Topic: Low-rank and structured matrices

Robert Webber connects partial pivoted Cholesky, effective for nearly low-rank matrices, with Vecchia approximation, effective when inverse Cholesky factors are nearly sparse. Combining a partial Cholesky approximation with a Vecchia approximation of its residual produces another Vecchia approximation of the original matrix with an enlarged sparsity pattern. This unifies several factored matrix-approximation approaches and explains the broader applicability of the Vecchia framework.

Alice Cortinovis presents Adaptive Randomized Pivoting for selecting representative matrix columns through adaptive leverage-score sampling. Its expected Frobenius approximation error matches the optimal existence guarantee. The method is a randomized counterpart to an approach by Osinsky and offers a simpler, less costly alternative to volume sampling with the same theoretical guarantee. The talk extends the strategy to the Discrete Empirical Interpolation Method, cross or skeleton approximation, and Nyström approximation of positive-semidefinite matrices.

Chris Musco studies how to approximate an unknown matrix by a structured one using a limited, adaptively chosen sequence of matrix-vector products. This models operator learning in scientific machine learning as well as computational algorithms. Randomized SVD provides strong guarantees for low-rank targets; analogous results for sparse and hierarchical structures are less developed. The talk presents progress on efficient algorithms for these classes and a broader complexity theory. Joint work with Noah Amsel, Pratyush Avi, Tyler Chen, Prathamesh Dharangutte, Chinmay Hegde, Feyza Duman Keles,

Sherry Xiaoye Li surveys randomized construction of hierarchically low-rank matrices, including H/H2, HODLR, HSS, and butterfly formats with different off-diagonal structures. Applications include integral equations, boundary elements, discretized PDEs, and statistical or machine-learning kernel matrices, using either iterative matrix-vector products or direct factorization and solves. Constructing these representations from an implicit dense operator is often the main cost. The talk offers a unified view of sketch distributions, sketch sizes, approximation error bounds, high-performance imple

Per-Gunnar Martinsson presents randomized black-box algorithms that compress rank-structured matrices, including H-matrices and HSS matrices, into data-sparse representations. Access is through matrix-vector products, which suits Schur-complement construction and matrix multiplication. When both the operator and its transpose admit O(N) application, the overall compression can also have linear complexity. A featured method combines compression and factorization of an H-matrix under strong admissibility.

Yuji Nakatsukasa explains how CUR decompositions approximate a matrix using selected columns and rows, without inspecting every entry once the indices are chosen. Near-optimal CUR approximations exist relative to the truncated singular value decomposition, and efficient algorithms make them useful for large problems. The talk covers computation and theoretical guarantees before exploring applications to approximation theory, model reduction, and parameter-dependent problems.