Fast deterministic and randomized algorithms for low-rank approximation, matrix functions, and trace estimation

In this thesis we propose and analyze algorithms for some numerical linear algebra tasks: finding low-rank approximations of matrices, computing matrix functions, and estimating the trace of matrices.

In the first part, we consider algorithms for building low-rank approximations of a matrix from some rows or columns of the matrix itself. We prove a priori error bounds for a greedy algorithm for cross approximation and we develop a faster and more efficient variant of an existing algorithm for column subset selection. Moreover, we present a new deterministic polynomial-time algorithm that gives a cross approximation which is quasi-optimal in the Frobenius norm.

The second part of the thesis is concerned with matrix functions. We develop a divide-and-conquer algorithm for computing functions of matrices that are banded, hierarchically semiseparable, or have some other off-diagonal low-rank structure. An important building block of our approach is an existing algorithm for updating the function of a matrix that undergoes a low-rank modification (update), for which we present new convergence results. The convergence analysis of our divide-and-conquer algorithm is related to polynomial or rational approximation of the function.

In the third part we consider the problem of approximating the trace of a matrix which is available indirectly, through matrix-vector multiplications. We analyze a stochastic algorithm, the Hutchinson trace estimator, for which we prove tail bounds for symmetric (indefinite) matrices. Then we apply our results to the computation of the (log)determinants of symmetric positive definite matrices.