Geometric inference for measures based on distance functions The DTM-signature for a geometric comparison of metric-measure spaces from samples

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1 Distance to Geometric inference for measures based on distance functions The - for a geometric comparison of metric-measure spaces from samples the Ohio State University wan.252@osu.edu 1/31

2 Geometric inference problem Question Given a noisy point cloud approximation C of a compact set K R d, how can we recover geometric and topological informations about K, such as its curvature, boundaries, Betti numbers, etc. knowing only the point cloud C? 2/31

3 Inference using distance functions One idea to retrieve information of a point cloud is to consider the R-offset of the point cloud - that is the union of balls of radius R whose center lie in the point cloud. This offset makes good estimation of the topology, normal cones, and curvature measures of the underlying object, shown in previous literature. The main tool used is a notion of distance function. 3/31

4 Inference using distance functions For a compact K R d, d K : R d R x dist(x, K) 1 d K is 1-Lipschitz. 2 d 2 K is 1-semiconcave. 3 d K d K d H (K, K ). 4/31

5 Unfortunately, offset-based methods do not work well at all in the presence of outliers. For example, the number of connected components will be overestimated if one adds just a single data point far from the original point cloud. 5/31

6 Solution to outliers Replace the distance function to a set K by a distance function to a measure. (Chazal, et al 2010) 6/31

7 Distance to a Measure Notice d K (x) = min y K x y = min{r > 0 : B(x, r) K }. Given a probability measure µ on R d, we mimick the formula above: δ µ,m : x R d inf{r > 0; µ( B(x, r)) > m}, which is 1-Lipschitz but not semi-concave. 7/31

8 Distance to a Measure Definition For any measure µ with finite second moment and a positive mass parameter m 0 > 0, the distance function to measure () µ is defined by the formula: d 2 µ,m 0 : R n R, x 1 m0 δ µ,m (x) 2 dm. m 0 0 Recall δ µ,m (x) = inf{r > 0; µ( B(x, r)) > m}. 8/31

9 Example Let C = {p 1,, p n } be a point cloud and µ C = 1 n i δ p i. Then function δ µc,m 0 with m 0 = k/n evaluated at x R d equal to the distance between x and its kth nearest neighbor in C. Given S C with S = k, define Vor C (S) = {x R d : p i / S, d(x, p i ) > d(x, S).}, which means its elements take S as their k first nearest neighbors in C. x Vor C (S), d 2 (x) = n x p 2. µ C, k n k p S 9/31

10 Proposition Equivalent formulation 1 is the minimal cost of the following problem: { ( 1 d µ,m0 (x) = min W2 δx, µ ) ; µ(r d ) = m 0, µ µ } µ m 0 2 Denote the set of minimizers as R µ,m0 (x). Then for each µ x,m0 R µ,m0 (x), supp( µ x,m0 ) B(x, δ µ,m0 (x)); µ x,m0 B(x,δµ,m0 (x)) = µ B(x,δµ,m0 (x)) ; µ x,m0 µ. 3 For any µ x,m0 R µ,m0 (x), d 2 µ,m 0 (x) = 1 m 0 h R d h x 2 d µ x,m0 = W 2 2 ( 1 ) δ x, µ x,m0. m 0 10/31

11 Regularity Properties Proposition 1 d 2 µ,m 0 is semiconcave, which means x 2 d 2 µ,m 0 is convex; 2 d 2 µ,m 0 is differentiable at a point x iff supp(µ) B(x, δ µ,m0 (x)) contains at most 1 point; 3 d 2 µ,m 0 is differentiable almost everywhere in R d in Lebesgue measure. (directly from item 1) 4 d µ,m0 is 1-Lipschitz. 11/31

12 Stability of Theorem ( stability theorem) If µ, ν are two probability measures on R d and m 0 > 0, then d µ,m0 d ν,m0 1 m0 W 2 (µ, ν). 12/31

13 Uniform Convergence of Lemma If µ is a compactly-supported measure, then d S is the uniform limit of d µ,m0 as m 0 converges to 0, where S = supp(µ), i.e., lim d µ,m 0 d S m 0 0 = 0. Remark If µ has dimension at most k > 0, i.e. µ(b(x, ɛ)) Cɛ k, x S when ɛ is small, then we can control the convergence speed: d µ,m0 d S = O(m 1/k 0 ). 13/31

14 Reconstruction from noisy data If µ is a probability measure of dimension at most k > 0 with compact support K R d, and µ is another probability measure, one has d K d µ,m 0 d K d µ,m0 + d µ,m0 d µ,m 0 O(m 1/k 0 ) + 1 m0 W 2 (µ, µ ). 14/31

15 Reconstruction from noisy data Define α-reach of K, α (0, 1] as r α (K) = inf{d K (x) > 0 : x d K α}. Theorem Suppose µ has dimension at most k with compact support K R d such that r α (K) > 0 for some α. For any 0 < η < r α (K), m 1 = m 1 (µ, α, η) > 0 and C = C(m 1 ) > 0 such that: for any m 0 < m 1 and µ satisfying W 2 (µ, µ ) < C m 0, d 1 µ,m 0 ([0, η]) is homotopy equivalent to the offset d 1 K ([0, r]) for 0 < r < r α(k). 15/31

