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From Data to Combinatorial Multivector field Through an Optimization-Based Framework

Dominic Desjardins Côté, Donald Woukeng

TL;DR

The paper presents an optimization-based framework to construct combinatorial multivector fields from finite vector data, enabling data-driven global dynamical analysis without explicit differential equations. It defines a binary-variable optimization with cosine-similarity costs and convexity constraints, and introduces two models—the Generalized CDS model and a simplified One-Toplex-per-Multivector model—to generate convex multivectors from data lying on a simplicial complex. The pipeline covers data preprocessing, complex construction, and interpretation via Morse decompositions and Conley indices, with theoretical notes on NP-hardness and post-processing when necessary. Empirical validation on Vanderpol and Lorenz systems demonstrates the method's ability to recover invariant structures and attractor-like behavior through interpretable multivector fields and strongly connected components.

Abstract

This paper extends and generalizes previous works on constructing combinatorial multivector fields from continuous systems (see [10]) and the construction of combinatorial vector fields from data (see [2]) by introducing an optimization based framework for the construction of combinatorial multivector fields from finite vector field data. We address key challenges in convexity, computational complexity and resolution, providing theoretical guarantees and practical methodologies for generating combinatorial representation of the dynamics of our data.

From Data to Combinatorial Multivector field Through an Optimization-Based Framework

TL;DR

The paper presents an optimization-based framework to construct combinatorial multivector fields from finite vector data, enabling data-driven global dynamical analysis without explicit differential equations. It defines a binary-variable optimization with cosine-similarity costs and convexity constraints, and introduces two models—the Generalized CDS model and a simplified One-Toplex-per-Multivector model—to generate convex multivectors from data lying on a simplicial complex. The pipeline covers data preprocessing, complex construction, and interpretation via Morse decompositions and Conley indices, with theoretical notes on NP-hardness and post-processing when necessary. Empirical validation on Vanderpol and Lorenz systems demonstrates the method's ability to recover invariant structures and attractor-like behavior through interpretable multivector fields and strongly connected components.

Abstract

This paper extends and generalizes previous works on constructing combinatorial multivector fields from continuous systems (see [10]) and the construction of combinatorial vector fields from data (see [2]) by introducing an optimization based framework for the construction of combinatorial multivector fields from finite vector field data. We address key challenges in convexity, computational complexity and resolution, providing theoretical guarantees and practical methodologies for generating combinatorial representation of the dynamics of our data.
Paper Structure (23 sections, 7 theorems, 25 equations, 5 figures)

This paper contains 23 sections, 7 theorems, 25 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Overview of main results
  4. Preliminaries
  5. Relations and posets
  6. Topological spaces
  7. Combinatorial multivector fields
  8. Construction of a combinatorial multivector field from data
  9. Optimization problem : Objective function
  10. Model 1 : Generalization of the CDS Model
  11. Model 2 : One Toplex per Multivector
  12. Complete Pipeline and interpretation
  13. Construction of a simplicial complex on data
  14. Multivector field construction
  15. One toplex per multivector's model
  16. ...and 8 more sections

Key Result

Proposition 2.1

En1989. Assume $A$ is a subset of a topological space $X$. Then the following conditions are equivalent. It is easy to see that the intersection of a finite family of locally closed subsets is locally closed.

Figures (5)

  • Figure 1: The combinatorial multivector field obtained from the optimization problem (\ref{['eq:Intro']}) with data from the dynamical system (\ref{['eq:dynSysRepOrb']}).
  • Figure 2: The red arrows are the the suppositions of the Example (\ref{['ex:constConvex']}). By the constraints (\ref{['eq:ConvConst1']}) (\ref{['eq:ConvConst2']}), and (\ref{['eq:ConvConst3']}), we need to set some variables to $1$ which are represented by blue arrows.
  • Figure 3: This is a feasible solution of (\ref{['eq:MinOptGenCDS']}) that does not induce a combinatorial vector field. The convexity condition is not satisfied.
  • Figure 4: The first Figure is the combinatorial multivector field $\mathcal{V}$ obtained from the Vanderpol oscillator's equation. The second Figure is the gradient part of $\mathcal{V}$. The third Figure is the only strongly connected components of $\mathcal{V}$.
  • Figure 5: On the left, we have the set of cluster and on the right, we have the strongly connected component.

Theorems & Definitions (10)

  • Proposition 2.1
  • Theorem 2.2
  • Example 3.1
  • Example 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 3.6
  • Lemma 3.7
  • Conjecture 3.8