Optimal quadrature-sparsification for integral operator approximation
Résumé
The design of sparse quadratures for the approximation of integral operators related to symmetric positive-semidefinite kernels is addressed. Particular emphasis is placed on the approximation of the main eigenpairs of the initial operator. A special attention is drawn to the design of sparse quadratures with support included in fixed finite sets of points (that is, quadrature-sparsification), this framework in particular encompassing the approximation of kernel matrices. For a given kernel, the accuracy of a quadrature approximation is assessed through the squared Hilbert-Schmidt norm (for operators acting 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, the underlying criterion is referred to as the squared-kernel discrepancy between the two measures. In the quadrature-sparsification framework, sparsity of the approximate quadrature is promoted through the introduction of an L1-type penalisation, and the computation of a penalised squared-kernel-discrepancy-optimal approximation then consists in a convex quadratic minimisation problem; such quadratic programs can in particular be interpreted as the Lagrange dual formulations of distorted one-class support-vector machines related to the squared kernel. Error bounds on the induced spectral approximations are derived, and the connection between penalisation and accuracy of the approximation is investigated. Numerical strategies for solving large-scale penalised squared- kernel-discrepancy minimisation problems are discussed, and the efficiency of the approach is illustrated by a series of examples. In particular, the ability of the proposed methodology to lead to accurate approximations of the main eigenpairs of kernel matrices related to large-scale datasets is demonstrated.
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