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Testing Positive Semi-Definiteness via Random Submatrices

We study the problem of testing whether a matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ with bounded entries ($\|\mathbf{A}\|_\infty \leq 1$) is positive semi-definite (PSD), or $ε$-far in Euclidean distance from the PSD cone, meaning that $\min_{\mathbf{B} \succeq 0} \|\mathbf{A} - \mathbf{B}\|_F^2 > εn^2$, where $\mathbf{B} \succeq 0$ denotes that $\mathbf{B}$ is PSD. Our main algorithmic contribution is a non-adaptive tester which distinguishes between these cases using only $\tilde{O}(1/ε^4)$ queries to the entries of $\mathbf{A}$. If instead of the Euclidean norm we considered the distance in spectral norm, we obtain the "$\ell_\infty$-gap problem", where $\mathbf{A}$ is either PSD or satisfies $\min_{\mathbf{B}\succeq 0} \|\mathbf{A}- \mathbf{B}\|_2 > εn$. For this related problem, we give a $\tilde{O}(1/ε^2)$ query tester, which we show is optimal up to $\log(1/ε)$ factors. Our testers randomly sample a collection of principal submatrices and check whether these submatrices are PSD. Consequentially, our algorithms achieve one-sided error: whenever they output that $\mathbf{A}$ is not PSD, they return a certificate that $\mathbf{A}$ has negative eigenvalues. We complement our upper bound for PSD testing with Euclidean norm distance by giving a $\tildeΩ(1/ε^2)$ 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 $\tildeΩ(1/ε^4)$ sampling lower bound for testing the Schatten-$1$ norm with a $εn^{1.5}$ gap, extending a result of Balcan, Li, Woodruff, and Zhang [SODA'19]. In addition, it yields new sampling lower bounds for estimating the Ky-Fan Norm, and the cost of the best rank-$k$ approximation.