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 @math . Our results give generalization guarantees for regression of infinite-dimensional signals from infinite-dimensional data. Next, we adapt the representer theorem of Abernethy 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 @math -Schatten--von Neumann operators is probably approximately correct (PAC)-learnable via a practical convex program for any @math .
We present a general approach for collaborative filtering (CF) using spectral regularization to learn linear operators mapping a set of "users" to a set of possibly desired "objects". In particular, several recent low-rank type matrix-completion methods for CF are shown to be special cases of our proposed framework. Unlike existing regularization-based CF, our approach can be used to incorporate additional information such as attributes of the users objects---a feature currently lacking in existing regularization-based CF approaches---using popular and well-known kernel methods. We provide novel representer theorems that we use to develop new estimation methods. We then provide learning algorithms based on low-rank decompositions and test them on a standard CF data set. The experiments indicate the advantages of generalizing the existing regularization-based CF methods to incorporate related information about users and objects. Finally, we show that certain multi-task learning methods can be also seen as special cases of our proposed approach.