Testing Matrix Rank, Optimally.

We show that for the problem of testing if a matrix A ∈ Fn × n has rank at most d, or requires changing an ϵ-fraction of entries to have rank at most d, there is a non-adaptive query algorithm making Õ(d2/ϵ) queries. Our algorithm works for any field F. This improves upon the previous O(d2/ϵ2) bound (Krauthgamer and Sasson, SODA '03), and bypasses an Ω(d2/ϵ2) lower bound of (Li, Wang, and Woodruff, KDD '14) which holds if the algorithm is required to read a submatrix. Our algorithm is the first such algorithm which does not read a submatrix, and instead reads a carefully selected non-adaptive pattern of entries in rows and columns of A. We complement our algorithm with a matching [MATH HERE] query complexity lower bound for non-adaptive testers over any field. We also give tight bounds of [MATH HERE] queries in the sensing model for which query access comes in the form of 〈Xi, A〉 ≔ tr(Xi⊤A); perhaps surprisingly these bounds do not depend on ϵ. Testing rank is only one of many tasks in determining if a matrix has low intrinsic dimensionality We next develop a novel property testing framework for testing numerical properties of a real-valued matrix A more generally, which includes the stable rank, Schatten-p norms, and SVD entropy. Specifically, we propose a bounded entry model, where A is required to have entries bounded by 1 in absolute value. Such a model provides a meaningful framework for testing numerical quantities and avoids trivialities caused by single entries being arbitrarily large. It is also well-motivated by recommendation systems. We give upper and lower bounds for a wide range of problems in this model, and discuss connections to the sensing model above. We obtain several results for estimating the operator norm that may be of independent interest. For example, we show that if the stable rank is constant, ||A||F = Ω(n), and the singular value gap σ1(A)/σ2(A) = (1/ϵ)γ for any constant γ > 0, then the operator norm can be estimated up to a (1 ± ϵ)-factor non-adaptively by querying O(1/ϵ2) entries. This should be contrasted to adaptive methods such as the power method, or previous non-adaptive sampling schemes based on matrix Bernstein inequalities which read a 1/ϵ2 × 1/ϵ2 submatrix and thus make Ω(1/ϵ4) queries. Similar to our non-adaptive algorithm for testing rank, our scheme instead reads a carefully selected pattern of entries.