Adaptive randomized pivoting for column subset selection, DEIM, and low-rank approximation
Alice Cortinovis, Daniel Kressner
TL;DR
This paper introduces Adaptive Randomized Pivoting (ARP), a simple, adaptive leverage-score sampling method for the Column Subset Selection Problem that guarantees, in expectation, an error bound of the form $\\mathbb{E}\\|A - \\Pi_J A\\|_F^2 \\le (r+1) \\|A - A V V^T\\|_F^2$, matching the optimal rank-$r$ approximation up to a constant factor. ARP leverages adaptive sampling based on the row norms of a provided row-space basis $V$, avoiding the need to form $A V$ and enabling efficient applications to DEIM, cross/skeleton approximation, and Nyström for SPSD matrices; a derandomized variant recovers Osinsky’s deterministic algorithm with favorable bounds. The method is implemented stably via Householder reflections, with computational cost $\\mathcal{O}(n r^2)$, and is supported by a thorough analysis using oblique projections and conditional expectations. Numerical experiments validate ARP and its variants across CSSP, DEIM, cross approximation, and Nyström, showing competitive accuracy and clear advantages in simplicity and efficiency. Derandomization further yields deterministic algorithms with strong error guarantees, particularly for SPSD Nyström, making ARP a versatile tool for large-scale matrix approximations.
Abstract
We derive a new adaptive leverage score sampling strategy for solving the Column Subset Selection Problem (CSSP). The resulting algorithm, called Adaptive Randomized Pivoting, can be viewed as a randomization of Osinsky's recently proposed deterministic algorithm for CSSP. It guarantees, in expectation, an approximation error that matches the optimal existence result in the Frobenius norm. Although the same guarantee can be achieved with volume sampling, our sampling strategy is much simpler and less expensive. To show the versatility of Adaptive Randomized Pivoting, we apply it to select indices in the Discrete Empirical Interpolation Method, in cross/skeleton approximation of general matrices, and in the Nystroem approximation of symmetric positive semi-definite matrices. In all these cases, the resulting randomized algorithms are new and they enjoy bounds on the expected error that match -- or improve -- the best known deterministic results. A derandomization of the algorithm for the Nystroem approximation results in a new deterministic algorithm with a rather favorable error bound.