Automatic Probabilistic Modelling of Dynamical Systems Based on Global Geometry & Topology of Data

Max Snijders,

December 7th, 2018

Opening

The Team

Prof. Dr. Ulrich Bauer Prof. Dr. Oliver Junge Prof. Dr. Konstantin Mischaikow Max Snijders
Technical University of Munich Technical University of Munich Rutgers University Technical University of Munich
&
Ludwig-Maximilians University Munich
Introduction

Dynamical System Decompositions

Introduction

Preview of Results

Key Insights

Local 1-dimensionality

Key Insights

Cubical Covering

Key Insights

Homology Intuition

Example complex

Cycles

Boundaries

Reps. for Homology

Key Insights

Cohomology Intuition

1 By the Universal Coefficient Theorem
Key Insights

Circle-valued Maps

Key Insights

Homotopy-equivalence

Key Insights

First Integral Cohomology and Coordinate Maps

\[ H^1(X; \mathbb{Z}) \simeq \lbrack X, S^1 \rbrack \]

Original Work

Lifting the coordinate function (1)

Original Work

Lifting the coordinate function (2)

Coordinate Independence

Original Work
Original Work

Coordinate Alignment

Original Work

Box Macro-states

Example

The Lorenz Attractor

\[ \frac{\partial}{\partial t} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \sigma (y - x) \\ x (\rho - z) - y \\ x y - \beta z \end{pmatrix} \]

We use: \( \beta = \nicefrac{8}{3}, \sigma = 10, \rho = 28 \)

Original Work

Results on Lorenz System

Original Work

Markov-Model from Lorenz System Data

Original Work

Scaling of Method Effectiveness

Closing

Previous Work

This work relies on a large body of pre-existing work, including:

Closing

Outlook

Paper
Persistent Homology
Comparison against competition
Real-world data
Closing

A Note of Gratitude

Automatic Probabilistic Modelling of Dynamical Systems Based on Global Geometry & Topology of Data

Closing

Backup Material

Thesis refs.:

Backup slides:

Normal slides:

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Rips Complexes

Def. For a given point metric space \( (X, d) \) we denote by \( \mathrm{Rips}_r(X,d) \) the Rips complex of radius \(r \in \mathbb{R}\) with vertex set \(X\) and as simplices all subsets of \(X\) whose constituent vertices are pairwise within distance \(r\).
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Coordinate Smoothing (1/2)

Starting from a cocycle \( \eta \) which represents a cohomology class, which can be seen as a map on the complex' edges, we perform the following optimisation programme: \[ \begin{align} \bar{f} &= \mathrm{argmin}_{f \in C^0(\bar{X}, \Phi; \mathbb{R})} ||\eta + d_0 f||^2_2 \\ ||\eta||_2^2 &= \sum_{ab \in (\bar{X}, \Phi)} || \eta(ab) ||^2 \end{align} \]

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Coordinate Smoothing (2/2)

\( \eta_{\alpha}, || \eta_{\alpha} ||^2_2 = 1 \)
\( \eta_{\beta}, || \eta_{\beta} ||^2_2 = \nicefrac{1}{2} \)
\( \eta_{\gamma}, || \eta_{\gamma} ||^2_2 = \nicefrac{1}{7} \)
Coordinate \( \alpha \) s.t. \( \eta_{\gamma} = \eta_{\alpha} + d_0 \alpha\)
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Cohomology

Cohomology is defined by dualizing the chains and the boundary maps between them by applying the \( \mathrm{Hom}\left( \_, G \right) \) functor as follows:

Intuitively, whereas an \(n\)-chain is a sum of \(n\)-simplices, an \(n\)-cochain is a function taking values on each \(n\)-simplex.

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Simplicial Complexes

Def. A simplicial complex \( (K, \Phi) \) is a totally ordered vertex set \( K \) together with a family of nonempty subsets \(\Psi\) of \( K \) called the simplices such that the following hold:
  1. \( k \in K \implies \{ k \} \in \Phi \)
  2. \( a, \in \Phi, b \subseteq a \implies b \in \Phi \)
We call a simplex \(a \in \Phi\) an \(n\)-simplex if \(|a| = n+1\).
Def. We define the \(n\)-chains of a simplicial complex \( (K, \Phi) \), \(C_n(K,\Phi)\) to be the group generated by the \(n\)-simplices of \((K,\Phi)\) under formal summation.
Def. For a given \(n \in \mathbb{Z}\) we define the \(n\)th boundary operator \( \partial_n: C_n(K, \Phi) \rightarrow C_{n-1}(K,\Phi)\) on individual \(n\)-simplices: \[ \partial_n: \{ v_0, ..., v_N \} \mapsto \sum_{i=0}^n (-1)^i \{ v_0, ..., v_{i-1}, v_{i+1}, ..., v_N \} \] and extend it linearly w.r.t. the summation on simplices.

