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. We first consider one-sided testers, 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 $\widetilde{\Theta}(1/\epsilon^{p/(2p+1)})$ bound, while in the vector-matrix-vector query model we show a tight $\widetilde{\Theta}(d^{1-1/p}/\epsilon)$ 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 $\widetilde{\Omega}(\sqrt{d}/\epsilon)$ queries. However we introduce a bilinear sketch for two-sided testing from which we construct a Frobenius norm tester achieving the optimal $\widetilde{O}(1/\epsilon^2)$ queries. We also give a number of additional separations between adaptive and non-adaptive testers. Our techniques have implications beyond testing, providing new methods to approximate the spectrum of a matrix with Frobenius norm error using dimensionality reduction in a way that preserves the signs of eigenvalues.

翻译：我们研究检验对称 $d \times d$ 输入矩阵 $A$ 是否为对称半正定（PSD），或是否与 PSD 锥 $\epsilon$-远离（即 $\lambda_{\min}(A) \leq - \epsilon \|A\|_p$，其中 $\|A\|_p$ 为 $A$ 的 Schatten-$p$ 范数）的问题。在应用中，常需快速判断输入矩阵是否 PSD，且与 PSD 锥的微小距离可被容忍。我们考虑两种衡量效率的经典查询模型：矩阵-向量查询模型和向量-矩阵-向量查询模型。首先研究单侧检验器（即能正确分类所有 PSD 输入，但对非 PSD 输入可能以微小失败概率出错的检验器）。在对数因子范围内，我们证明：对于任意 $p \geq 1$，矩阵-向量查询模型具有紧界 $\widetilde{\Theta}(1/\epsilon^{p/(2p+1)})$，而向量-矩阵-向量查询模型具有紧界 $\widetilde{\Theta}(d^{1-1/p}/\epsilon)$。我们还展示了向量-矩阵-向量模型中单侧与双侧检验器的显著分离：双侧检验器允许对 PSD 和非 PSD 输入均以微小概率失败。特别地，对于 Frobenius 范数这一重要情形，我们证明任何单侧检验器需要 $\widetilde{\Omega}(\sqrt{d}/\epsilon)$ 次查询。然而，我们引入一种用于双侧检验的双线性草图（bilinear sketch），并据此构建了达到最优 $\widetilde{O}(1/\epsilon^2)$ 查询复杂度的 Frobenius 范数检验器。此外，我们给出了自适应与非自适应检验器之间的一系列分离结果。我们的技术方法超越了检验本身，提供了利用降维方法近似矩阵谱的新途径——在保持特征值符号的前提下，以 Frobenius 范数误差进行逼近。