Testing Matrix Rank, Optimally

Abstract

We show that for the problem of testing if a matrix A ∈ F n×n has rank at most d, or requires changing an -fraction of entries to have rank at most d, there is a non-adaptive query algorithm making O(d 2 / ) queries. Our algorithm works for any field F. This improves upon the previous O(d 2 / 2 ) bound (Krauthgamer and Sasson, SODA '03), and bypasses an Ω(d 2 / 2 ) 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 Ω(d 2 / ) query complexity lower bound for non-adaptive testers over any field. We also give tight bounds of Θ(d 2 ) queries in the sensing model for which query access comes in the form of X i , A := tr(X i A); perhaps surprisingly these bounds do not depend on .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 σ 1 (A)/σ 2 (A) = (1/ ) γ for any constant γ > 0, then the operator norm can be estimated up to a (1 ± )-factor non-adaptively by querying O(1/ 2 ) 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/ 2 × 1/ 2 submatrix and thus make Ω(1/ 4 ) queries. Similar to our non-adaptive algorithm for testing rank, our scheme instead reads a carefully selected pattern of entries.

Optimal Estimation of Schatten Norms of a rectangular Matrix

“…In this noiseless problem, one aims at recovering functionals of A such as a Schatten norm or the rank using the smallest possible budget. See [2,19,26]. As in our noisy framework, [26] emphasizes that estimating even Schatten norms using sketches seems easier than estimating non-even Schatten norms.…”

“…Most work around effective ranks consider noiseless setting [33], but the matrix A is sometimes allowed to be only partially observed (e.g [2]). In this work, we tackle the general problem of estimating the effective rank of A relying on a single noisy observation Y of A.…”

“…The latter problem is classicalsee e.g. [9] and our simple estimator U 1 for A 2 2 has been already analyzed in the Gaussian sequence model. Still, we provide here a self-contained analysis as a warming-up for general even Schatten norms.…”

“…Still, we provide here a self-contained analysis as a warming-up for general even Schatten norms. Since Y 2 2 follows a non-central χ 2 distribution, we derive from standard computations for the normal random variables that E[ Y 2 2 ] = A 2 2 + pq. This leads us to the following unbiased estimator…”

Querying a Matrix through Matrix-Vector Products

“…We note that for non-adaptive queries, there is an Ω(n 2 ) sketching lower bound over the reals given in [19], and an Ω(n 2 / log p) lower bound for finite fields (of size p) in [3]. There is also a property testing lower bound in [6], though such a lower bound makes additional assumptions on the input. Thus, our model gives a new lens to study this problem from, from which we are able to derive strong lower bounds for adaptive queries.…”