16 Example Figure: On the left, a point cloud sampled on a mechanical part to which 10% of outliers have been added- the outliers are uniformly distributed in a box enclosing the original point cloud. On the right, the reconstruction of an isosurface of the distance function d µc,m 0 to the uniform probability measure on this point cloud. 16/31

17 How to determine that two N-samples are from the same underlying space? based asymptotic statistical test. (Brecheteau 2017) 17/31

18 - Definition (-) The - associated to some mm-space (X, δ, µ), denoted d µ,m (µ), is the distribution of the real valued random variable d µ,m (Y ) where Y is some random variable of law µ. 18/31

19 Stability of Proposition Given two mm-spaces (X, δ X, µ), (Y, δ Y, ν), we have W 1 (d µ,m (µ), d ν,m (ν)) 1 m GW 1(X, Y ). Proposition If (X, δ X, µ), (Y, δ Y, ν) are embedded into some metric space (Z, δ), then we can upper bound W 1 (d µ,m (µ), d ν,m (ν)) by W 1 (µ, ν)+min{ d µ,m d ν,m,supp(µ), d µ,m d ν,m,supp(ν) }, and more generally by (1 + 1 m )W 1(µ, ν). 19/31

20 Non discriminative example There are non isomorphic (X, δ, µ), (X, δ, ν) with d µ,m (µ) = d ν,m (ν). Figure: Each cluster has the same weight 1/3. 20/31

21 Discriminative results Proposition Let (O, 2, µ O ), (O, 2, µ O ) be two mm-spaces, for O, O two non-empty bounded open subset of R d satisfying O = (Ō) and O = (Ō ), µ O, µ O uniform measures. A lower bound for W 1 (d µo,m(µ O ), d µo,m(µ O )) is given by: C Leb d (O) 1 d Lebd (O ) 1 d, where C depends on m, ɛ, O, O, d. Remark can be discriminative under some conditions. 21/31

22 Statistic test Given two N-samples from the mm-spaces (X, δ, µ), (Y, γ, ν), we want to build a algorithm using these two samples to test the null hypothesis: H 0 two mm-spaces X, Y are isomorphic, against its alternative: H 1 two mm-spaces X, Y are not isomorphic, 22/31

23 The test proposed in the paper is based on the fact that the - associated to two isomorphic mm-spaces are equal, which leads to W 1 (d µ,m (µ), d ν,m (ν)) = 0. 23/31

24 Idea Given two N-samples from the mm-spaces (X, δ, µ), (Y, γ, ν), choose randomly two n-samples from them respectively, which gives four empirical measures, ˆµ n, ˆµ N, ˆν n, ˆν N. Test statistic: T N,n,m (µ, ν) = nw 1 (dˆµn,m(ˆµ n ), dˆνn,m(ˆν n )). Denote the law of T N,n,m (µ, ν) as L N,n,m (µ, ν). 24/31

25 Lemma If two mm-spaces are isomorphic, then L N,n,m (µ, ν) = L N,n,m (ν, ν) = L N,n,m (µ, µ) = 1 2 L N,n,m(µ, µ) L N,n,m(ν, ν). Remark 1 2 L N,n,m(µ, µ) L N,n,m(ν, ν) is the distribution of ZT N,n,m (µ, µ) + (1 Z)T N,n,m (ν, ν), where Z is another independent random variable with Bernoulli distribution. 25/31

26 The α-quantile q α,n,n of 1 2 L N,n,m(µ, µ) L N,n,m(ν, ν) will be approximated by the α-quantile ˆq α,n,n of 1 2 L N,n,m (ˆµ N, ˆµ N ) L N,n,m (ˆν N, ˆν N ). Here L N,n,m (ˆµ N, ˆµ N ) stands for the distribution of T N,n,m (ˆµ N, ˆµ N ) = nw 1 (dˆµn,m(µ n), dˆµn,m(µ n)) conditionally to ˆµ N, where µ n and µ n are two independent n-samples of law ˆµ N. We deal with the test: φ N = 1 TN,n,m (µ,ν) ˆq α,n,n. 26/31

27 Bootstrap method 27/31

28 Asymptotic level α For properly chosen n depending on N, for example, N = cn ρ, with ρ > max{d,2} 2, test is of asymptotic level α, i.e. lim sup N P (µ,ν) H0 (φ N = 1) α. 28/31

29 Numerical illustrations µ v : distribution of (R sin(vr) M, R cos(vr) M ) with R, M, M independent variables; M and M from the standard normal distribution and R uniform on (0, 1). Sample N = 2000 points from two measure, choose α = 0.05, m = 0.05, n = 20, N MC = /31

30 Numerical illustrations Figure: Left: - estimates. Right: Bootstrap validity, v = 10. Figure: Type 1 error and power approximations by repeating 1000 times. 30/31

31 Thank you! 31/31