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Motivated by the least squares solver Blendenpik, we investigate threestrategies for uniform sampling of rows from m x n matrices Q with orthonormalcolumns. The goal is to determine, with high probability, how many rows arerequired so that the sampled matrices have full rank and are well-conditionedwith respect to inversion.  Extensive numerical experiments illustrate that the three sampling strategies(without replacement, with replacement, and Bernoulli sampling) behave almostidentically, for small to moderate amounts of sampling. In particular, sampledmatrices of full rank tend to have two-norm condition numbers of at most 10.  We derive a bound on the condition number of the sampled matrices in terms ofthe coherence {\mu} of Q. This bound applies to all three different samplingstrategies; it implies a, not necessarily tight, lower bound of O(m {\mu}ln(n)) for the number of sampled rows; and it is realistic and informative evenfor matrices of small dimension and the stringent requirement of a 99 percentsuccess probability.  For uniform sampling with replacement we derive a potentially tightercondition number bound in terms of the leverage scores of Q. To obtain a moreeasily computable version of this bound, in terms of just the largest leveragescores, we first derive a general bound on the two-norm of diagonally scaledmatrices.  To facilitate the numerical experiments and test the tightness of the bounds,we present algorithms to generate matrices with user-specified coherence andleverage scores. These algorithms, the three sampling strategies, and a largevariety of condition number bounds are implemented in the Matlab toolboxkappa_SQ_v3.

The Effect of Coherence on Sampling from Matrices with Orthonormal Columns, and Preconditioned Least Squares Problems