Two-Manifold Problems with Applications to Nonlinear System Identification
This work addresses a specific limitation in spectral manifold learning for researchers in machine learning and system identification, offering an incremental improvement in noise robustness.
The paper tackles the problem of noise sensitivity in kernel eigenmap methods for manifold learning by introducing two-manifold problems, where two related manifolds are reconstructed simultaneously to suppress noise, and demonstrates its application to nonlinear system identification with improved robustness.
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.