Submanifold Reconstruction Jean-Daniel Boissonnat DataShape, INRIA http://www-sop.inria.fr/datashape Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 1 / 36
Geometric data analysis Images, text, speech, neural signals, GPS traces,... Geometrisation : Data = points + distances between points Hypothesis : Data lie close to a structure of “small” intrinsic dimension Problem : Infer the structure from the data Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 2 / 36
Submanifolds of R d A compact subset M ⊂ R d is a submanifold without boundary of (intrinsic) dimension k < d , if any p ∈ M has an open (topological) k -ball as a neighborhood in M R N M W φ R m U Intuitively, a submanifold of dimension k is a subset of R d that looks locally like an open set of an affine space of dimension k A curve a 1 -dimensional submanifold A surface is a 2 -dimensional submanifold More generally, manifolds are defined in an intrinsic way, Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 3 / 36 d
Triangulation of a submanifold We call triangulation of a submanifold M ⊂ R d a simplicial complex ˆ M such that ˆ M is embedded in R d its vertices are on M it is homeomorphic to M Submanifold reconstruction The problem is to construct a triangulation ˆ M of some unknown submanifold M given a finite set of points P ⊂ M Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 4 / 36
Triangulation of a submanifold We call triangulation of a submanifold M ⊂ R d a simplicial complex ˆ M such that ˆ M is embedded in R d its vertices are on M it is homeomorphic to M Submanifold reconstruction The problem is to construct a triangulation ˆ M of some unknown submanifold M given a finite set of points P ⊂ M Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 4 / 36
Issues in high-dimensional geometry Dimensionality severely restricts our intuition and ability to visualize data ⇒ need for automated and provably correct methods methods Complexity of data structures and algorithms rapidly grow as the dimensionality increases ⇒ no subdivision of the ambient space is affordable ⇒ data structures and algorithms should be sensitive to the intrinsic dimension (usually unknown) of the data Inherent defects : sparsity, noise, outliers Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 5 / 36
Issues in high-dimensional geometry Dimensionality severely restricts our intuition and ability to visualize data ⇒ need for automated and provably correct methods methods Complexity of data structures and algorithms rapidly grow as the dimensionality increases ⇒ no subdivision of the ambient space is affordable ⇒ data structures and algorithms should be sensitive to the intrinsic dimension (usually unknown) of the data Inherent defects : sparsity, noise, outliers Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 5 / 36
Issues in high-dimensional geometry Dimensionality severely restricts our intuition and ability to visualize data ⇒ need for automated and provably correct methods methods Complexity of data structures and algorithms rapidly grow as the dimensionality increases ⇒ no subdivision of the ambient space is affordable ⇒ data structures and algorithms should be sensitive to the intrinsic dimension (usually unknown) of the data Inherent defects : sparsity, noise, outliers Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 5 / 36
Looking for small and faithful simplicial complexes Need to compromise Size of the complex ◮ can we have dim ˆ M = dim M ? Efficiency of the construction algorithms and of the representations ◮ can we avoid the exponential dependence on d ? ◮ can we minimize the number of simplices ? Quality of the approximation ◮ Homotopy type & homology (Cech and α complexes, persistence) ◮ Homeomorphism (Delaunay-type complexes) Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 6 / 36
Sampling and distance functions [Niyogi et al.], [Chazal et al.] d K : x → inf p ∈ K � x − p � Distance to a compact K : Stability If the data points C are close (Hausdorff) to the geometric structure K , the topology and the geometry of the offsets K r = d − 1 ([ 0 , r ]) and C r = d − 1 ([ 0 , r ]) are close Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 7 / 36
