A dynamical systems based framework for dimension reduction

TL;DR

This paper introduces a novel nonlinear dynamical systems framework called dynamical dimension reduction (DDR) for learning low-dimensional data representations by evolving points towards a subspace, with training via gradient-based optimization.

Contribution

The paper develops a new DDR framework that models data embedding as a flow towards a subspace, with a well-posed optimization and gradient computation using optimal control theory.

Findings

DDR outperforms PCA, t-SNE, and Umap on synthetic datasets.

The method is theoretically grounded with proven properties.

Gradient-based training effectively learns the low-dimensional embedding.

Abstract

We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems, which we call dynamical dimension reduction (DDR). In the DDR model, each point is evolved via a nonlinear flow towards a lower-dimensional subspace; the projection onto the subspace gives the low-dimensional embedding. Training the model involves identifying the nonlinear flow and the subspace. Following the equation discovery method, we represent the vector field that defines the flow using a linear combination of dictionary elements, where each element is a pre-specified linear/nonlinear candidate function. A regularization term for the average total kinetic energy is also introduced and motivated by optimal transport theory. We prove that the resulting optimization problem is well-posed and establish several properties of the DDR method. We also show how the DDR method can be trained using a gradient-based optimization method, where the gradients are computed using the adjoint method from optimal control theory. The DDR method is implemented and compared on synthetic and example datasets to other dimension reductions methods, including PCA, t-SNE, and Umap.