ( x ) ≤ ϕ ( n − 1 , p − , k ) . Notice that ( n − 1) p − = (1 − ε ) k < k . Then, according to Equation (22), P ( A − i ) ≤ exp  (1 + log(1 − ε ) − 1 + ε ) k  ≤ exp  − ε 2 2 k  . If the graph G r − ( X n ) is not included in G k ( X n ) , then there exists tw o sample p oin ts x i and x j that are at most r − apart but such that neither of them is one of the k nearest neigh b ors of the other, and in particular A − i holds. Then P  G r − ( X n ) ⊂ G k ( X n )  ≤ P n [ i =1 A − i ! ≤ n exp  − ε 2 2 k  . (24) F or all x ∈ M denote p + x = µ  B ( x, r + )  , which is low er b ounded by p + = c µ ( r + ) d b y Ahlfors assumption. Denote A + i the ev ent that B ( x i , r + ) contai ns at most k − 1 p oin ts from X n \ { x i } . Then, since each of the n − 1 other sample p oin ts has a probability p + x ≥ p + of falling within B ( x, r + ) , P ( A + i ) = Z M P ( A + i | x i = x ) d µ ( x ) = Z M ψ ( n − 1 , p + x , k − 1) d µ ( x ) ≤ ψ ( n − 1 , p + , k ) . Notice that k = (1 − ε )( n − 1) p + < ( n − 1) p + . Then, according to Equation (23), P ( A + i ) ≤ exp  1 + log  1 1 − ε  − 1 1 − ε  k  ≤ exp  − ε 2 2 k  . J. T aupin 19 If the graph G k ( X n ) is not included in G r + ( X n ) , then there exists tw o sample p oin ts x i and x j that are more than r − apart but suc h that x j is one of the k nearest neigh b ors of x i , and in particular A + i holds. Then P  G k ( X n ) ⊂ G r + ( X n )  ≤ P n [ i =1 A + i ! ≤ n exp  − ε 2 2 k  . (25) Finally , it follows from Equations (24) and (25) that P  G r − ( X n ) ⊂ G k ( X n ) ⊂ G r + ( X n )  ≥ 1 − 2 n exp  − ε 2 2 k  whic h concludes the pro of of Prop osition 4.4. B Deterministic T echnical Pro ofs This section is devoted to technical pro ofs inv olving paths of p ositiv e reach and the ob jects studied in Section 3. B.1 Length of P aths In this section we state some useful inequalities deduced from Equation (2) along with lemmas for comparing the conformal length of paths. The following quantities are all v alid upp er b ounds of arcsin( t ) − t for all 0 < t ≤ 1 2 , which is deduced from elementary computations. arcsin( t ) 3 6 , 4  1 − 3 π  arcsin( t ) t 2 , 4  π 3 − 1  t 3 , 1 8 (arcsin(2 t ) − 2 t ) . (26) The low er b ounds arcsin( t ) − t ≥ arcsin( t ) t 2 6 ≥ t 3 6 (27) also hold for all 0 ≤ t ≤ 1 . Applying the upp er b ounds from Equation (26) to Equation (2) with t = ∥ x − y ∥ / 2 τ M yields the following inequalities. ▶ Lemma B.1. If ∥ x − y ∥ ≤ τ M then ∥ x − y ∥ ≥ D M ( x, y ) − D M ( x, y ) 3 24 τ 2 M , (28) D M ( x, y ) ≤ ∥ x − y ∥ +  π 3 − 1  ∥ x − y ∥ 3 τ 2 M ≤ π 3 ∥ x − y ∥ , (29) 1 − ∥ x − y ∥ D M ( x, y ) ≤  1 − 3 π  ∥ x − y ∥ 2 τ 2 M , (30) D M ( x, y ) − ∥ x − y ∥ ≤ 1 4  τ M arcsin  ∥ x − y ∥ τ M  − ∥ x − y ∥  . (31) 20 Estimation of Conformal Metrics Figure 3 Comparison b etw een the path and the straight line. Note that in Lemma B.1 the set M ma y b e the whole domain as well as an y given curve γ b et ween endp oints x and y and with p ositiv e reach, in which case D γ ( x, y ) = | γ | . The follo wing results sho w that a path with length close to the Euclidean distance betw een its endp oin ts must remain close to the straight line at all time. Since f is Lipsc hitz, it follows that t w o paths with same endp oin ts and length close to the Euclidean distance b etw een the endp oin ts also hav e similar conformal lengths. ▶ Lemma B.2. L et x, y ∈ R N and γ ∈ Γ( x, y ) b e a p ath