Testing Matrix Rank, Optimally
This work addresses efficient matrix property testing for computational mathematics and data analysis, offering optimal algorithms that are incremental improvements over prior methods.
The paper tackles the problem of testing whether a matrix has rank at most d or is far from it, achieving a non-adaptive query algorithm with O~(d^2/ε) queries, improving upon the previous O(d^2/ε^2) bound and bypassing a lower bound for submatrix-based algorithms. It also provides matching lower bounds and extends the framework to test other matrix properties like stable rank and Schatten norms with tight bounds.
We show that for the problem of testing if a matrix $A \in F^{n \times n}$ 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 $\widetilde{O}(d^2/ε)$ queries. Our algorithm works for any field $F$. This improves upon the previous $O(d^2/ε^2)$ bound (SODA'03), and bypasses an $Ω(d^2/ε^2)$ 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 $A$. We complement our algorithm with a matching query complexity lower bound for non-adaptive testers over any field. We also give tight bounds of $\widetildeΘ(d^2)$ queries in the sensing model for which query access comes in the form of $\langle X_i, A\rangle:=tr(X_i^\top A)$; perhaps surprisingly these bounds do not depend on $ε$. 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. We give upper and lower bounds for a wide range of problems in this model, and discuss connections to the sensing model above.