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

TL;DR

This paper introduces an optimization-based framework to construct combinatorial multivector fields from finite vector field data, addressing key challenges and providing theoretical guarantees for representing complex dynamics.

Contribution

It generalizes previous methods by offering a unified optimization approach for deriving combinatorial multivector fields from continuous and data-driven systems.

Findings

Provides a new optimization framework for multivector field construction

Addresses convexity, complexity, and resolution challenges

Offers theoretical guarantees and practical methodologies

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.