p ar ameterize d over [0 , 1] with c onstant velo city. Denote γ : t 7→ (1 − t ) x + ty the str aight p ath fr om x to y . Then ∥ γ − γ ∥ 1 ≤ 1 √ 6 p | γ | 2 − | γ | 2 . (32) Pro of. Fix t ∈ [0 , 1] . Denote a ≤ | γ | t and b ≤ | γ | (1 − t ) the resp ectiv e distances from γ ( t ) to x and y . Denote τ ∈ R suc h that γ ( τ ) is the orthogonal pro jection of γ ( t ) onto the line passing through x and y . The fact that τ ma y not b elong to [0 , 1] do es not inﬂuence the follo wing. Finally denote h = ∥ γ ( t ) − γ ( τ ) ∥ . See Figure 3 for an illustration of the ab o ve. By Pythagoras theorem, it holds that h 2 = a 2 − | γ | 2 τ 2 = b 2 − | γ | 2 (1 − τ ) 2 = ∥ γ ( t ) − γ ( t ) ∥ 2 − | γ | 2 ( t − τ ) 2 . In particular, ∥ γ ( t ) − γ ( t ) ∥ 2 = (1 − t )  a 2 − | γ | 2 τ 2  + t  b 2 − | γ | 2 (1 − τ ) 2  + | γ | 2 ( t − τ ) 2 = (1 − t ) a 2 + tb 2 + | γ | 2  ( t − 1) τ 2 − t + 2 tτ − tτ 2 + t 2 − 2 tτ + τ 2  = (1 − t ) a 2 + tb 2 − t (1 − t ) | γ | 2 ≤ (1 − t ) t 2 | γ | 2 + t (1 − t ) 2 | γ | 2 − t (1 − t ) | γ | 2 = t (1 − t )  | γ | 2 − | γ | 2  . In tegrating the ab ov e then yields ∥ γ − γ ∥ 1 ≤ ∥ γ − γ ∥ 2 ≤ s Z 1 0 t (1 − t )  | γ | 2 − | γ | 2  d t = 1 √ 6 p | γ | 2 − | γ | 2 whic h concludes the pro of. ◀ ▶ Corolla ry B.3. L et x, y ∈ R N and γ , ω ∈ Γ M ( x, y ) . Then     | γ | f | γ | − | ω | f | ω |     ≤ κ √ 6  p | γ | 2 − ∥ x − y ∥ 2 + p | ω | 2 − ∥ x − y ∥ 2  . J. T aupin 21 Pro of. Consider parameterizations of γ and ω o v er [0 , 1] with constant velocity . Using in order the κ -Lipsc hitz prop ert y of f , the triangular inequality of ∥ · ∥ 1 and Lemma B.2 yields     | γ | f | γ | − | ω | f | ω |     ≤ Z 1 0   f ( γ ( t )) − f ( ω ( t ))   d t ≤ κ ∥ γ − ω ∥ 1 ≤ κ √ 6  p | γ | 2 − ∥ x − y ∥ 2 + p | ω | 2 − ∥ x − y ∥ 2  whic h concludes the pro of. ◀ B.2 Regula rity of Geo desics In this section we prov e Prop osition 2.4 using the results prov en in App endix B.1. Assume b y contradiction that τ M ,f < T M ,f . Then according to Prop osition 2.3 there exists 0 < r < T M ,f = τ M 2 ∧ f min 8 κ and x, y ∈ M suc h that, letting γ ∈ Γ ⋆ M ,f ( x, y ) b e a geo desic and γ = t 7→ (1 − t ) x + ty b e the straight path from x to y , one has | γ | = ∥ x − y ∥ < 2 r and | γ | ≥ 2 r arcsin  | γ | 2 r  . (33) Let ω ∈ Γ ⋆ M ( x, y ) b e a geo desic w.r.t. the induced metric and let us show that | ω | f < | γ | f , whic h contradicts the geo desic nature of γ . First, rewrite | ω | f − | γ | f =  | ω | | γ | − 1  | γ | f + | ω |  | ω | f | ω | − | γ | f | γ |  . (34) The ﬁrst term in the right-hand side of Equation (34) is negative as | ω | = D M ( x, y ) ≤ | γ | hence can b e upp er b ounded by noticing that | γ | f ≥ | γ | f min . As for the second term, it can b e upp er b ounded according to Corollary B.3. Moreov er, recall that | γ | < 2 r < τ M , hence Equation (29) may b e rewritten as | ω | ≤ | γ | +  π 3 − 1  | γ | 3 τ 2 M ≤ π 3 | γ | . (35) Com bining these three inequalities yields | ω | f − | γ | f ≤  | ω | − | γ |  f min + π κ 3 √ 6 | γ |  p | ω | 2 − | γ | 2 + p | γ | 2 − | γ | 2  . (36) In the following, b oth terms are upp er b ounded in a wa y that factors out | γ | − | γ | . Using in order Equation (31), the fact that 2 r ≤ τ M and t 7→ t arcsin ( | γ | /t ) is non-increasing, and the initial assumption given by Equation (33), it holds that | ω | − | γ | ≤ 1 4  τ M arcsin  | γ | τ M  − | γ |  ≤ 1 4  2 r arcsin  | γ | 2 r  − | γ |  ≤ 1 4  | γ | − | γ |  and | ω | − | γ | = | ω | − | γ | + | γ | − | γ | ≤ − 3 4  | γ | − | γ |  . (37) On the other hand, applying Equation (27) to t = | γ | / 2 r , then Equation (33), | γ | 3 ≤ 6(2 r ) 3  arcsin  | γ | 2 r  − | γ | 2 r  ≤ 24 r 2  | γ | − | γ |  . (38) 22 Estimation of Conformal Metrics Then, using Equation (35) again, then Equation (38), | γ | p | ω | 2 − | γ | 2 = | γ | p | ω | + | γ | p | ω | − | γ | ≤ | γ | r  π 3 + 1  | γ | s  π 3 − 1  | γ | 3 τ 2 M = r π 2 9 − 1 | γ | 3 τ M ≤ r π 2 9 − 1 24 r 2 τ M  | γ | − | γ |  . Hence, since 2 r ≤ τ M , | γ | p | ω | 2 − | γ | 2 ≤ 12 r π 2 9 − 1 r  | γ | − | γ |  . (39) Lik ewise, Equation (38) and | γ | ≤ 2 r imply that | γ | p | γ | 2 − | γ | 2 = q 2 | γ | 3 + | γ | 2  | γ | − | γ |  p | γ | − | γ | ≤ p 48 r 2 + (2 r ) 2  | γ | − | γ |  . (40) Finally , combining Equations (36), (37), (39), and (40) yields | ω | f − | γ | f ≤ − 3 4 f min + π 3 √ 6 12 r π 2 9 − 1 + 2 √ 13 ! κr !  | γ | − | γ |  . By assumption, r < f min 8 κ ≤ 12 r π 2 9 − 1 + 2 √ 13 ! − 1 9 √ 6 4 π f min κ , whic h concludes that | ω | f − | γ | f is negativ e, hence the contradiction and Prop osition 2.4 is pro v en. The choice of the constant 1 8 instead of the one app earing in the right-hand side of the last inequality is purely for cosmetic reasons. B.3 W eight F unction In this section w e state the Lipsc hitz prop ert y of the weigh t function deﬁned in Deﬁnition 3.2 w.r.t. the function f and the endp oints ( x, y ) . The latter is inherited from the Lipschitz nature of f . W e also quantify the conv ergence of w f ,q to w f , ∞ as q → ∞ and then prov e Prop osition 3.3. The following three lemmas are obtained through elementary computations that are not detailed here. Recall that w f ,q ( x, y ) = ∥ x − y ∥ 2( q − 1) f ( x ) + 2 q − 1 X k =2 f  q − k q − 1 x + k − 1 q − 1 y  + f ( y ) ! and w f , ∞ ( x, y ) = ∥ x − y ∥ Z 1 0 f  (1 − t ) x + ty  d t . ▶ Lemma B.4. L et f : R N → R ∗ + b e a κ -Lipschitz function and q ∈ { 2 , . . . , ∞} . Then the weight function w f ,q satisﬁes for al l x, y , x ′ , y ′ ∈ M     w f ,q ( x, y ) ∥ x − y ∥ − w f ,q ( x ′ , y ′ ) ∥ x ′ − y ′ ∥     ≤ κ ∥ x − x ′ ∥ + ∥ y − y ′ ∥ 2 . J. T aupin 23 ▶ Lemma B.5. L et f , g : R N → R ∗ + b e two functions and q ∈ { 2 , . . . , ∞} . Then for al l x, y ∈ M   w f ,q ( x, y ) − w g ,q ( x, y )   ≤ ∥ x − x ∥∥ f − g ∥ ∞ . ▶ Lemma B.6. L et f : R N → R ∗ + b e a κ -Lipschitz function and q ≥ 2 an inte ger. Then for al l x, y ∈ M | w f ,q ( x, y ) − w f , ∞ ( x, y ) | ≤ κ 4 ∥ x − y ∥ 2 q − 1 . Lemma B.7 b elo w states the upp er b ound on the distortion of the weigh t function when q = ∞ . The case of ﬁnite q is then deduced from Lemma B.6. Indeed, according to Lemmas B.6 and B.7, for all ∥ x − y ∥ ≤ τ M ,f one has     1 − w f ,q ( x, y ) D M ,f ( x, y )     ≤     w f , ∞ ( x, y ) − w f ,q ( x, y ) D M ,f ( x, y )     +     1 − w f , ∞ ( x, y ) D M ,f ( x, y )     ≤ κ 4 ∥ x − y ∥ 2 ( q − 1) D M ,f ( x, y ) + ∥ x − y ∥ 2 16 T 2 M ,f ≤ κ 4 f min ∥ x − y ∥ ( q − 1) + ∥ x − y ∥ 2 16 T 2 M ,f since D M ,f ( x, y ) ≥ f min ∥ x − y ∥ . This concludes Prop osition 3.3. ▶ Lemma B.7. L et x, y ∈ M such that ∥ x − y ∥ ≤ τ M ,f . Then     1 − w f , ∞ ( x, y ) D M ,f ( x, y )     ≤ ∥ x − y ∥ 2 16 T 2 M ,f . (41) Pro of. Let γ ∈ Γ ⋆ M ,f ( x, y ) b e a geodesic and γ b e the straigh t path from x to y . By deﬁnition D M ,f ( x, y ) = | γ | f , w f , ∞ ( x, y ) = | γ | f and | γ | = ∥ x − y ∥ . First rewrite D M ,f ( x, y ) − w f , ∞ ( x, y ) =  1 − | γ | | γ |  | γ | f + | γ |  | γ | f | γ | − | γ | f | γ |  . (42) Regarding the ﬁrst term, as ∥ x − y ∥ ≤ τ M ,f ≤ τ γ , applying Equation (30) to the induced metric D γ yields 0 ≤ 1 − | γ | | γ | = 1 − ∥ x − y ∥ D γ ( x, y ) ≤  1 − 3 π  ∥ x − y ∥ 2 τ 2 γ . (43) As for the second term, applying Corollary B.3 yields     | γ | f | γ | − | γ | f | γ |     ≤ κ √ 6 p | γ | 2 − | γ | 2 = κ √ 6 | γ | s 1 + | γ | | γ | s 1 − | γ | | γ | and using the fact that | γ | f ≥ | γ | f min and | γ | ≤ | γ | along with Equation (43) further implies that     | γ | f | γ | − | γ | f | γ |     ≤ κ √ 6 | γ | f f min √ 2 s  1 − 3 π  ∥ x − y ∥ 2 τ 2 γ . (44) 24 Estimation of Conformal Metrics Finally , combining Equations (42)–(44) together with the fact that τ γ ≥ T M ,f and f min 8 κ ≥ T M ,f yields     1 − w f , ∞ ( x, y ) D M ,f ( x, y )     ≤  1 − 3 π  1 τ 2 γ + r 1 3 − 1 π κ f min τ γ ! ∥ x − y ∥ 2 ≤ 1 − 3 π + 1 8 r 1 3 − 1 π ! ∥ x − y ∥ 2 T 2 M ,f ≤ ∥ x − y ∥ 2 16 T 2 M ,f where the last inequality is purely cosmetic and concludes the pro of. ◀ B.4 Discrete Appro ximation of the Conformal Metric In this section w e prov e Theorem 3.5. Let X ⊂ M b e a point cloud and denote ρ = d H ( M , X ) . Assume that 4 ρ ≤ r ≤ T M ,f and let x, y ∈ M . Lo wer b ound Let ( x 0 , . . . , x K ) b e a p olygonal path in G r ( X ∪ { x, y } ) from x = x 0 to y = x K ac hieving the inﬁmum in Deﬁnition 3.4. According to Prop osition 3.3, b D X,f ( x, y ) = K − 1 X k =1 w f ,q ( x k , x k +1 ) ≥ K − 1 X k =1  1 − δ q ( ∥ x k − x k +1 ∥ )  D M ,f ( x k , x k +1 ) . Using the fact that δ q is non-decreasing and that ∥ x k − x k +1 ∥ ≤ r for all k , along with the triangular inequality of D M ,f , it follows that b D X,f ( x, y ) ≥  1 − δ q ( r )  D M ,f ( x, y ) . (45) Lo cal upp er b ound If ∥ x − y ∥ ≤ r , the edge ( x, y ) b elongs to G r ( X ∪ { x, y } ) , hence according to Prop osition 3.3, b D X,f ( x, y ) ≤ w f ,q ( x, y ) ≤  1 + δ q ( r )  D M ,f ( x, y ) . (46) Global upp er b ound Assume that ∥ x − y ∥ ≥ r . Let γ ∈ Γ ⋆ M ,f ( x, y ) b e a geo desic and decomp ose it into K = ⌈| γ | / ( r − 2 ρ ) ⌉ sections of equal arc-length l = | γ | /K . Denote ( x = x 0 , x 1 , . . . , x K = y ) the intermediate p oin ts resulting from this decomp osition and ( x = x ′ 0 , x ′ 1 , . . . , x ′ K = y ) their resp ective nearest neighbors in the p oint cloud X ∪ { x, y } . Then 1 ≤ | γ | r − 2 ρ ≤ K ≤ 2 | γ | r − 2 ρ and r 4 ≤ r 2 − ρ ≤ l ≤ r − 2 ρ . In particular, ∥ x ′ k − x ′ k +1 ∥ ≤ ∥ x ′ k − x k ∥ + ∥ x k − x k +1 ∥ + ∥ x k +1 − x ′ k +1 ∥ ≤ ρ + l + ρ ≤ r so that ( x ′ 0 , . . . , x ′ K ) is indeed a path in G r ( X ∪ { x, y } ) . Denote for short ε k = x ′ k − x k , w k = w f ,q ( x k , x k +1 ) , w ′ k = w f ,q ( x ′ k , x ′ k +1 ) , u k = x k − x k +1 ∥ x k − x k +1 ∥ , w k = w f ,q ( x k , x k +1 ) ∥ x k − x k +1 ∥ , w ′ k = w f ,q ( x ′ k , x ′ k +1 ) ∥ x ′ k − x ′ k +1 ∥ . J. T aupin 25 Let us sho w that the w eigh t P k w ′ k of the p olygonal path is not muc h greater than the conformal distance D M ,f ( x, y ) . First, rewrite w ′ k = w k + ∥ x ′ k − x ′ k +1 ∥ ( w ′ k − w k ) +  ∥ x ′ k − x ′ k +1 ∥ − ∥ x k − x k +1 ∥  w k . (47) In order to upp er b ound the last term in Equation (47), we use the following inequality from the pro of of Theorem 2.1 in [2]: ∥ x ′ k − x ′ k +1 ∥ ≤ ∥ x