Testing Positive Semidefiniteness Using Linear Measurements

William Joseph Swartworth, David Woodruff, Deanna Needell

Abstract: We study the problem of testing whether a symmetric $d \times d$ input matrix $A$ is symmetric positive semidefinite (PSD), or is $\epsilon$-far from the PSD cone, meaning that $\lambda_{\min}(A) \leq - \epsilon \|A\|_p$, where $\|A\|_p$ is the Schatten-$p$ norm of $A$. In applications one often needs to quickly tell if an input matrix is PSD, and a small distance from the PSD cone may be tolerable. We consider two well-studied query models for measuring efficiency, namely, the matrix-vector and vector-matrix-vector query models, and focus on one-sided testers, the latter of which are testers that correctly classify any PSD input, but may fail on a non-PSD input with a tiny failure probability. Up to logarithmic factors, in the matrix-vector query model we show a tight $\tildeTheta(1/\epsilon^{p/(2p+1)})$ bound, while in the vector-matrix-vector query model we show a tight $\tildeTheta(n^{1-1/p}/\epsilon)$ bound, for every $p \geq 1$. We also show a strong separation between one-sided and two-sided testers in the vector-matrix-vector model, where a two-sided tester can fail on both PSD and non-PSD inputs with a tiny failure probability. In particular, for the important case of the Frobenius norm, we show that any one-sided tester requires $\tildeOmega(\sqrt{n}/\epsilon)$ queries, but there is a two-sided tester with $\tildeO(1/\epsilon^8)$ queries. We also give a number of additional separations between adaptive and non-adaptive testers.