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Small-Sample Statistical Estimates for Matrix Norms

This paper extends a recent statistically based vector-norm estimator to matrices. The new estimator requires only a few matrix-vector multiplications and can be applied when the matrix is not known explicitly. It is useful for efficiently estimating the sensitivity of vector-valued functions and can be applied to many problems where the power method runs into difficulties. Lower bounds for the probability that an estimate is within a given factor of the correct norm are derived. These bounds are straightforward to compute and show that a very inaccurate estimate is extremely unlikely in most cases. A conservative lower bound has been derived and a tighter bound is given in the form of a conjecture. This conjecture is true in some important special cases and the general case is supported by considerable empirical evidence.