k − x k +1 ∥ + ⟨ u k , ε k − ε k +1 ⟩ + 1 2 ∥ ε k − ε k +1 ∥ 2 ∥ x k − x k +1 ∥ , whic h implies that  ∥ x ′ k − x ′ k +1 ∥ − ∥ x k − x k +1 ∥  w k ≤ ⟨ w k u k , ε k − ε k +1 ⟩ + w k 2 ∥ ε k − ε k +1 ∥ 2 ∥ x k − x k +1 ∥ 2 and from Equation (47) it follo ws that w ′ k ≤ w k |{z} (i) + ∥ x ′ k − x ′ k +1 ∥ ( w ′ k − w k ) | {z } (ii) + ⟨ w k u k , ε k − ε k +1 ⟩ | {z } (iii) + w k 2 ∥ ε k − ε k +1 ∥ 2 ∥ x k − x k +1 ∥ 2 | {z } (iv) . (48) Let us now upp er b ound all four terms in this upp er b ound. (i) Prop osition 3.3 ensures that w k ≤  1 + δ q ( ∥ x k − x k +1 ∥ )  D M ,f ( x k , x k +1 ) ≤  1 + δ q ( r )  D M ,f ( x k , x k +1 ) . (49) (ii) Lemma B.4 ensures that ∥ x ′ k − x ′ k +1 ∥ ( w ′ k − w k ) ≤ ∥ x ′ k − x ′ k +1 ∥ κ 2  ∥ x ′ k − x k ∥ + ∥ x ′ k +1 − x k +1 ∥  ≤ ( l + 2 ρ ) κρ , whic h implies since ρ ≤ r 4 ≤ l and D M ,f ( x k , x k +1 ) ≥ lf min that ∥ x ′ k − x ′ k +1 ∥ ( w ′ k − w k ) ≤ 3 κ f min ρ D M ,f ( x k , x k +1 ) . (50) (iii) Recall that ε 0 = ε K = 0 . Then when summing ov er k , the terms can b e rearranged so that K − 1 X k =0 ⟨ w k u k , ε k − ε k +1 ⟩ = K − 1 X k =1 ⟨ w k u k − w k − 1 u k − 1 , ε k ⟩ = K − 1 X k =1  w k ⟨ u k − u k − 1 , ε k ⟩ + ( w k − w k − 1 ) ⟨ u k − 1 , ε k ⟩  . A ccording to Lemma 2.5, ∥ u k − u k +1 ∥ ≤ 1 τ γ ∥ x k − x k +1 ∥ since r − 2 ρ ≤ T M ,f ≤ τ γ . A ccording to Lemma B.4, w k − w k − 1 ≤ κ 2  ∥ x k − x k +1 ∥ + ∥ x k +1 − x k +2 ∥  ≤ κl . T hen, using Cauch y-Sch wartz inequality and D M ,f ( x k , x k +1 ) ≥ lf min , w k ⟨ u k − u k − 1 , ε k ⟩ + ( w k − w k − 1 ) ⟨ u k − 1 , ε k ⟩ ≤ w k τ γ ∥ x k − x k +1 ∥ ρ + κlρ ≤ w k τ γ ρ + κ f min D M ,f ( x k , x k +1 ) ρ . A dding the analogous term corresp onding to k = 0 , K − 1 X k =0 ⟨ w k u k , ε k − ε k +1 ⟩ ≤ K − 1 X k =0  w k τ γ + κ f min D M ,f ( x k , x k +1 )  ρ . (51) 26 Estimation of Conformal Metrics (iv) Since ∥ x k − x k +1 ∥ ≤ D γ ( x k , x k +1 ) = l ≤ τ γ and r ≤ 4 l , applying Equation (28) to the metric D γ yields ∥ x k − x k +1 ∥ ≥ l − l 3 24 τ 2 γ ≥ 23 24 l ≥ 23 96 r , hence w k 2 ∥ ε k − ε k +1 ∥ 2 ∥ x k − x k +1 ∥ 2 ≤ w k 2  96 23  2 4 ρ 2 r 2 . (52) Finally , since ( x 0 , . . . , x K ) are in termediate p oints on the geodesic γ from x to y , D M ,f ( x, y ) = P K − 1 k =0 D M ,f ( x k , x k +1 ) . T ogether with Equations (48)–(52), this yields K − 1 X k =0 w ′ k ≤  1 + δ q ( r ) + 4 κ f min ρ  D M ,f ( x, y ) + ρ τ γ + 2  96 23  2 ρ 2 r 2 ! K − 1 X k =0 w k In particular, recall that κ f min ≤ 1 8 T M,f and r ≤ T M ,f , hence δ q ( r ) ≤ 9 64 and Equation (49) further implies that w k ≤ 73 64 D M ,f ( x k , x k +1 ) , so that b D X,f ( x, y ) ≤ K − 1 X k =0 w ′ k ≤ 1 + δ q ( r ) + 105 64 ρ T M ,f + 73 32  96 23  2 ρ 2 r 2 ! D M ,f ( x, y ) . Upp er b ounding 105 64 ≤ 2 and 73 32  96 23  2 ≤ 40 to get cleaner constants, it ﬁnally holds that b D X,f ( x, y ) ≤  1 + δ q ( r ) + 2 ρ T M ,f + 40 ρ 2 r 2  D M ,f ( x, y ) . (53) Conclusion According to the low er b ound Equation (45) and to the upp er b ounds Equa- tions (46) and (53) dep ending on the case, the distortion b et w een the p olygonal metric and the original metric is upp er b ounded as l ∞ ,M  b D X,f | D M ,f  ≤ δ q ( r ) + 2 ρ T M ,f + 40 ρ 2 r 2 = κ 4 f min r q − 1 + r 2 16 T 2 M ,f + 2 · r 4 T M ,f · 4 ρ r + 40 ρ 2 r 2 ≤ 1 32 T M ,f r q − 1 + r 2 8 T 2 M ,f + 56 ρ 2 r 2 where we used the inequality 2 ab ≤ a 2 + b 2 . This concludes the pro of of Theorem 