Simplicial Multivalued Maps and the Witness Complex Zachary - - PowerPoint PPT Presentation

Simplicial Multivalued Maps and the Witness Complex

Zachary Alexander+, Elizabeth Bradley∗, James D. Meiss∗∗, and Nikki Sandersono

University of Colorado, Boulder *Professor, Computer Science, **Professor, Applied Mathematics,

• graduate student, Mathematics

+ data scientist, Microsoft

Erwin Schr¨

• dinger International Institute for Mathematical

Physics

February 23, 2015

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Overview

• What we did: Develop a new computational topology tool.
• Why it’s useful: Computer-based proofs of existence of regular

and chaotic invariant sets of dynamical systems.

• How we did it: Discretization of state space, novel outer

approximation of dynamics, computation of discrete Conley index.

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Set-up

• Underlying dynamical system is a flow ϕ : R × Rn → Rn or a

map f : Rn → Rn from the state space to itself.

• Only knowledge of dynamics is Γ = {x0, . . . xT−1}, a finite time

series from a state space trajectory, with xi ∈ Rn.

• Goal: characterize properties of f |Λ (i.e. number and type of
• rbits, topological entropy)
• Issue: need to locate isolating neighborhoods of f to compute

Conley index.

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How We Did It

How do we get this information from a (*scalar) time series?

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How We Did It

How do we get this information from a (*scalar) time series? 1) (*Delay coordinate embedding) 2) Discretization of state space 3) Construct outer approximation of dynamics as multivalued map 4) Compute discrete Conley index using this multivalued map

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Multivalued Maps

*Only have access to finite time series; need to locate isolating neighborhoods of f to compute Conley index.

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Multivalued Maps

*Only have access to finite time series; need to locate isolating neighborhoods of f to compute Conley index. A grid on X is a family of nonempty compact sets A such that:

1 Geometrical realization of A, denoted |A|, is equal to X. 2 Each set in A is equal to the closure of its interior. 3 Distinct sets of A can only intersect in their boundaries or not

at all.

4 Each compact S ⊂ X is covered by a finite subset of A.

• Previous work: cubical grid, cubical multivalued map (Kacyznski,

Mischaikow, Mrozek) *Computationally expensive: throw away lots of cells, number of boundary cells scales badly with dimension

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Multivalued Maps

STEP 1: GENERALIZE Definition (Cellular Multivalued Map, CMM) A cellular multivalued map FA : |A| → |A| is an outer approximation of f defined on the geometrical realization of an arbitrary grid A by FA(x) :=

• B∈A:x∈|B|

{A ∈ A : A ∩ f (B) = ∅}.

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Intermediary Ideas

To get a discretization of our state space into cells, we utilize the following constructions from computational geometry:

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Intermediary Ideas

To get a discretization of our state space into cells, we utilize the following constructions from computational geometry: The Voronoi diagram of a set L, denoted V (L) = {Vℓ : ℓ ∈ L}, is the covering of Rn by the cells Vℓ := {x ∈ Rn : d(x, ℓ) ≤ d(x, ℓ′), ∀ℓ′ ∈ L}.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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Intermediary Ideas

The α-diagram of a set L, denoted Aα(L) = {Aℓ : ℓ ∈ L}, is the collection of cells Aℓ := {x ∈ Rn : Vℓ ∩ Bα(ℓ)}. (The geometrical realization of an α-diagram is called an α-grid.) The α-complex, Kα, is the nerve of an α-diagram: Let σ = l0, . . . , lk, where {l0, . . . , lk} is any finite subset of L. Consider the intersection of cells Aσ :=

li∈σ Ali. A simplex σ is in

the α-complex, Kα, if Aσ = ∅. (The geometrical realization of an α-complex is called an α-shape.)

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SMM and the WC

In particular, we can define a CMM on the α-grid: FA(x) :=

• B∈A:x∈|B|

{A ∈ A : A ∩ f (B) = ∅}.

