Learning latent representations in high-dimensional state spaces using polynomial manifold constructions

TL;DR

This paper introduces a nonlinear dimension reduction framework using polynomial manifolds to learn efficient latent representations in high-dimensional state spaces, demonstrated on the Korteweg-de Vries equation.

Contribution

It proposes two methods for learning polynomial manifold embeddings, improving representation accuracy by capturing nonlinear interactions.

Findings

Reduced representation error by up to two orders of magnitude

Effective nonlinear dimension reduction for high-dimensional data

Applicable to complex dynamical systems like Korteweg-de Vries

Abstract

We present a novel framework for learning cost-efficient latent representations in problems with high-dimensional state spaces through nonlinear dimension reduction. By enriching linear state approximations with low-order polynomial terms we account for key nonlinear interactions existing in the data thereby reducing the problem's intrinsic dimensionality. Two methods are introduced for learning the representation of such low-dimensional, polynomial manifolds for embedding the data. The manifold parametrization coefficients can be obtained by regression via either a proper orthogonal decomposition or an alternating minimization based approach. Our numerical results focus on the one-dimensional Korteweg-de Vries equation where accounting for nonlinear correlations in the data was found to lower the representation error by up to two orders of magnitude compared to linear dimension reduction techniques.