Optimal quadrature-sparsification for integral operator approximation
Résumé
We address the problem of designing sparse quadratures for the approximation of integral operators related to symmetric positive-semidefinite kernels. We more specifically focus on computing sparse quadratures with support included in fixed finite sets of points (quadrature-sparsification), this framework being of particular interest since it encompasses the Nyström approximation of kernel-matrices. For a given kernel, the accuracy of a quadrature approximation is assessed through the squared Hilbert-Schmidt norm (for operators on the underlying reproducing kernel Hilbert space) of the difference between the integral operators related to the initial and approximate measures; by analogy with the notion of kernel discrepancy, we refer to the underlying criterion as the squared-kernel discrepancy between two measures. Sparsity of the approximate quadrature is achieved through the introduction of an L1-type penalisation, and the computation of a penalised squared-kernel-discrepancy-optimal approximation thus consists in a convex quadratic minimisation problem. The penalisation can be introduced under the form of a regularisation term or of a constraint, both formulations being equivalent. The quadratic programs related to the regularised and constrained problems can in particular be interpreted as the Lagrange dual formulations of distorted one-class support-vector machines related to the squared kernel and the initial measure. Numerical strategies for solving large-scale squared-kernel discrepancy minimisation problems are investigated and the efficiency of the approach is illustrated on a series of examples. We in particular demonstrate the ability of the proposed methodology to lead to accurate sparse representations of the main eigenpairs of kernel-matrices related to large-scale datasets.
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