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

arXiv: https://arxiv.org/abs/1604.04492 · HTML: https://mlsanity.com/paper/1604.04492/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Apr 2016.

## Built on by (most cited first, 4 of 4)

- [Spatiotemporal Analysis Using Riemannian Composition of Diffusion Operators](https://mlsanity.com/paper/2201.08530/) (2022)
- [Analyzing Deep Transformer Models for Time Series Forecasting via Manifold Learning](https://mlsanity.com/paper/2410.13792/) (2024)
- [Geometric path augmentation for inference of sparsely observed stochastic nonlinear systems](https://mlsanity.com/paper/2301.08102/) (2023)
- [Geometric constraints improve inference of sparsely observed stochastic dynamics](https://mlsanity.com/paper/2304.00423/) (2023)

## Abstract (arXiv)

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.