3.5. B.5 App roximation of the Confo rmal Facto r In this section we pro v e Lemma 3.6. Fix x, y ∈ M and denote b D g = b D X,g ( x, y ) , b D f = b D X,f ( x, y ) and D = D M ,f ( x, y ) for short. Denote G the graph considered for the paths in b oth b D g and b D f and q the resolution. First, notice that the assumption that ∥ g − f ∥ ∞ ≤ 1 2 f min implies the low er b ound g min = 1 2 f min on g . F or any edge ( u, v ) in G , Lemma B.5 yields   w g ,q ( u, v ) − w f ,q ( u, v )   ≤ ∥ u − v ∥∥ g − f ∥ ∞ . (54) J. T aupin 27 Then, since w g ,q ( u, v ) ≥ ∥ u − v ∥ g min , summing Equation (54) ov er a path that realizes b D g yields b D f ≤  1 + ∥ g − f ∥ ∞ g min  b D g ≤ 2 b D g . (55) On the other hand, since w f ,q ( u, v ) ≥ ∥ u − v ∥ f min , summing Equation (54) ov er a path that realizes b D f yields b D g ≤  1 + ∥ g − f ∥ ∞ f min  b D f , and it follows that    b D g − b D f    ≤ max  ∥ g − f ∥ ∞ g min b D g , ∥ g − f ∥ ∞ f min b D f  = 2 ∥ g − f ∥ ∞ f min b D g . (56) Then, Equations (55) and (56) imply that    b D g − D    ≤    b D g − b D f    +    b D f − D    ≤ 2 ∥ g − f ∥ ∞ f min b D g + ℓ ∞ ,M  b D X,f , D M ,f  b D f ∨ D  ≤ 2 ∥ g − f ∥ ∞ f min b D g + 2 ℓ ∞ ,M  b D X,f , D M ,f  b D g ∨ D  . T aking the supremum ov er x and y ﬁnally yields ℓ ∞ ,M  b D X,g , D M ,f  ≤ 2  ∥ g − f ∥ ∞ f min + ℓ ∞ ,M  b D X,f , D M ,f   whic h concludes the pro of of Lemma 3.6. C Probabilistic T echnical Pro ofs C.1 Convergence on the Ball Graph In this section we prov e Theorem 4.2. In the following we say that something is true for n large enough when it is true for all n larger than some constant dep ending on L µ , T M ,f and d . Assume that 0 < r ≤ T M ,f and q ∈ { 2 , . . . , ∞} . Then E h ℓ ∞ ,M  b D X n ,f , D M ,f  i ≤ P  4 d H ( M , X n ) > r  + E h 1 l (4 d H ( M ,X n ) ≤ r ) l ∞ ,M  b D X n ,f | D M ,f  i since ℓ ∞ ,M is upp er b ounded b oth b y 1 and by l ∞ ,M according to Equation (9). According to Lemma A.1, P  4 d H ( M , X n ) > r  ≤ 16 d c µ r d e − nc µ ( r 8 ) d =  16 L µ r  d e − n  r 8 L µ  d . (57) On the other hand, assuming that n ≥ 8 , Theorem 3.5 and Corollary A.2 yield E h 1 l (4 d H ( M ,X n ) ≤ r ) ℓ ∞ ,M  b D X n ,f , D M ,f  i ≤ r 32( q − 1) T M ,f + r 2 8 T 2 M ,f + 448 L 2 µ r 2  log( n ) n  2 d (58) 28 Estimation of Conformal Metrics so that E h ℓ ∞ ,M  b D X n ,f , D M ,f  i ≤ A n ( r ) + B n ( r , q ) (59) where A n ( r ) and B n ( r , q ) resp ectively denote the righ t-hand sides in Equation (57) and in Equation (58). W e no w consider the tw o cases for the choice of the parameters r and q sp eciﬁed in Theorem 4.2. La rge resolution Let r = 8 p L µ T M ,f ( log ( n ) /n ) 1 / 2 d , which is indeed smaller than T M ,f for n large enough. Let q ≥ 1 + 4 T M,f r . Then A n ( r ) = 2 d  L µ T M ,f  d 2 r n log( n ) exp −  T M ,f L µ  d 2 p n log( n ) ! ≤ 1 2 L µ T M ,f  log( n ) n  1 d for n large enough and B n ( r , q ) ≤ B n  1 + 4 T M ,f r , r  =  1 2 + 8 + 7  L µ T M ,f  log( n ) n  1 d , so that according to Equation (59), E h ℓ ∞ ,M  b D X n ,f , D M ,f  i ≤ 16 L µ T M ,f  log( n ) n  1 d whic h concludes the pro of of Equation (15). Small resolution Let r = 8 L 2 / 3 µ T 1 / 3 M ,f ( log ( n ) /n ) 2 / 3 d , which is indeed smaller than T M ,f for n large enough. Let q = 2 . Then A n ( r ) = 2 d  L µ T M ,f  d 3  n log( n )  2 3 exp −  T M ,f L µ  d 3 n 1 3 log( n ) 2 3 ! ≤ 1 2  L µ T M ,f  2 3  log( n ) n  2 3 