Distance functions and triangulations ˇ Nerve theorem (Leray) The nerve of the balls (Cech complex) and the union of balls have the same homotopy type (same result for the α -complex) Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 8 / 36
Questions + The homotopy type of a compact set X can be computed from the ˘ Cech complex of a sample of X + The same is true for the α -complex – The ˘ Cech and the α -complexes are huge ( O ( n d ) and O ( n ⌈ d / 2 ⌉ ) ) and difficult to compute in high dimensions – Both complexes are not in general homeomorphic to X (i.e. not a triangulation of X ) – The ˘ Cech complex cannot be realized in general in the same space as X Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 9 / 36
˘ Cech and Rips complexes The Rips complex is easier to compute but still very big, and less precise in approximating the topology α Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 10 / 36
An example where no offset has the right topology ! 1. Manifold + small noise assumption 2. Call persistent homology at rescue ! Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 11 / 36
The curses of Delaunay triangulations in higher dimensions Complexity depends exponentially on the ambient dimension. Robustness issues become very tricky Higher dimensional Delaunay triangulations are not thick even if the vertices are well-spaced The restricted Delaunay triangulation is no longer a good approximation of the manifold even under strong sampling conditions (for d > 2 ) Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 12 / 36
3D Delaunay Triangulations are not thick even if the vertices are well-spaced Each square face can be circumscribed by an empty sphere This remains true if the grid points are slightly perturbed therefore creating thin simplices Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 13 / 36
Badly-shaped simplices Badly-shaped simplices lead to bad geometric approximations which in turn may lead to topological defects in Del |M ( P ) [Oudot] see also [Cairns], [Whitehead], [Munkres], [Whitney] Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 14 / 36
Tangent space approximation M is a smooth k -dimensional manifold ( k > 2 ) embedded in R d Bad news [Oudot 2005] The Delaunay triangulation restricted to M may be a bad approximation of the manifold even if the sample is dense t = ∆ + δ / 2 t = ∆ c w v p p 0 c 0 u t t y y z z x x Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 15 / 36
Thickness and tangent space approximation Lemma [Whitney 1957] If σ is a j -simplex whose vertices all lie within a distance η from a hyperplane H ⊂ R d , then sin ∠ ( aff ( σ ) , H ) ≤ 2 j η D ( σ ) Corollary If σ is a j -simplex, j ≤ k , vert ( σ ) ⊂ M , ∆( σ ) ≤ δ rch ( M ) δ ∀ p ∈ σ, sin ∠ ( aff ( σ ) , T p ) ≤ Θ( σ ) ∆( σ ) 2 ( η ≤ 2 rch ( M ) by the Chord Lemma) Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 16 / 36
The assumptions M is a differentiable submanifold of positive reach of R d The dimension k of M is small P is an ε -net of M , i.e. ∀ x ∈ M , ∃ p ∈ P , � x − p � ≤ ε rch ( M ) ◮ ◮ ∀ p , q ∈ P , � p − q � ≥ ¯ η ε ε is small enough Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 17 / 36
The tangential Delaunay complex [B. & Ghosh 2010] M T p p Construct the star of p ∈ P in the Delaunay triangulation Del Tp ( P ) 1 of P restricted to T p Del T M ( P ) = � p ∈ P star ( p ) 2 Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 18 / 36
+ Del T M ( P ) ⊂ Del ( P ) + star ( p ) , Del T p ( P ) and therefore Del T M ( P ) can be computed without computing Del ( P ) – Del T M ( P ) is not necessarily a triangulated manifold Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 19 / 36
Construction of Del T p ( P ) Given a d -flat H ⊂ R , Vor ( P ) ∩ H is a weighted Voronoi diagram in H p ′ j � x − p i � 2 ≤ � x − p j � 2 x i � 2 + � p i − p ′ i � 2 ≤ � x − p ′ j � 2 + � p j − p ′ p ′ � x − p ′ j � 2 ⇔ i p j H p i Corollary: construction of Del T p Ψ p ( p i ) = ( p ′ i , −� p i − p ′ i � 2 ) (weighted point) project P onto T p which requires O ( Dn ) time 1 construct star (Ψ p ( p i )) in Del (Ψ p ( p i )) ⊂ T p i 2 star ( p i ) ≈ star (Ψ p ( p i )) (isomorphic ) 3 Algorithmic Geometry Submanifold reconstruction J-D. Boissonnat 20 / 36
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