A randomized algorithm to solve reduced rank operator regression
This provides an efficient method for high-dimensional operator regression, but it is incremental as it adapts existing reduced rank techniques with randomization.
They tackled vector-valued regression with infinite-dimensional inputs/outputs by developing a randomized algorithm (R4) based on reduced rank regression and Gaussian sketching, proving it achieves near-optimal empirical risk and demonstrating advantages in neuroscience and dynamical systems datasets.
We present and analyze an algorithm designed for addressing vector-valued regression problems involving possibly infinite-dimensional input and output spaces. The algorithm is a randomized adaptation of reduced rank regression, a technique to optimally learn a low-rank vector-valued function (i.e. an operator) between sampled data via regularized empirical risk minimization with rank constraints. We propose Gaussian sketching techniques both for the primal and dual optimization objectives, yielding Randomized Reduced Rank Regression (R4) estimators that are efficient and accurate. For each of our R4 algorithms we prove that the resulting regularized empirical risk is, in expectation w.r.t. randomness of a sketch, arbitrarily close to the optimal value when hyper-parameteres are properly tuned. Numerical expreriments illustrate the tightness of our bounds and show advantages in two distinct scenarios: (i) solving a vector-valued regression problem using synthetic and large-scale neuroscience datasets, and (ii) regressing the Koopman operator of a nonlinear stochastic dynamical system.