Robust and Sample Optimal Algorithms for PSD Low-Rank Approximation

Recently, Musco and Woodruff (FOCS, 2017) showed that given an $n \times n$ positive semidefinite (PSD) matrix $A$, it is possible to compute a $(1+\epsilon)$-approximate relative-error low-rank approximation to $A$ by querying $O(nk/\epsilon^{2.5})$ entries of $A$ in time $O(nk/\epsilon^{2.5} +n k^{{\omega-1}/\epsilon{2(\omega-1)})$.} They also showed that any relative-error low-rank approximation algorithm must query $\Omega(nk/\epsilon)$ entries of $A$, this gap has since remained open. Our main result is to resolve this question by obtaining an optimal algorithm that queries $O(nk/\epsilon)$ entries of $A$ and outputs a relative-error low-rank approximation in $O(n(k/\epsilon)^{\omega-1})$ time. Note, our running time improves that of Musco and Woodruff, and matches the information-theoretic lower bound if the matrix-multiplication exponent $\omega$ is $2$. We then extend our techniques to negative-type distance matrices. Bakshi and Woodruff (NeurIPS, 2018) showed a bi-criteria, relative-error low-rank approximation which queries $O(nk/\epsilon^{2.5})$ entries and outputs a rank-$(k+4)$ matrix. We show that the bi-criteria guarantee is not necessary and obtain an $O(nk/\epsilon)$ query algorithm, which is optimal. Our algorithm applies to all distance matrices that arise from metrics satisfying negative-type inequalities, including $\ell1, \ell2,$ spherical metrics and hypermetrics. Next, we introduce a new robust low-rank approximation model which captures PSD matrices that have been corrupted with noise. While a sample complexity lower bound precludes sublinear algorithms for arbitrary PSD matrices, we provide the first sublinear time and query algorithms when the corruption on the diagonal entries is bounded. As a special case, we show sample-optimal sublinear time algorithms for low-rank approximation of correlation matrices corrupted by noise.