Learning Schatten--von Neumann Operators
This work addresses the learnability of operators crucial in learning theory and inverse problems, offering theoretical guarantees for regression of infinite-dimensional signals, but it appears incremental as it adapts existing representer theorems.
The paper tackles the problem of learning Schatten--von Neumann operators, which are compact operators between infinite-dimensional function spaces, by providing an upper bound on the sample complexity for generalization and showing that empirical risk minimization can be converted into a convex finite-dimensional optimization problem. The result is that this class of operators is PAC-learnable via a practical convex program for any p < ∞.
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 $p < \infty$.