Randomized least squares is a promising method but not yet widely used in practice. We show an example of its use for finding low-rank canonical polyadic (CP) tensor decompositions for large sparse tensors. This involves solving a sequence of overdetermined least problems with special (Khatri-Rao product) structure.

In this work, we present an application of randomized algorithms to fitting the CP decomposition of sparse tensors, solving a significantly smaller sampled least squares problem at each iteration with probabilistic guarantees on the approximation errors. We perform sketching through leverage score sampling, crucially relying on the fact that the problem structure enable efficient sampling from overestimates of the leverage scores with much less work. We discuss what it took to make the algorithm practical, including general-purpose improvements.

Numerical results on real-world large-scale tensors show the method is faster than competing methods without sacrificing accuracy.

*This is joint work with Brett Larsen, Stanford University.

This work considers weighted and preconditioned GMRES. The objective is to provide a way of choosing the preconditioner and the inner product, also called weight, that ensure fast convergence. The main focus of the article is on Hermitian preconditioning (even for non-Hermitian problems).

It is indeed proposed to choose a Hermitian preconditioner H, and to apply GMRES in the inner product induced by H. If moreover, the problem matrix A is positive definite, then a new convergence bound is proved that depends only on how well H preconditions the Hermitian part of A, and on a measure of how non-Hermitian A is. In particular, if a scalable preconditioner is known for the Hermitian part of A, then the proposed method is also scalable. I will also illustrate this result numerically.