# Sharp analysis of low-rank kernel matrix approximations

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Francis Bach
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Proceedings of the 26th Annual Conference on Learning Theory, PMLR 30:185-209, 2013.

#### Abstract

We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces, a common practical limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n^2). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p^2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the \emphdegrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such methods, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same \emphpredictive performance than existing algorithms, for any given problem instance, and not only for worst-case situations.

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