We study the problem of testing whether a matrix A \in R^{n \times n} with bounded entries (\| A \|_\infty \leq 1) is positive semi-definite (PSD), or \epsilon-far in \ell_2^2-distance from the PSD cone, i.e. \min _{ B \succeq 0} \| A - B \|_F^2 = \sum_{i : \lambda_i(A) < 0 \lambda_i^2(A) } > \epsilon n^2  \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/\epsilon^4) queries to the entries of A. For the related "\ell_\infty-gap problem", where A is either PSD or has an eigenvalue satisfying \lambda_i(A) <\succeq 0} \|\mathbf{A}} - \epsilon n, our algorithm only requires\|_2 > \epsilon n. For this related problem, we give a } \tilde{O}(1/\epsilon^2) queries, which is optimal up to \log(1/\epsilon) factors. Our testers randomly sample a collection of principle 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 \ell_2^2-gap 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  ^{1.5} gap, extending a result of Balcan, Li, Woodruff, and Zhang [SODA'19]. 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. \|A - A_k\|_F^2 = \sum_{i > k \sigma_i^2(A)}  .
We study the problem of testing whether a matrix \mathbf{A \in R^{n \times n} with bounded entries (\| \mathbf{A \|_\infty \leq 1) is positive semi-definite (PSD), or \epsilon-far in Euclidean distance from the PSD cone, meaning that \min _{ \mathbf{B \succeq 0} \| \mathbf{A - \mathbf{B \|_F^2 \lambda_i^2(A) } > \epsilon n^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/\epsilon^4) queries to the entries of \mathbf{A "\ell_\infty-gap problem", where \mathbf{A is either PSD or satisfies \min_{\mathbf{B\succeq 0} \|\mathbf{A}} - \mathbf{B\|_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. 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 \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 [SODA'19]. In addition, it yields new sampling lower bounds for estimating the Ky-Fan Norm, and the cost of the best rank-k \sigma_i^2(A)} approximation .
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