A bridge between continuous and discrete multiD persistence N. - PowerPoint PPT Presentation

A bridge between continuous and discrete multiD persistence N. Cavazza 1 , M. Ethier 2 , P. Frosini 1 , T. Kaczynski 2 , Claudia Landi 3 1 Universit` a di Bologna 2 Universit´ e de Sherbrooke 3 Universit` a di Modena e Reggio Emilia Applied and Computational Algebraic Topology Bremen, July 15-19, 2013

Motivation Real object Models • How accurately does rank invariant comparison on discrete models approximate that on continuous objects? • To which extent can data resolution be coarsened in order to maintain a certain error threshold on rank invariants comparison? 2 of 15

Outline • Multidimensional persistence of a filtration ◦ sub-level set filtrations ◦ simplicial complex filtrations • From discrete to continuous filtrations: ◦ an obstacle: topological aliasing ◦ a way round: axis-wise linear interpolation • From continuous to discrete: ◦ stable comparison of multi-D persistence • Application: ◦ a procedure to predetermine the model precision required to reach a given error threshold. 3 of 15

1-D vs. multi-D Persistence 1-D persistence captures the topology of a one-parameter filtration. X 1 X 2 X 3 X 4 mass X 1 X 2 X 3 X 4 darkness 4 of 15

1-D vs. multi-D Persistence Multi -D persistence captures the topology of a family of spaces filtered along multiple geometric dimensions. darkness X 4 , 1 X 4 , 2 X 4 , 3 X 4 , 4 X 3 , 1 X 3 , 2 X 3 , 3 X 3 , 4 X 2 , 1 X 2 , 2 X 2 , 3 X 2 , 4 X 1 , 1 X 1 , 2 X 1 , 3 X 1 , 4 mass 4 of 15

Filtrations • Sublevelset filtrations: Any continuous function f = ( f 1 ,..., f k ) : X → R k induces sub-level sets: k f − 1 (( − ∞ , α i ]) , α = ( α 1 ,..., α k ) ∈ R k . � X α = i i =1 Setting α = ( α i ) � β = ( β i ) iff α i ≤ β i for every i we get a k -parameter filtration of X by sub-level sets: α � β implies X α ⊆ X β . • Discrete filtrations: Given a simplicial complex K and a function ϕ : V ( K ) → R k , for any α ∈ R k let K α = { σ ∈ K | ϕ ( v ) � α for all vertices v ≤ σ } . 5 of 15

Rank invariant For a filtration F = { X α } α ∈ R k on a triangulable subspace of some R d , ρ F : { ( α , β ) ∈ R k × R k | α ≺ β } → N , ρ F ( α , β ) = dim im H ∗ ( X α ֒ → X β ) . z z X f = ( y , z ) y y x 6 of 15

Rank invariant For a filtration F = { X α } α ∈ R k on a triangulable subspace of some R d , ρ F : { ( α , β ) ∈ R k × R k | α ≺ β } → N , ρ F ( α , β ) = dim im H ∗ ( X α ֒ → X β ) . z z X β α f = ( y , z ) y y ρ f ( α , β ) = 2 x 6 of 15

Rank invariant For a filtration F = { X α } α ∈ R k on a triangulable subspace of some R d , ρ F : { ( α , β ) ∈ R k × R k | α ≺ β } → N , ρ F ( α , β ) = dim im H ∗ ( X α ֒ → X β ) . z z X β α f = ( y , z ) y y ρ f ( α , β ) = 1 x 6 of 15

Rank invariant For a filtration F = { X α } α ∈ R k on a triangulable subspace of some R d , ρ F : { ( α , β ) ∈ R k × R k | α ≺ β } → N , ρ F ( α , β ) = dim im H ∗ ( X α ֒ → X β ) . z z β X α f = ( y , z ) y y ρ f ( α , β ) = 1 x 6 of 15

Continuous vs discrete setting • Sub-level set filtrations are those for which stability results hold: ∀ f , f ′ : X → R k continuous functions, D ( ρ f , ρ f ′ ) ≤ � f − f ′ � ∞ . 7 of 15

