Testing Positive Semi-Definiteness via Random Submatrices

We study the problem of testing whether a matrix $\mathrm{A}\in \mathbb{R}^{n\times n}$ with bounded entries ($\Vert \mathrm{A}\Vert_{\infty}\leq 1$) is positive semidefinite (PSD), or $\epsilon$-far in Euclidean distance from the PSD cone, meaning that $\min\nolimits_{\mathrm{B}\succeq 0}\Vert \mathrm{A}-\mathrm{B}\Vert_{F}^{2} > \epsilon n^{2}$, where $\mathrm{B}\succeq 0$ denotes that B is PSD. Our main algorithmic contribution is a non-adaptive tester which distinguishes between these cases using only $\tilde{O}(1/\epsilon^{4})$ queries to the entries of A.11Throughout the paper, $\tilde{O}(\cdot)$ hides $\log(1/\epsilon)$ factors. If instead of the Eucledian norm we considered the distance in spectral norm, we obtain the “$\ell_{\infty}$-gap problem”, where A is either PSD or satisfies $\min\nolimits_{\mathrm{B}\succ 0}\Vert \mathrm{A}-\mathrm{B}\Vert_{2} > \epsilon n$. For this related problem, we give a $\tilde{O}(1/\epsilon^{2})$ query tester, which we show is optimal up to $\log(1/\epsilon)$ factors. Both our testers randomly sample a collection of principal sub-matrices and check whether these sub-matrices are PSD. Consequentially, our algorithms achieve one-sided error: whenever they output that A is not PSD, they return a certificate that A has negative eigenvalues. We complement our upper bound for PSD testing with Eucledian norm distance by giving a $\tilde{\Omega}(1/\epsilon^{2})$ lower bound for any non-adaptive algorithm. Our lower bound construction is general, and can be used to derive lower bounds for a number of spectral testing problems. As an example of the applicability of our construction, we obtain a new $\tilde{\Omega}(1/\epsilon^{4})$ sampling lower bound for testing the Schatten-1 norm with a $\epsilon n^{1.5}$ gap, extending a result of Balcan, Li, Woodruff, and Zhang [11]. In addition, our hard instance results in new sampling lower bounds for estimating the Ky-Fan Norm, and the cost of rank-$k$ approximations, i.e. $\Vert\mathrm{A}-\mathrm{A}_{k}\Vert_{F}^{2}=\sum\nolimits_{i > k}\sigma_{i}^{2}(\mathrm{A})$.