@article{holodnak_ipsen_smith_2018, title={A PROBABILISTIC SUBSPACE BOUND WITH APPLICATION TO ACTIVE SUBSPACES}, volume={39}, ISSN={["1095-7162"]}, url={https://doi.org/10.1137/17M1141503}, DOI={10.1137/17M1141503}, abstractNote={Given a real symmetric positive semi-definite matrix E, and an approximation S that is a sum of n independent matrix-valued random variables, we present bounds on the relative error in S due to randomization. The bounds do not depend on the matrix dimensions but only on the numerical rank (intrinsic dimension) of E. Our approach resembles the low-rank approximation of kernel matrices from random features, but our accuracy measures are more stringent. 
In the context of parameter selection based on active subspaces, where S is computed via Monte Carlo sampling, we present a bound on the number of samples so that with high probability the angle between the dominant subspaces of E and S is less than a user-specified tolerance. This is a substantial improvement over existing work, as it is a non-asymptotic and fully explicit bound on the sampling amount n, and it allows the user to tune the success probability. It also suggests that Monte Carlo sampling can be efficient in the presence of many parameters, as long as the underlying function f is sufficiently smooth.}, number={3}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Holodnak, John T. and Ipsen, Ilse C. F. and Smith, Ralph C.}, year={2018}, pages={1208–1220} }
 @article{holodnak_ipsen_wentworth_2015, title={CONDITIONING OF LEVERAGE SCORES AND COMPUTATION BY QR DECOMPOSITION}, volume={36}, ISSN={["1095-7162"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84944583307&partnerID=MN8TOARS}, DOI={10.1137/140988541}, abstractNote={The leverage scores of a full-column rank matrix A are the squared row norms of any orthonormal basis for range(A). We show that corresponding leverage scores of two matrices A and A + \Delta A are close in the relative sense, if they have large magnitude and if all principal angles between the column spaces of A and A + \Delta A are small. We also show three classes of bounds that are based on perturbation results of QR decompositions. They demonstrate that relative differences between individual leverage scores strongly depend on the particular type of perturbation \Delta A. The bounds imply that the relative accuracy of an individual leverage score depends on: its magnitude and the two-norm condition of A, if \Delta A is a general perturbation; the two-norm condition number of A, if \Delta A is a perturbation with the same norm-wise row-scaling as A; (to first order) neither condition number nor leverage score magnitude, if \Delta A is a component-wise row-scaled perturbation. Numerical experiments confirm the qualitative and quantitative accuracy of our bounds.}, number={3}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Holodnak, John T. and Ipsen, Ilse C. F. and Wentworth, Thomas}, year={2015}, pages={1143–1163} }
 @article{holodnak_ipsen_2015, title={RANDOMIZED APPROXIMATION OF THE GRAM MATRIX: EXACT COMPUTATION AND PROBABILISTIC BOUNDS}, volume={36}, ISSN={["1095-7162"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84925298413&partnerID=MN8TOARS}, DOI={10.1137/130940116}, abstractNote={Given a real matrix A with n columns, the problem is to approximate the Gram product AA^T by c = rank(A) columns depend on the right singular vector matrix of A. For a Monte-Carlo matrix multiplication algorithm by Drineas et al. that samples outer products, we present probabilistic bounds for the 2-norm relative error due to randomization. The bounds depend on the stable rank or the rank of A, but not on the matrix dimensions. Numerical experiments illustrate that the bounds are informative, even for stringent success probabilities and matrices of small dimension. We also derive bounds for the smallest singular value and the condition number of matrices obtained by sampling rows from orthonormal matrices.}, number={1}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Holodnak, John T. and Ipsen, Ilse C. F.}, year={2015}, pages={110–137} }