Two-Manifold Problems with Applications to Nonlinear System Identification

TL;DR

This paper introduces two-manifold algorithms that jointly reconstruct related manifolds from different data views, improving robustness to noise and aiding nonlinear system identification.

Contribution

It proposes a spectral decomposition-based approach for two-manifold problems, enhancing noise robustness and demonstrating applications in nonlinear system learning.

Findings

Two-manifold algorithms outperform single-view methods in noisy settings.

Joint reconstruction reduces bias and improves manifold learning accuracy.

Application to nonlinear system identification from limited data is successful.

Abstract

Recently, there has been much interest in spectral approaches to learning manifolds---so-called kernel eigenmap methods. These methods have had some successes, but their applicability is limited because they are not robust to noise. To address this limitation, we look at two-manifold problems, in which we simultaneously reconstruct two related manifolds, each representing a different view of the same data. By solving these interconnected learning problems together, two-manifold algorithms are able to succeed where a non-integrated approach would fail: each view allows us to suppress noise in the other, reducing bias. We propose a class of algorithms for two-manifold problems, based on spectral decomposition of cross-covariance operators in Hilbert space, and discuss when two-manifold problems are useful. Finally, we demonstrate that solving a two-manifold problem can aid in learning a nonlinear dynamical system from limited data.