We show that for the problem of testing if a matrix @math has rank at most @math , or requires changing an @math -fraction of entries to have rank at most @math , there is a non-adaptive query algorithm making @math queries. Our algorithm works for any field @math . This improves upon the previous @math bound (SODA'03), and bypasses an @math lower bound of (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 @math . We complement our algorithm with a matching query complexity lower bound for non-adaptive testers over any field. We also give tight bounds of @math queries in the sensing model for which query access comes in the form of @math ; perhaps surprisingly these bounds do not depend on @math . We next develop a novel property testing framework for testing numerical properties of a real-valued matrix @math more generally, which includes the stable rank, Schatten- @math norms, and SVD entropy. Specifically, we propose a bounded entry model, where @math is required to have entries bounded by @math in absolute value. We give upper and lower bounds for a wide range of problems in this model, and discuss connections to the sensing model above.
In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix concentration inequalities, research has advanced to the point where we can conquer many (formerly) challenging problems with a page or two of arithmetic. The aim of this monograph is to describe the most successful methods from this area along with some interesting examples that these techniques can illuminate. As the Netflix Prize competition has demonstrated, matrix factorization models are superior to classic nearest neighbor techniques for producing product recommendations, allowing the incorporation of additional information such as implicit feedback, temporal effects, and confidence levels.