Continuous vs discrete setting • Sub-level set filtrations are those for which stability results hold: ∀ f , f ′ : X → R k continuous functions, D ( ρ f , ρ f ′ ) ≤ � f − f ′ � ∞ . • Discrete filtrations are those actually used in computations: Laser Projector CCD scanner Stable comparison of rank invariants obtained from discrete data? 7 of 15

From discrete to continuous filtrations Question: How to extend ϕ : V ( K ) → R k to a continuous function K → R k so that its sub-level set filtration coincides with { K α } α ∈ R k ? 8 of 15

From discrete to continuous filtrations Question: How to extend ϕ : V ( K ) → R k to a continuous function K → R k so that its sub-level set filtration coincides with { K α } α ∈ R k ? Answer: 1-D persistence: use linear interpolation [Morozov, 2008] α 8 of 15

From discrete to continuous filtrations Question: How to extend ϕ : V ( K ) → R k to a continuous function K → R k so that its sub-level set filtration coincides with { K α } α ∈ R k ? Answer: Multi-D persistence: linear interpolation yields topological aliasing ϕ 2 ϕ ( v 1 ) v 1 α ϕ v 0 ϕ ( v 0 ) ϕ 1 8 of 15

Topological Aliasing: numerical experiments Original Linear int. % Diff cat0 vs. cat0-tran1-1 0.046150 0.040576 H 1 -13.737185 0.225394 0.207266 H 0 -8.746249 cat0-tran1-2 vs. cat0-tran2-1 0.034314 0.029188 H 1 -17.562012 0.208451 0.204511 H 0 -1.926547 cat0-tran2-1 vs. cat0-tran2-2 0.045545 0.037061 H 1 -22.891989 0.212733 0.208097 -2.227807 H 0 9 of 15

Axis-wise linear interpolation • Given any σ ∈ K , set µ ( σ ) = max { ϕ ( v ) | v is a vertex of σ } . • Use induction to define ϕ � : K → R k on σ and a point w σ ∈ σ s.t. ◦ For all x ∈ σ , ϕ � ( x ) � ϕ � ( w σ ) = µ ( σ ) ; ◦ ϕ � is linear on any line segment [ w σ , y ] with y ∈ ∂σ . 10 of 15

Axis-wise linear interpolation • Given any σ ∈ K , set µ ( σ ) = max { ϕ ( v ) | v is a vertex of σ } . • Use induction to define ϕ � : K → R k on σ and a point w σ ∈ σ s.t. ◦ For all x ∈ σ , ϕ � ( x ) � ϕ � ( w σ ) = µ ( σ ) ; ◦ ϕ � is linear on any line segment [ w σ , y ] with y ∈ ∂σ . ϕ 2 ϕ ( v 1 ) = µ ( σ ) = ϕ � ( w σ ) v 1 = w σ ϕ ϕ ( v 0 ) v 0 ϕ 1 10 of 15

Axis-wise linear interpolation • Given any σ ∈ K , set µ ( σ ) = max { ϕ ( v ) | v is a vertex of σ } . • Use induction to define ϕ � : K → R k on σ and a point w σ ∈ σ s.t. ◦ For all x ∈ σ , ϕ � ( x ) � ϕ � ( w σ ) = µ ( σ ) ; ◦ ϕ � is linear on any line segment [ w σ , y ] with y ∈ ∂σ . ϕ 2 µ ( σ ) = ϕ � ( w σ ) v 1 ϕ ϕ ( v 1 ) w σ ϕ ( v 0 ) v 0 ϕ 1 10 of 15

Axis-wise linear interpolation • Given any σ ∈ K , set µ ( σ ) = max { ϕ ( v ) | v is a vertex of σ } . • Use induction to define ϕ � : K → R k on σ and a point w σ ∈ σ s.t. ◦ For all x ∈ σ , ϕ � ( x ) � ϕ � ( w σ ) = µ ( σ ) ; ◦ ϕ � is linear on any line segment [ w σ , y ] with y ∈ ∂σ . ϕ 2 µ ( σ ) = ϕ � ( w σ ) v 1 ϕ ϕ ( v 1 ) w σ ϕ ( v 0 ) v 0 ϕ 1 Theorem For any α ∈ R k , K α is a strong deformation retract of K ϕ � � α . 10 of 15