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Homology

Def. For a given simplicial complex \( K, \Phi \) and integer \(n \in \mathbb{Z} \) we denote two special subgroups of \(C_n(K,\Phi)\), namely the \(n\)-cycles \( Z_n(K, \Phi) = \mathrm{ker}\left( \partial_n: C_n(K, \Phi) \rightarrow C_{n-1}(K, \Phi)\right)\) and the \(n\)-boundaries \( B_n(K, \Phi) = \mathrm{im}\left( \partial_{n+1}: C_{n+1}(K, \Phi) \rightarrow C_{n}(K, \Phi)\right)\).
We note that \( B_n(K, \Phi) \subseteq Z_n(K, \Phi) \) as \( \partial_{n} \circ \partial_{n+1} = 0 \) and so the following group is well-defined: \[ H_n(K, \Phi) = \nicefrac{Z_n(K, \Phi)}{B_n(K, \Phi)} \] we call this group the \(n\)th homology group for \((K, \Phi)\).

We will only need the first two homology groups, \( H_0 \) and \( H_1 \).

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Monotonic, Cycling, Transient Definitions

Def. A lifted coordinate function \( \theta: \{ t_1, ..., t_N \} \rightarrow \mathbb{R} \) is said to be \(\epsilon\)-increasing on \( J \subseteq \{ t_1, ..., t_N \} \) iff \( \frac{\theta(t') - \theta(t)}{t' - t} \ge \epsilon \) for all \(t, t' \in J\) whenever \(t' > t\).
Def. A lifted coordinate function \( \theta: \{ t_1, ..., t_N \} \rightarrow \mathbb{R} \) is said to be \(\epsilon\)-cycling on an interval \( (a, b) \subseteq \mathbb{R} \) for \( a,b \in \{ t_1, ..., t_N \} \) if \( \theta \) is \(\epsilon\)-increasing on \( \{t_1, ..., t_N\} \big|_{(a,b)} \) and \( \theta(b) - \theta(a) \ge 1\).
Def. A cube \( \xi \) in the cover is said to be \(\epsilon\)-cycling w.r.t. the function \(f: \bar{X} \rightarrow \mathbb{R}\) and the boxed trajectory \( \gamma: \{t_1, ..., t_N\} \rightarrow \bar{X} \) if there is some interval \( (a,b) \subseteq \mathbb{R} \) s.t. \( \gamma((a,b)) \ni \xi \) where \( f \circ \gamma: \{ t_1, ..., t_N \} \rightarrow \mathbb{R} \) is \( \epsilon \)-cycling on the interval \( (a, b) \).
Def. A cube \( \xi \) in the cover is said to be \(\epsilon\)-transient w.r.t. the function \(f: \bar{X} \rightarrow \mathbb{R}\) and the boxed trajectory \( \gamma: \{t_1, ..., t_N\} \rightarrow \bar{X} \) if for a maximal interval \( (a,b) \subseteq \mathbb{R} \) s.t. \( \gamma((a,b)) \ni \xi \) and \( f \circ \gamma: \{ t_1, ..., t_N \} \rightarrow \mathbb{R} \) is \( \epsilon \)-cycling on the interval \( (a, b) \) we have that \( f(\xi) - f(\gamma(b)) \le 1 \), i.e. the box is in the last cycle of the interval .

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Dynamical Systems and Decompositions

Def. A dynamical system is some system with time-dependent state, i.e. some system \( S \) with a transition map \(f: S \rightarrow S\) or a continuous-time evolution map \(f: \mathbb{R} \times S \rightarrow S\).

In our case, our state-space consists of \( \mathbb{R}^n \) and our time-evolution map is provided by an ODE.

We want to decompose the state-space of a given dynamical system into regions where we can understand behaviour and then study how the system transitions between these regions.

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Homotopy-equivalence

Def. Let \(f, g: X \rightarrow Y\) be two maps between topological spaces. \(f\) and \(g\) are said to be homotopy-equivalent or homotopic if there exists a continuous map \(H: X \times \lbrack 0, 1 \rbrack \rightarrow Y\) such that \(H \big|_{X \times \{0\}} = f \) and \( H \big|_{X \times \{1\}} = g \).
Def. Two topological spaces \(X\) and \(Y\) are said to be homotopy-equivalent or homotopic if there exist continuous maps \(f : X \rightarrow Y \) and \(g: Y \rightarrow X\) such that \(f \circ g: Y \rightarrow Y\) is homotopic to the identity map on \(Y\) and \(g \circ f: X \rightarrow X\) is homotopic to the identity map on \(X\).
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