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Norm and trace estimation with random rank-one vectors

A few matrix-vector multiplications with random vectors are often sufficient to obtain reasonably good estimates for the norm of a general matrix or the trace of a symmetric positive semi-definite matrix. Several such probabilistic estimators have been proposed and analyzed for standard Gaussian and Rademacher random vectors. In this talk, we discuss the use of rank-one random vectors, that is, Kronecker products of (smaller) Gaussian or Rademacher vectors. It is not only cheaper to sample such vectors but it can sometimes also be much cheaper to multiply a matrix with a rank-one vector instead of a general vector. We provide theoretical and numerical evidence that the use of rank-one instead of unstructured random vectors still leads to good estimates. In particular, we show that our rank-one estimators multiplied with a modest constant constitute, with high probability, both upper and lower bounds of the quantity of interest. We illustrate the application of our techniques to condition number estimation for matrix functions.

matrix norm ; matrix trace ; random vector ; probabilistic estimator ; Kronecker product

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