Manifold Learning With Contracting Observers for Data-Driven Time-Series Analysis

Analyzing signals arising from dynamical systems typically requires many modeling assumptions. 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.