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We present and analyze an algorithm designed for addressing vector-valuedregression problems involving possibly infinite-dimensional input and outputspaces. The algorithm is a randomized adaptation of reduced rank regression, atechnique to optimally learn a low-rank vector-valued function (i.e. anoperator) between sampled data via regularized empirical risk minimization withrank constraints. We propose Gaussian sketching techniques both for the primaland dual optimization objectives, yielding Randomized Reduced Rank Regression(R4) estimators that are efficient and accurate. For each of our R4 algorithmswe prove that the resulting regularized empirical risk is, in expectationw.r.t. randomness of a sketch, arbitrarily close to the optimal value whenhyper-parameteres are properly tuned. Numerical expreriments illustrate thetightness of our bounds and show advantages in two distinct scenarios: (i)solving a vector-valued regression problem using synthetic and large-scaleneuroscience datasets, and (ii) regressing the Koopman operator of a nonlinearstochastic dynamical system.

A randomized algorithm to solve reduced rank operator regression