d for n large enough and B n ( q , r ) =  1 4 + 7   L µ T M ,f  2 3  log( n ) n  2 3 d + 8  L µ T M ,f  4 3  log( n ) n  4 3 d ≤ 15 2  L µ T M ,f  2 3  log( n ) n  2 3 d for n large enough, so that according to Equation (59), E h ℓ ∞ ,M  b D X n ,f , D M ,f  i ≤ 8  L µ T M ,f  2 3  log( n ) n  2 3 d whic h concludes the pro of of Equation (16). J. T aupin 29 C.2 Convergence on the Nea rest-Neighb ors Graph In this section we prov e Theorem 4.5 using Theorem 4.2 and Prop osition 4.4. Consider k =  p n log( n )  , q = ⌈ n 1 / 4 ⌉ and let r − =  k 2 C µ ( n − 1)  1 d and r + =  2 k c µ ( n − 1)  1 d , so that according to Prop osition 4.4, the even t A 1 def =  G r − ( X n ) ⊂ G k ( X n ) ⊂ G r + ( X n )  satisﬁes P  A 1  ≤ 2 n exp  − 1 8 p n log( n )  . (60) Denote b D r − , b D r + and b D k the three diﬀerent instances of b D X n ,f deﬁned for the r − -ball graph, the r + -ball graph and the k -NN graph resp ectiv ely , all with resolution q . Also denote D = D M ,f for short. Under the even t A 1 , for all x, y ∈ X n it holds that b D r + ( x, y ) ≤ b D k ( x, y ) ≤ b D r − ( x, y ) as the graphs are equipp ed with the same weigh t function w f ,q and the inﬁmum o v er paths increases when fewer paths are allow ed due to the graph b eing smaller. It follows that l ∞ ,X n  b D k | D  = sup x  = y ∈ M      D ( x, y ) − b D k ( x, y ) D ( x, y )      ≤ sup x  = y ∈ M      D ( x, y ) − b D r − ( x, y ) D ( x, y )      ∨      D ( x, y ) − b D r + ( x, y ) D ( x, y )      = l ∞ ,X n  b D r − | D  ∨ l ∞ ,X n  b D r + | D  . Under the even t A 2 def =  ℓ ∞ ,X  b D r − , D  ≤ 1 2  ∩  ℓ ∞ ,X n  b D r + , D  ≤ 1 2  , it then holds according to Equation (9) that ℓ ∞ ,X n  b D k , D  ≤ l ∞ ,X n  b D k | D  ≤ l ∞ ,X n  b D r − | D  ∨ l ∞ ,X n  b D r + | D  ≤ 2 ℓ ∞ ,X n  b D r − , D  + 2 ℓ ∞ ,X n  b D r + , D  . This implies that E h 1 l A 1 ∩ A 2 ℓ ∞ ,X n  b D k , D  i ≤ 2 E h ℓ ∞ ,X n  b D r − , D  i + 2 E h ℓ ∞ ,X n  b D r + , D  i . (61) The even t A 1 is already known to hav e high probability . Regarding A 2 , Marko v inequality yields P  A 2  ≤ 2 E h ℓ ∞ ,X n  b D r − , D  i + 2 E h ℓ ∞ ,X n  b D r + , D  i . (62) 30 Estimation of Conformal Metrics Since r − ≍ r + ≍ ( log ( n ) /n ) 1 / 2 d when n is large and q = ⌈ n 1 / 4 ⌉ ≳ 1 /r + ≥ 1 /r − , there exists according to Theorem 4.2 a constant c dep ending on L µ and T M ,f suc h that E h ℓ ∞ ,X n  b D r − , D  i + E h ℓ ∞ ,X n  b D r + , D  i ≤ c  log( n ) n  1 d . (63) Finally , as ℓ ∞ ,X n ≤ 1 , combining Equations (60)–(63) yields E h ℓ ∞ ,X n  b D k , D  i ≤ E h 1 l A 1 ∩ A 2 ℓ ∞ ,X n  b D k , D  i + P  A 2  + P  A 1  ≤ 4 E h ℓ ∞ ,X n  b D r − , D  i + 4 E h ℓ ∞ ,X n  b D r + , D  i + 2 ne − 1 8 √ n log( n ) ≤ C  log( n ) n  1 d where C is a constant dep ending on L µ and T M ,f . This concludes the pro of of Theorem 4.5. C.3 Minimax Low er Bound In this section we prov e Theorem 4.6 using Le Cam’s metho d [ 19 ]. W e ﬁrst adapt this standard result to our context. ▶ Lemma C.1. L et n ≥ 2 and M b e some set of distributions over R N such that ther e exists two me asur es µ 1 , µ 2 ∈ M with r esp e ctive supp orts M 1 and M 2 and at total variation distanc e d TV ( µ 1 , µ 2 ) ≤ 1 n fr om e ach other. Then, for al l x, y ∈ M 1 ∩ M 2 the minimax risk asso ciate d with the estimation of the induc e d metric may b e lower b ounde d by the distortion b etwe en the two metrics D M 1 and D M 2 at endp oints x