Manifold Learning with Contracting Observers for Data-driven Time-series Analysis

TL;DR

This paper introduces a data-driven framework combining manifold learning and contracting observers to uncover intrinsic variables of high-dimensional dynamical systems from measurements, demonstrated on toy and music analysis examples.

Contribution

It presents a novel integration of diffusion maps and control theory to estimate latent variables without extensive modeling assumptions.

Findings

Successfully revealed intrinsic variables in toy and music data

Demonstrated effectiveness of the method in high-dimensional settings

Provided a new approach for data-driven time-series analysis

Abstract

Analyzing signals arising from dynamical systems typically requires many modeling assumptions and parameter estimation. In high dimensions, this modeling is particularly difficult due to the "curse of dimensionality". In this paper, we propose a method for building an intrinsic representation of such signals in a purely data-driven manner. First, we apply a manifold learning technique, diffusion maps, to learn the intrinsic model of the latent variables of the dynamical system, solely from the measurements. Second, we use concepts and tools from control theory and build a linear contracting observer to estimate the latent variables in a sequential manner from new incoming measurements. The effectiveness of the presented framework is demonstrated by applying it to a toy problem and to a music analysis application. In these examples we show that our method reveals the intrinsic variables of the analyzed dynamical systems.