Invited Speakers

Schatten norms extend familiar matrix norms such as the Frobenius, spectral, and nuclear norms, but their computation typically requires a full singular value decomposition and is therefore too expensive for large-scale matrices. I will discuss a randomized estimator for Schatten-2p norms, based on an approach by Kong and Valiant. The focus will be on improved variance bounds that provide a more accurate characterization of the estimator’s behavior for moderate sketch sizes and values of p, for matrices which are numerically low-rank. I will also compare the estimator with the Girard-Hutchinson's method, highlighting the trade-offs in terms of variance and matrix access. Numerical experiments illustrate how the theory reflects practice and indicate scenarios where these estimators are useful.

The growing availability of lower precision arithmetics has led to the development of mixed precision algorithms in numerical linear algebra. Most recently we have seen the emergence of AI-specialized hardware such as GPU accelerators implementing extremely fast matrix multiply--accumulate operations, but using extremely low precisions such as 16-, 8- or even 4-bit floating-point or integer formats. This motivates a new class of mixed precision algorithms aiming to emulate high accuracy with low precision computations. In this talk, we will review the dense and rapidly growing literature on precision emulation. We will notably compare two competing classes of methods respectively based on multiword and multimodular decompositions, and discuss in which situation each method is preferable.

Low-rank approximation methods for matrices are based on projecting the columns (or rows) to suitable low-dimensional subspaces. Taking subspaces spanned by individual columns or by random linear combinations of all columns are common alternatives to the computation of optimal subspaces via the SVD, especially for large or implicitly given matrices. Well-known results on volume sampling or randomized SVD show that such approaches indeed achieve quasi-optimal approximation errors in expectation. In this talk, we discuss generalizations of such results to low-rank approximation of Hilbert-Schmidt operators between infinite-dimensional Hilbert spaces. In the first part, we consider the approximation of vector valued L2 functions in subspaces spanned by point samples, and show existence of quasi-optimal sample points based on a continuous version of volume sampling. In the second part, we discuss infinite-dimensional extensions of the randomized SVD as recently proposed by Boullé and Townsend, for which we present an alternative approach. This also includes a novel extension of the Nyström approximation for self-adjoint positive semi-definite trace class operators. Based on joint work with D. Kressner, T. Ni, and D. Persson.