and y . inf ˆ D ∈ D n sup µ ∈M E X ∼ µ ⊗ n h ℓ ∞ ,M µ  b D X , D M µ  i ≥ 1 8     D M 1 ( x, y ) − D M 2 ( x, y ) D M 1 ( x, y ) ∨ D M 2 ( x, y )     (64) wher e we r e c al l D n is the set of p ossible estimators deﬁne d in The or em 4.6 and M µ denotes the supp ort of µ . Pro of. Denote R n the left-hand side of Equation (64) to b e low er b ounded. The supremum o v er all measures in M is low er b ounded by the suprem um ov er the tw o measures µ 1 and µ 2 and the loss ℓ ∞ ,M µ b y the distortion b etw een endp oints x and y . Finally , the supremum is lo w er b ounded by the av erage. These inequalities yield R n ≥ inf b D sup µ ∈{ µ 1 ,µ 2 } E µ ⊗ n h ℓ ∞ ,M µ  b D X , D M  i ≥ inf b D sup µ ∈{ µ 1 ,µ 2 } E µ ⊗ n "      b D X ( x, y ) − D M µ ( x, y ) b D X ( x, y ) ∨ D M µ ( x, y )      # ≥ 1 2 inf b D ( E µ ⊗ n 1 "      b D X ( x, y ) − D M 1 ( x, y ) b D X ( x, y ) ∨ D M 1 ( x, y )      # + E µ ⊗ n 2 "      b D X ( x, y ) − D M 2 ( x, y ) b D X ( x, y ) ∨ D M 2 ( x, y )      #) ≥ 1 2 inf b D E µ ⊗ n 1 "      b D X ( x, y ) − D M 1 ( x, y ) b D X ( x, y ) ∨ D M 1 ( x, y )      +      b D X ( x, y ) − D M 2 ( x, y ) b D X ( x, y ) ∨ D M 2 ( x, y )      !  1 ∧ dµ ⊗ n 2 dµ ⊗ n 1  # and it can b e shown using elementary computations that for all a, b > 0 , min δ ≥ 0      δ − a δ ∨ a     +     δ − b δ ∨ b      =     a − b a ∨ b     , J. T aupin 31 hence R n ≥ 1 2 E µ ⊗ n 1      D M 1 ( x, y ) − D M 2 ( x, y ) D M 1 ( x, y ) ∨ D M 2 ( x, y )      1 ∧ dµ ⊗ n 2 dµ ⊗ n 1  . Moreo v er, using the prop erties of the total v ariation distance along with the assumption that d TV ( µ 1 , µ 2 ) ≤ 1 n and n ≥ 2 , E µ ⊗ n 1  1 ∧ dµ ⊗ n 2 dµ ⊗ n 1  = 1 − d TV ( µ ⊗ n 1 , µ ⊗ n 2 ) =  1 − d TV ( µ 1 , µ 2 )  n ≥  1 − 1 n  n ≥ 1 4 , hence R n ≥ 1 8     D M 1 ( x, y ) − D M 2 ( x, y ) D M 1 ( x, y ) ∨ D M 2 ( x, y )     whic h concludes the pro of. ◀ Let us now prov e Theorem 4.6 using Lemma C.1 with the measures µ 1 and µ 2 describ ed in Section 4.3. The total v ariation distance b et ween µ 1 , µ 2 ∈ M ( d, L, τ ) is the fraction of v olume of M 1 that was carved out to create M 2 . The carved vol ume is upp er b ounded by ∥ x − y ∥ ρ d − 1 where ρ is the distance of the middle of the black arcs in Figure 2 to the missing edge, that is ρ = ( τ − δ ) / √ d − 1 where δ = √ τ 2 − ε 2 is the distance from the edge to the cen ter of the ball used to carve M 2 . Then, d TV ( µ 1 , µ 2 ) ≤ ∥ x − y ∥ ρ d − 1 (2 αL ) d = 2 ε ( αL ) d ( d − 1) d − 1 2 τ d − 1 1 − r 1 − ε 2 τ 2 ! d − 1 ≤ 2 α d ( d − 1) d − 1 2 ετ d − 1 L d  ε 2 τ 2  d − 1 = β ε 2 d − 1 L d τ d − 1 where β = 2 α − d ( d − 1) − ( d − 1) / 2 is a constant dep ending only on d . Moreov er, since a geo desic b et ween x and y is a straight line in M 1 and an arc of radius τ in M 2 , the induced distortion is D M 2 ( x, y ) − D M 1 ( x, y ) D M 2 ( x, y ) ∨ D M 1 ( x, y ) = arcsin  ∥ x − y ∥ 2 τ  − ∥ x − y ∥ 2 τ arcsin  ∥ x − y ∥ 2 τ  ≥ 1 6  ∥ x − y ∥ 2 τ  2 = ε 2 6 τ 2 using Equation (27) to get the lo w er b ound. Then, by letting ε =  L d τ d − 1 β n  1 2 d − 1 , whic h is indeed smaller than b oth αL and τ for n large enough dep ending on d , L and τ , one has d TV ( µ 1 , µ 2 ) ≤ 1 n and D M 2 ( x, y ) − D M 1 ( x, y ) D M 2 ( x, y ) ∨ D M 1 ( x, y ) ≥ 1 6  L d β τ d n  2 2 d − 1 . Theorem 4.6 then follows from Lemma C.1.

Estimation of Conformal Metrics

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