Towards a robust vision of geometric inference Claire Brcheteau - PowerPoint PPT Presentation

Towards a robust vision of geometric inference Claire Brécheteau Université Paris-Sud 11, Laboratoire de Mathématiques d’Orsay and Inria Select Team – Inria Saclay, Data Shape Team Under the supervision of Pascal Massart (Université Paris-Sud 11) and Frédéric Chazal (Inria Saclay) September 24, 2018 Claire Brécheteau Towards a robust vision of geometric inference

❳ Towards a robust vision of geometric inference Geometric Inference : Recover geometric information from a point cloud sampled around some shape. Claire Brécheteau Towards a robust vision of geometric inference

Towards a robust vision of geometric inference Geometric Inference : Recover geometric information from a point cloud sampled around some shape. Global setting : ( X , δ ) , metric space P , probability distribution supported on X That is, ( X , δ , P ) is a metric-measure space. Q , probability distribution (close to P somehow) ❳ n = { X 1 , X 2 ,..., X n } , n -sample from Q Claire Brécheteau Towards a robust vision of geometric inference

Towards a robust vision of geometric inference Robustness : Robustness to outliers or Trimming : 1 Getting rid of a proportion 1 − η of the probability (resp. of the data-points). Claire Brécheteau Towards a robust vision of geometric inference

Towards a robust vision of geometric inference Robustness : Robustness to outliers or Trimming : 1 Getting rid of a proportion 1 − η of the probability (resp. of the data-points). t ∗ ˜ P γ ( t ,.) η ∈ arg min inf t η ˜ P ≤ P Claire Brécheteau Towards a robust vision of geometric inference

Towards a robust vision of geometric inference Robustness : Robustness to outliers or Trimming : 1 Getting rid of a proportion 1 − η of the probability (resp. of the data-points). t ∗ ˜ P γ ( t ,.) η ∈ arg min inf t η ˜ P ≤ P P η such that : P γ ( t ∗ η ,.) = P η γ ( t ∗ ˜ inf η ,.) η ˜ P ≤ P Claire Brécheteau Towards a robust vision of geometric inference

Towards a robust vision of geometric inference Robustness : Robustness to outliers or Trimming : 1 Getting rid of a proportion 1 − η of the probability (resp. of the data-points). t ∗ ˜ P γ ( t ,.) η ∈ arg min inf t η ˜ P ≤ P P η such that : P γ ( t ∗ η ,.) = P η γ ( t ∗ ˜ inf η ,.) η ˜ P ≤ P Stability (e.g. according to a Wasserstein metric W p ). 2 Small W p ( P , Q ) � Roughly the same geometric information in P and Q . Claire Brécheteau Towards a robust vision of geometric inference

Towards an implementable robust vision of geometric inference Claire Brécheteau Towards a robust vision of geometric inference

Main questions How to compare two datasets ? Claire Brécheteau Towards a robust vision of geometric inference

Main questions How to compare two datasets ? How to make clusters from a dataset ? Claire Brécheteau Towards a robust vision of geometric inference

Main questions How to compare two datasets ? How to make clusters from a dataset ? How to infer the distance to a compact set, with a fixed budget ? Claire Brécheteau Towards a robust vision of geometric inference

A multifunction tool : the distance-to-measure function Claire Brécheteau Towards a robust vision of geometric inference

A definition for the DTM � � � � δ P , h ( x ) = inf r > 0 | P B( x , r ) > h Claire Brécheteau Towards a robust vision of geometric inference

A definition for the DTM � � � � δ P , h ( x ) = inf r > 0 | P B( x , r ) > h The distance-to-measure (DTM) [Chazal, Cohen-Steiner, Mérigot 09’] is defined for all x ∈ X and h ∈ [0,1] by : � h d P , h ( x ) = 1 δ P , l ( x )d l h 0 Claire Brécheteau Towards a robust vision of geometric inference

A definition for the DTM � � � � δ P , h ( x ) = inf r > 0 | P B( x , r ) > h The distance-to-measure (DTM) [Chazal, Cohen-Steiner, Mérigot 09’] is defined for all x ∈ X and h ∈ [0,1] by : � 1 � 1 � h p d ( p ) δ p P , h ( x ) = P , l ( x )d l h 0 Claire Brécheteau Towards a robust vision of geometric inference

❳ The DTM for Stable – Geometric Inference When h = 0 , d P , 0 = d X . 1 ∞ ≤ h − 1 � � d ( p ) P , h − d ( p ) � p W p ( P , Q ) [Chazal, Cohen-Steiner, Mérigot 09’]. 2 � � Q , h � Claire Brécheteau Towards a robust vision of geometric inference

The DTM for Stable – Geometric Inference When h = 0 , d P , 0 = d X . 1 ∞ ≤ h − 1 � � d ( p ) P , h − d ( p ) � p W p ( P , Q ) [Chazal, Cohen-Steiner, Mérigot 09’]. 2 � � Q , h � 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 −0.5 0.0 0.5 1.0 1.5 −0.5 0.0 0.5 1.0 1.5 −0.5 0.0 0.5 1.0 1.5 Distance to X Distance to ❳ n DTM with h = 0.2 Claire Brécheteau Towards a robust vision of geometric inference

