Distance to the Measure Offset Recon- struction Geometric - PowerPoint PPT Presentation

Distance to the Measure Zhengchao Wan DTM Distance to the Measure Offset Recon- struction Geometric inference for measures based on distance DTM functions signature The DTM-signature for a geometric comparison of Statistical test metric-measure spaces from samples End Zhengchao Wan the Ohio State University wan.252@osu.edu 1/31

Distance to the Measure Geometric inference problem Zhengchao Wan DTM Offset Recon- struction DTM signature Statistical test Question End 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

Distance to the Measure Inference using distance functions Zhengchao Wan DTM Offset Recon- struction One idea to retrieve information of a point cloud is to consider DTM signature the R -offset of the point cloud - that is the union of balls of Statistical test radius R whose center lie in the point cloud. End 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

Distance to the Measure Inference using distance functions Zhengchao Wan DTM Offset Recon- struction For a compact K ⊂ R d , DTM signature d K : R d → R Statistical test End 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

Distance to the Measure Zhengchao Wan DTM Offset Recon- struction DTM Unfortunately, offset-based methods do not work well at all in signature the presence of outliers. For example, the number of connected Statistical test End components will be overestimated if one adds just a single data point far from the original point cloud. 5/31

Distance to the Measure Solution to outliers Zhengchao Wan DTM Offset Recon- struction DTM signature Statistical test End Replace the distance function to a set K by a distance function to a measure . (Chazal, et al 2010) 6/31

Distance to the Measure Distance to a Measure Zhengchao Wan DTM Offset Recon- struction DTM Notice signature d K ( x ) = min y ∈ K � x − y � = min { r > 0 : B ( x , r ) ∩ K � = ∅} . Statistical test End 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

Distance to the Measure Distance to a Measure Zhengchao Wan DTM Offset Recon- struction Definition DTM signature For any measure µ with finite second moment and a positive Statistical test mass parameter m 0 > 0 , the distance function to measure End (DTM) µ is defined by the formula: � m 0 µ, m 0 : R n → R , x �→ 1 d 2 δ µ, m ( x ) 2 dm . m 0 0 Recall δ µ, m ( x ) = inf { r > 0; µ ( ¯ B ( x , r )) > m } . 8/31

Distance to the Measure Zhengchao Wan DTM Example Offset Recon- Let C = { p 1 , · · · , p n } be a point cloud and µ C = 1 � struction i δ p i . n Then function δ µ C , m 0 with m 0 = k / n evaluated at x ∈ R d equal DTM signature to the distance between x and its kth nearest neighbor in C. Statistical test Given S ⊂ C with | S | = k, define End 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. n ( x ) = n � x − p � 2 . ∀ x ∈ Vor C ( S ) , d 2 � µ C , k k p ∈ S 9/31

Distance to the Measure Equivalent formulation Zhengchao Wan Proposition DTM Offset Recon- 1 DTM is the minimal cost of the following problem: struction DTM signature δ x , 1 µ ( R d ) = m 0 , ˜ � � � � d µ, m 0 ( x ) = min W 2 µ ˜ ; ˜ µ ≤ µ Statistical test m 0 µ ˜ End 2 Denote the set of minimizers as R µ, m 0 ( x ) . Then for each µ x , m 0 ∈ R µ, m 0 ( x ) , ˜ µ x , m 0 ) ⊂ ¯ • supp (˜ B ( x , δ µ, m 0 ( x )) ; � � • ˜ µ x , m 0 B ( x ,δ µ, m 0 ( x )) = µ B ( x ,δ µ, m 0 ( x )) ; � � • ˜ µ x , m 0 ≤ µ . µ x , m 0 ∈ R µ, m 0 ( x ) , 3 For any ˜ µ, m 0 ( x ) = 1 � δ x , 1 h ∈ R d � h − x � 2 d ˜ � � d 2 µ x , m 0 = W 2 µ x , m 0 ˜ . 2 m 0 m 0 10/31

