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 multi plications and can be applied when the matrix is not known explicitly. It is useful for efficiently estimating the sensitivity of vector-val ued functions and can be applied to many problems where the power meth od runs into difficulties. Lower bounds for the probability that an es timate 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 bo und has been derived and a tighter bound is given in the form of a con jecture. This conjecture is true in some important special cases and t he general case is supported by considerable empirical evidence.