Bridging stability from continuous to discrete persistence • X and Y homeomorphic triangulable spaces (real objects); • f : X → R k , g : Y → R k continuous functions (real measurements); • K ′ and L ′ simplicial complexes with | K ′ | = K , | K ′ | = L (approximated object); ψ : L → R k continuous functions (approximated ϕ : K → R k , ˜ • ˜ measurements); Theorem: If two homeomorphisms ξ : K → X , ζ : L → Y exist s.t. ϕ − f ◦ ξ � ∞ ≤ ε / 4 , � ˜ ψ − g ◦ ζ � ∞ ≤ ε / 4 � ˜ then, for any sufficiently fine subdivision K of K ′ and L of L ′ , � D ( ρ f , ρ g ) − D ( ρ ϕ , ρ ψ ) � ≤ ε , � � ϕ : V ( K ) → R k , ψ : V ( L ) → R k being restrictions of ˜ ϕ and ˜ ψ . 11 of 15

Sketch of the proof • ∃ δ > 0 s.t. max { diam σ | σ ∈ K or σ ∈ L } < δ = ⇒ | D ( ρ ˜ ϕ , ρ ˜ ψ ) − D ( ρ ϕ � , ρ ψ � ) | < ε / 2 . 12 of 15

Sketch of the proof • ∃ δ > 0 s.t. max { diam σ | σ ∈ K or σ ∈ L } < δ = ⇒ | D ( ρ ˜ ϕ , ρ ˜ ψ ) − D ( ρ ϕ � , ρ ψ � ) | < ε / 2 . • ρ ϕ = ρ ϕ � , ρ ψ = ρ ψ � . 12 of 15

Sketch of the proof • ∃ δ > 0 s.t. max { diam σ | σ ∈ K or σ ∈ L } < δ = ⇒ | D ( ρ ˜ ϕ , ρ ˜ ψ ) − D ( ρ ϕ � , ρ ψ � ) | < ε / 2 . • ρ ϕ = ρ ϕ � , ρ ψ = ρ ψ � . • max { diam σ | σ ∈ K or σ ∈ L } < δ = ⇒ | D ( ρ ˜ ϕ , ρ ˜ ψ ) − D ( ρ ϕ , ρ ψ ) | < ε / 2 . 12 of 15

Sketch of the proof • ∃ δ > 0 s.t. max { diam σ | σ ∈ K or σ ∈ L } < δ = ⇒ | D ( ρ ˜ ϕ , ρ ˜ ψ ) − D ( ρ ϕ � , ρ ψ � ) | < ε / 2 . • ρ ϕ = ρ ϕ � , ρ ψ = ρ ψ � . • max { diam σ | σ ∈ K or σ ∈ L } < δ = ⇒ | D ( ρ ˜ ϕ , ρ ˜ ψ ) − D ( ρ ϕ , ρ ψ ) | < ε / 2 . • D ( ρ f , ρ g ) D ( ρ f , ρ f ◦ ξ )+ D ( ρ f ◦ ξ , ρ ˜ ϕ )+ D ( ρ ˜ ϕ , ρ ˜ ≤ ψ ) D ( ρ ˜ ψ , ρ g ◦ ζ )+ D ( ρ g ◦ ζ , ρ g ) + . 12 of 15

Applications to model precision concerns • Aim: Calculate the model precision required to reach a given error threshold 13 of 15

Applications to model precision concerns • Aim: Calculate the model precision required to reach a given error threshold • Method: demonstrated using the following example 13 of 15

Applications to model precision concerns • Aim: Calculate the model precision required to reach a given error threshold • Method: demonstrated using the following example For a dataset of 5000 functions f i : T → R 2 on the torus T , given a set of triangulations of T with 2 2 N simplices (varying N ) we obtain the function ϕ i , N by sampling f i at the vertices of the triangulations. 13 of 15