Can matrix coherence be efficiently and accurately estimated?

Matrix coherence has recently been used to characterize the ability to extract global information from a subset of matrix entries in the context of low-rank approximations and other sampling-based algorithms. The significance of these results crucially hinges upon the possibility of efficiently and accurately testing this coherence assumption. This paper precisely addresses this issue. We introduce a novel sampling-based algorithm for estimating coherence, present associated estimation guarantees and report the results of extensive experiments for coherence estimation. The quality of the estimation guarantees we present depends on the coherence value to estimate itself, but this turns out to be an inherent property of sampling-based coherence estimation, as shown by our lower bound. In practice, however, we find that these theoretically unfavorable scenarios rarely appear, as our algorithm efficiently and accurately estimates coherence across a wide range of datasets, and these estimates are excellent predictors of the effectiveness of sampling-based matrix approximation on a case-by-case basis. These results are significant as they reveal the extent to which coherence assumptions made in a number of recent machine learning publications are testable.