Learning Schatten--von Neumann Operators

TL;DR

This paper investigates the learnability of Schatten--von Neumann operators, providing sample complexity bounds and a practical convex optimization approach for their regression in infinite-dimensional spaces.

Contribution

It establishes PAC-learnability of Schatten--von Neumann operators for any p<infinity and adapts the representer theorem to enable finite-dimensional optimization.

Findings

Sample complexity bounds for learning Schatten--von Neumann operators.

PAC-learnability of the class for any p<infinity.

Conversion of infinite-dimensional problems into finite convex programs.

Abstract

We study the learnability of a class of compact operators known as Schatten--von Neumann operators. These operators between infinite-dimensional function spaces play a central role in a variety of applications in learning theory and inverse problems. We address the question of sample complexity of learning Schatten-von Neumann operators and provide an upper bound on the number of measurements required for the empirical risk minimizer to generalize with arbitrary precision and probability, as a function of class parameter $p$. Our results give generalization guarantees for regression of infinite-dimensional signals from infinite-dimensional data. Next, we adapt the representer theorem of Abernethy \emph{et al.} to show that empirical risk minimization over an a priori infinite-dimensional, non-compact set, can be converted to a convex finite dimensional optimization problem over a compact set. In summary, the class of $p$-Schatten--von Neumann operators is probably approximately correct (PAC)-learnable via a practical convex program for any $p < \infty$.