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arXiv · 2304.09678

Column Subset Selection and Nystr\"om Approximation via Continuous Optimization

Anant Mathur·Sarat Moka·Zdravko Botev
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Abstract

We propose a continuous optimization algorithm for the Column Subset Selection Problem (CSSP) and Nystr\"om approximation. The CSSP and Nystr\"om method construct low-rank approximations of matrices based on a predetermined subset of columns. It is well known that choosing the best column subset of size $k$ is a difficult combinatorial problem. In this work, we show how one can approximate the optimal solution by defining a penalized continuous loss function which is minimized via stochastic gradient descent. We show that the gradients of this loss function can be estimated efficiently using matrix-vector products with a data matrix $X$ in the case of the CSSP or a kernel matrix $K$ in the case of the Nystr\"om approximation. We provide numerical results for a number of real datasets showing that this continuous optimization is competitive against existing methods.

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Anant Mathur, Sarat Moka, Zdravko Botev. 2023-04-19. Column Subset Selection and Nystr\"om Approximation via Continuous Optimization. https://arxiv.org/abs/2304.09678

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