A Dynamical System-Based Framework for Dimension Reduction

Abstract

We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems, which we call the 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 the 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 the optimal control theory. The DDR method is implemented and compared on synthetic and example data sets to other dimension reduction methods, including the PCA, t-SNE, and Umap.

Funding

Directorate for Mathematical and Physical Sciences(17-52202)