A4 A1 A2 A3 A6 A5 A8 A9 A7 α ℓ4 ℓ1 ℓ2 ℓ3 ℓ6 ℓ5 ℓ7 ℓ8 ℓ9 A2 A3 A6 A5 A8 A9 FA f(A3) f(A1) f(x) x 9 / 23

SMM and the WC

FK(σ) := {τ : Aτ ⊂ FA(Aσ)}.

A4 A1 A2 A3 A6 A5 A8 A9 A7 α ℓ4 ℓ1 ℓ2 ℓ3 ℓ6 ℓ5 ℓ7 ℓ8 ℓ9 A2 A3 A6 A5 A8 A9 FA f(A3) f(A1) f(x) x

〈ℓ3〉 〈ℓ1〉 〈ℓ2〉 〈ℓ4〉 〈ℓ7〉 〈ℓ9〉 〈ℓ8〉 〈ℓ5〉

〈ℓ5,ℓ6,ℓ8〉 〈ℓ3,ℓ5,ℓ6〉 〈ℓ6,ℓ8,ℓ9〉

〈ℓ6〉

〈 ℓ1 , ℓ3 〉 〈 ℓ3 , ℓ6 〉 〈ℓ6,ℓ9〉 〈ℓ8,ℓ9〉 〈 ℓ

5

, ℓ

8

〉 〈ℓ3,ℓ5〉 〈 ℓ2 , ℓ3 〉 〈 ℓ

2

, ℓ

4

〉 〈 ℓ

4

, ℓ

7

〉 〈ℓ7,ℓ9〉

FK

FK(〈ℓ1〉) FK(〈ℓ3〉) FK(〈ℓ1,ℓ3〉)

〈ℓ6〉 〈ℓ9〉 〈ℓ8〉 〈ℓ5〉 〈ℓ3〉 〈ℓ2〉

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SMM and the WC

When the nerve K of a grid A is a geometrical simplicial complex, an SMM, in turn, induces a CMM on the geometrical realization |K|: FK(x) :=

• σ∈K:x∈|σ|

{|FK(σ)|}. Theorem (Alexander, Bradley, Meiss, S. ) If a CMM on an α-grid is acyclic, then the corresponding SMM on the α-complex induces a CMM on the α-shape that induces the same map on homology as the CMM on the α-grid.

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SMM and the WC

• to use all of the points of Γ = {x0, . . . , xT−1} as the centers of

α-cells would be computationally expensive and not necessarily illuminating

• instead use a much smaller set of points, called landmarks,

L = {l1, . . . , lk}

• don’t know f on all points of state space
• define a CMM on the α-grid based on a witness relation between

the points of Γ and the landmarks of L.

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SMM and the WC

We never actually compute the α-diagram Aα(L) corresponding to the set of landmarks L!

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SMM and the WC

We never actually compute the α-diagram Aα(L) corresponding to the set of landmarks L! Yet we construct a CMM on the α-grid using the temporal

• rdering of Γ and a witness relation:

Definition (Witness Relation) We use a fuzzy version of the strong-witness relation (Carlsson, de Silva): Wǫ(Γ, ℓi) := {x ∈ Γ : d(x, ℓi) ≤ d(x, L) + ǫ}.

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We define an outer approximation of our dynamical system as follows: Definition (Cellular Witness Map) Suppose α and ǫ are as above. The witness map FW : |Aα(L)| → |Aα(L)| for the witness relation is the cellular multivalued map FW(x) :=

• Ai∈Aα(L):x∈Ai

{Aj ∈ Aα(L) : ∃ xt ∈ Wǫ(Γ, li) s.t. xt+1 ∈ Wǫ(Γ, lj)}.

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We define an outer approximation of our dynamical system as follows: Definition (Cellular Witness Map) Suppose α and ǫ are as above. The witness map FW : |Aα(L)| → |Aα(L)| for the witness relation is the cellular multivalued map FW(x) :=

• Ai∈Aα(L):x∈Ai

{Aj ∈ Aα(L) : ∃ xt ∈ Wǫ(Γ, li) s.t. xt+1 ∈ Wǫ(Γ, lj)}. We define witness complex to be the clique complex for the edge-set determined by pairwise intersections of the sets of witnesses to a landmark.