Distance to the Measure Regularity Properties Zhengchao Wan DTM Offset Recon- struction DTM Proposition signature µ, m 0 is semiconcave, which means � x � 2 − d 2 Statistical test 1 d 2 µ, m 0 is convex; End 2 d 2 µ, m 0 is differentiable at a point x iff supp ( µ ) ∩ ∂ B ( x , δ µ, m 0 ( x )) contains at most 1 point; µ, m 0 is differentiable almost everywhere in R d in 3 d 2 Lebesgue measure. (directly from item 1) 4 d µ, m 0 is 1-Lipschitz. 11/31

Distance to the Measure Stability of DTM Zhengchao Wan DTM Offset Recon- struction DTM signature Theorem (DTM stability theorem) Statistical test If µ, ν are two probability measures on R d and m 0 > 0 , then End 1 � d µ, m 0 − d ν, m 0 � ∞ ≤ √ m 0 W 2 ( µ, ν ) . 12/31

Distance to the Measure Uniform Convergence of DTM Zhengchao Wan DTM Lemma Offset Recon- struction If µ is a compactly-supported measure, then d S is the uniform DTM limit of d µ, m 0 as m 0 converges to 0, where S = supp ( µ ) , i.e., signature Statistical test m 0 → 0 � d µ, m 0 − d S � ∞ = 0 . lim End Remark If µ has dimension at most k > 0 , i.e. µ ( B ( x , ǫ )) ≥ C ǫ k , ∀ x ∈ S when ǫ is small, then we can control the convergence speed: � d µ, m 0 − d S � ∞ = O ( m 1 / k ) . 0 13/31

Distance to the Measure Reconstruction from noisy data Zhengchao Wan DTM Offset Recon- struction DTM If µ is a probability measure of dimension at most k > 0 with signature compact support K ⊂ R d , and µ ′ is another probability Statistical test measure, one has End � � � � � d K − d µ ′ , m 0 ∞ ≤ � d K − d µ, m 0 � ∞ + � d µ, m 0 − d µ ′ , m 0 � � ∞ 1 ≤ O ( m 1 / k W 2 ( µ, µ ′ ) . ) + √ m 0 0 14/31

Distance to the Measure Reconstruction from noisy data Zhengchao Wan DTM Offset Recon- struction Define α - reach of K , α ∈ (0 , 1] as DTM r α ( K ) = inf { d K ( x ) > 0 : �∇ x d K � ≤ α } . signature Statistical test Theorem End 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

Distance to the Measure Example Zhengchao Wan DTM Offset Recon- struction DTM signature Statistical test End 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 µ C , m 0 to the uniform probability measure on this point cloud. 16/31

Distance to the Measure Zhengchao Wan DTM Offset Recon- struction DTM How to determine that two N -samples are from the same signature Statistical test underlying space? End DTM based asymptotic statistical test. (Brecheteau 2017) 17/31

Distance to the Measure DTM-signature Zhengchao Wan DTM Offset Recon- struction DTM signature Statistical test Definition (DTM-signature) End The DTM-signature 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

Distance to the Measure Stability of DTM Zhengchao Wan DTM Proposition Offset Recon- struction Given two mm-spaces ( X , δ X , µ ) , ( Y , δ Y , ν ) , we have DTM signature W 1 ( d µ, m ( µ ) , d ν, m ( ν )) ≤ 1 Statistical test mGW 1 ( X , Y ) . End 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

Distance to the Measure Non discriminative example Zhengchao Wan DTM There are non isomorphic ( X , δ, µ ) , ( X , δ, ν ) with Offset Recon- struction d µ, m ( µ ) = d ν, m ( ν ). DTM signature Statistical test End Figure: Each cluster has the same weight 1 / 3. 20/31

Distance to the Measure Discriminative results Zhengchao Wan DTM Offset Recon- Proposition struction DTM Let ( O , �� 2 , µ O ) , ( O ′ , �� 2 , µ O ′ ) be two mm-spaces, for O , O ′ signature two non-empty bounded open subset of R d satisfying Statistical test O ) ◦ and O = ( ¯ O = ( ¯ O ′ ) ◦ , µ O , µ O ′ uniform measures. A lower End bound for W 1 ( d µ O , m ( µ O ) , d µ O ′ , m ( µ O ′ )) is given by: 1 1 d − Leb d ( O ′ ) d | , C | Leb d ( O ) where C depends on m , ǫ, O , O ′ , d. Remark DTM can be discriminative under some conditions. 21/31