A dynamical systems based framework for dimension reduction
This work addresses dimension reduction for data analysis, presenting a novel framework that is incremental as it builds on existing dynamical systems and optimal transport concepts.
The authors tackled the problem of learning low-dimensional data representations by proposing a dynamical dimension reduction (DDR) framework based on nonlinear dynamical systems, which evolves points towards a subspace for embedding and showed competitive performance on synthetic and example datasets compared to methods like PCA, t-SNE, and Umap.
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.