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SMM and the WC

For Γ, L, α and ǫ as specified below, we get that Wǫ(Γ, L) = Kα(L). Theorem (Alexander, Bradley, Meiss, S.) Kα(L) ⊆ Wǫ(Γ, L): For a time series Γ, a set of landmarks L, and α, ǫ > 0, let Kα(L) be the α-complex and Wǫ(Γ, L) be the witness

• complex. If there exists a δ > 0 with δ ≤ ǫ/2 such that Γ is

δ-dense of Aα(L), then Kα(L) ⊆ Wǫ(Γ, L). Theorem (Alexander, Bradley, Meiss, S.) Wǫ(Γ, L) ⊆ Kα(L): Suppose Kα(L) and Wǫ(Γ, L) are as above. Let M = maxx∈Γd(x, L) and β = mini=jd(li, lj). If α > 0 is selected such that M + ǫ ≤ α ≤

β √ 2 and Kα(L) is a clique complex, then

Wǫ(Γ, L) ⊆ Kα(L).

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SMM and the WC

We also get that: Theorem (Alexander, Bradley, Meiss, S.) Suppose that |Aα(L)| is compact and f is Lipschitz on |Aα(L)| with constant c. If there exists a δ > 0 with δ ≤ ǫ

2min{1, 1 c } such

that Γ is δ-dense on |Aα(L)|, then FW is an outer approximation of f . Since the nerve of Aα(L) is Kα(L) = Wǫ(Γ, L), the CMM FW induces a SMM on the witness complex: FW : Wǫ(Γ, L) → Wǫ(Γ, L).

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SMM and the WC

• We check the acyclicity of the CMM FW
• We then use the SMM FW to construct an explicit chain selector

ϕ : Wǫ(Γ, L) → Wǫ(Γ, L)

• It follows that |ϕ| : |Wǫ(Γ, L)| → |Wǫ(Γ, L)| is a continuous

selector of the CMM FW So we can compute the Conley index [F(N,E)∗] = [ϕ∗] and prove the existence of non-trivial invariant sets of dynamical systems!

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Example: Finding fixed point of H´ enon map

x y

0.4 0.2 1.5 −0.4 −0.2 0.5 1 −1.5 −1 −0.5

For f (x, y) = (y + 1 − 1.4x2, 0.3x) with initial condition z0 = (−0.4, 0.3) near attractor Λ, we generate a trajectory Γ of length T = 105. We select 216 landmarks L on a hexagonal grid with spacing β = 0.05. Choosing α = β/ √ 2 and ǫ = 0.005, we construct CMM FW using the fuzzy witness relation.

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Example: Finding fixed point of H´ enon map

x y

0.4 0.2 1.5 −0.4 −0.2 0.5 1 −1.5 −1 −0.5

Using the cell of the landmark nearest to x39,436 as the initial input to our algorithm, we find an isolating neighborhood for FW. We then use an algorithm to get index pair (N, E) for FW and represent index pair (N, E) with the corresponding witness complex W(N, E).

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Example: Finding fixed point of H´ enon map

We use FW|N to verify that FW is acyclic on N.

A2 A1 A3 A4 A5 A6 A7 A8 A9

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Example: Finding fixed point of H´ enon map

We construct a chain selector ϕ by determining the image of each vertex and then the image of edges, and compute the induced map

• n relative homology H(N, E).

l4 l5 l9 l3 l6 E Since [ϕ∗] = [id.] is not in the shift equivalence class of [0], we conclude that (N, E) has a non-trivial invariant set!

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Future Work

*embedding dimension *landmark selection *persistence *general metrics on submanifold itself *segmentation

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Thank you!

• Z. Alexander, E. Bradley, J. Meiss, N. Sanderson, Simplicial

Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series arXiv:1406.2245 (2015)

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