❘ The DTM contains information Theorem (Brécheteau) P O (uniform distribution on O ) can be recovered from d P O , h provided that h is small enough and O regular enough. Claire Brécheteau Towards a robust vision of geometric inference

The DTM contains information Theorem (Brécheteau) P O (uniform distribution on O ) can be recovered from d P O , h provided that h is small enough and O regular enough. Theorem (Brécheteau) P can be recovered from (d P , h ) h ∈ [ 0 , 1 ] provided that ( X , δ ) = ( ❘ d , �·� ) . Claire Brécheteau Towards a robust vision of geometric inference

The DTM, an implementable tool κ = nh 1 1 Empirical distribution P n = � n n δ X i 2 i = 1 X (1) , X (2) ,. . . X ( κ ) : κ nearest neighbours of x in ❳ n 3 Claire Brécheteau Towards a robust vision of geometric inference

The DTM, an implementable tool κ = nh 1 1 Empirical distribution P n = � n n δ X i 2 i = 1 X (1) , X (2) ,. . . X ( κ ) : κ nearest neighbours of x in ❳ n 3 κ d P n , h ( x ) = 1 � � X ( i ) , x � δ κ i = 1 � Easy implementation of the DTM at a point x in practice ! Claire Brécheteau Towards a robust vision of geometric inference

The DTM, an implementable tool κ = nh 1 1 Empirical distribution P n = � n n δ X i 2 i = 1 X (1) , X (2) ,. . . X ( κ ) : κ nearest neighbours of x in ❳ n 3 �� 1 � κ p 1 d ( p ) δ p � X ( i ) , x � P n , h ( x ) = κ i = 1 � Easy implementation of the DTM at a point x in practice ! Claire Brécheteau Towards a robust vision of geometric inference

Claire Brécheteau Towards a robust vision of geometric inference

1) A statistical test of isomorphism between mm-spaces Claire Brécheteau Towards a robust vision of geometric inference

A statistical test of isomorphism between mm-spaces Two mm-spaces ( X , δ , P ) and ( Y , δ ′ , P ′ ) are isomorphic [Gromov 81’] if : ∃ φ : X �→ Y a one-to-one isometry, s.t. for all Borel set A , P ′ ( φ ( A )) = P ( A ) . Claire Brécheteau Towards a robust vision of geometric inference

A statistical test of isomorphism between mm-spaces Two mm-spaces ( X , δ , P ) and ( Y , δ ′ , P ′ ) are isomorphic [Gromov 81’] if : ∃ φ : X �→ Y a one-to-one isometry, s.t. for all Borel set A , P ′ ( φ ( A )) = P ( A ) . How to build a test of level α > 0 to test the null hypothesis H 0 : “ ( X , δ , P ) and ( Y , δ ′ , P ′ ) are isomorphic” ? vs H 1 : “ ( X , δ , P ) and ( Y , δ ′ , P ′ ) are not isomorphic” ? ( Y , δ ′ , P ′ ) ( X , δ , P ) Claire Brécheteau Towards a robust vision of geometric inference

From the Gromov-Wasserstein distance to the DTM-signature The Gromov-Wasserstein distance [Mémoli 10’] GW is a metric such that GW (( X , δ , P ),( Y , δ ′ , P ′ )) = 0 iff the mm-spaces are isomorphic. � Too high computational cost. Definition The DTM-signature, d P , h ( P ) is the distribution of d P , h ( X ) when X ∼ P . Theorem (Brécheteau) ≤ 1 P ′ �� � � W 1 d P , h ( P ),d P ′ , h h GW ( X , Y ) Claire Brécheteau Towards a robust vision of geometric inference

Bootstrap approximation Definition Defined by d P N , h ( P n ) with P N from ( X 1 , X 2 ,..., X N ) and P n from ( X 1 , X 2 ,..., X n ) . Statistic : � P ′ n � � �� T = nW 1 d P N , h ( P n ),d P ′ N , h Subsampling distribution : L ∗ ( P ) = L ∗ � � ′ �� P ∗ P ∗ � � � � � nW 1 d P N , h ,d P N , h | P N n n 2 L ∗ ( P ) + 1 Under hypothesis H 0 , L ( T ) is approximated with L ∗ = 1 2 L ∗ � P ′ � . Claire Brécheteau Towards a robust vision of geometric inference

Bootstrap approximation Definition Defined by d P N , h ( P n ) with P N from ( X 1 , X 2 ,..., X N ) and P n from ( X 1 , X 2 ,..., X n ) . Statistic : � P ′ n � � �� T = nW 1 d P N , h ( P n ),d P ′ N , h Subsampling distribution : L ∗ ( P ) = L ∗ � � ′ �� P ∗ P ∗ � � � � � nW 1 d P N , h ,d P N , h | P N n n 2 L ∗ ( P ) + 1 Under hypothesis H 0 , L ( T ) is approximated with L ∗ = 1 2 L ∗ � P ′ � . 1.0 0.8 0.6 0.4 0.2 0.0 0.02 0.04 0.06 0.08 0.10 Wasserstein distance between DTM signatures Cdf of L ( T ) and L ∗ ( P ) (Bunny) N = 10000 , n = 100 , h = 0.1 Claire Brécheteau Towards a robust vision of geometric inference