This paper introduces a sparse matrix discrete interpolation method to effectively compute matrix approximations in the reduced order modeling framework. The sparse algorithm developed herein relies on the discrete empirical interpolation method and uses only samples of the nonzero entries of the matrix series. The proposed approach can approximate very large matrices, unlike the current matrix...

The problem of multiple kernel learning based on penalized empirical risk minimization is discussed. The complexity penalty is determined jointly by the empirical L2 norms and the reproducing kernel Hilbert space (RKHS) norms induced by the kernels with a data-driven choice of regularization parameters. The main focus is on the case when the total number of kernels is large, but only a relative...

Abstract. This paper presents a general coding method where data in a Hilbert space are represented by finite dimensional coding vectors. The method is based on empirical risk minimization within a certain class of linear operators, which map the set of coding vectors to the Hilbert space. Two results bounding the expected reconstruction error of the method are derived, which highlight the role...

We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsity-inducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual l1-norm and the group l1-norm by allowing the subsets to overlap. This leads to a specific set of allowed nonzero patterns for the solutions of such problem...

AbsIracf-In this paper, the generalization ability 01 empirical risk minimization algorithms is investigated in the cootext of dislribution-lm prohably appmxjmalely comecl (PAC) learning. We identily a class of empirical risk minimization algorithms that are PAC, and show that the generic version of the support vector regression method belongs lo the class lor any given Mercer kernel. Moreover,...

We develop an approach to risk minimization and stochastic optimization that pro-1vides a convex surrogate for variance, allowing near-optimal and computationally2efficient trading between approximation and estimation error. Our approach builds3off of techniques for distributionally robust optimization and Owen’s empirical4likelihood, and we provide a number of f...