# Accuracy of Suboptimal Solutions to Kernel Principal Component Analysis

Gnecco, Giorgio and Sanguineti, Marcello Accuracy of Suboptimal Solutions to Kernel Principal Component Analysis. Computational Optimization and Applications, 42 (2). pp. 265-287. ISSN 0926-6003 (2009)

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## Abstract

For Principal Component Analysis in Reproducing Kernel Hilbert Spaces (KPCA), optimization over sets containing only linear combinations of all n-tuples of kernel functions is investigated, where n is a positive integer smaller than the number of data. Upper bounds on the accuracy in approximating the optimal solution, achievable without restrictions on the number of kernel functions, are derived. The rates of decrease of the upper bounds for increasing number n of kernel functions are given by the summation of two terms, one proportional to n −1/2 and the other to n −1, and depend on the maximum eigenvalue of the Gram matrix of the kernel with respect to the data. Primal and dual formulations of KPCA are considered. The estimates provide insights into the effectiveness of sparse KPCA techniques, aimed at reducing the computational costs of expansions in terms of kernel units.

Item Type: Article 10.1007/s10589-007-9108-y Principal component analysis (PCA); Kernel methods; Suboptimal solutions; Primal and dual problems; Lagrangian; Regularized optimization problems Q Science > QA Mathematics > QA75 Electronic computers. Computer science Computer Science and Applications Giorgio Gnecco 17 Sep 2013 07:40 17 Sep 2013 07:40 http://eprints.imtlucca.it/id/eprint/1744