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Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop

Noah Amsel, Yves Baumann, Paul Beckman, Peter Bürgisser, Chris Camaño, Tyler Chen, Edmond Chow, Anil Damle, Michal Derezinski, Mark Embree, Ethan N. Epperly, Robert Falgout, Mark Fornace, Anne Greenbaum, Chen Greif, Diana Halikias, Zhen Huang, Elias Jarlebring, Yiannis Koutis, Daniel Kressner, Rasmus Kyng, Jörg Liesen, Jackie Lok, Raphael A. Meyer, Yuji Nakatsukasa, Kate Pearce, Richard Peng, David Persson, Eliza Rebrova, Ryan Schneider, Rikhav Shah, Edgar Solomonik, Nikhil Srivastava, Alex Townsend, Robert J. Webber, Jess Williams

TL;DR

The paper collects open questions from a Simons Institute workshop on Linear Systems and Eigenvalue Problems, spanning iterative solvers, eigenvalue computations, low-rank methods, sketching, and tensor/quantum problems. It highlights concrete formulations such as the $(n,k,l,\, ablaappa)$ spectrum-outlier model, MR$^3$ and its bidiagonal SVD challenges, and deterministic pseudospectral shattering to connect theory with practice. The document emphasizes the interplay between deterministic and randomized approaches, finite-precision considerations, and problem-specific preconditioning, aiming to advance both theory and scalable algorithms for large-scale linear systems and eigenvalue problems. Together, these questions illuminate pathways toward provable, efficient solvers in PDEs, data science, and quantum computation, with emphasis on robust performance under finite precision and structured matrix classes.

Abstract

This document presents a series of open questions arising in matrix computations, i.e., the numerical solution of linear algebra problems. It is a result of working groups at the workshop \emph{Linear Systems and Eigenvalue Problems}, which was organized at the Simons Institute for the Theory of Computing program on \emph{Complexity and Linear Algebra} in Fall 2025. The complexity and numerical solution of linear algebra problems %in matrix computations and related fields is a crosscutting area between theoretical computer science and numerical analysis. The value of the particular problem formulations here is that they were produced via discussions between researchers from both groups. The open questions are organized in five categories: iterative solvers for linear systems, eigenvalue computation, low-rank approximation, randomized sketching, and other areas including tensors, quantum systems, and matrix functions.

Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop

TL;DR

The paper collects open questions from a Simons Institute workshop on Linear Systems and Eigenvalue Problems, spanning iterative solvers, eigenvalue computations, low-rank methods, sketching, and tensor/quantum problems. It highlights concrete formulations such as the spectrum-outlier model, MR and its bidiagonal SVD challenges, and deterministic pseudospectral shattering to connect theory with practice. The document emphasizes the interplay between deterministic and randomized approaches, finite-precision considerations, and problem-specific preconditioning, aiming to advance both theory and scalable algorithms for large-scale linear systems and eigenvalue problems. Together, these questions illuminate pathways toward provable, efficient solvers in PDEs, data science, and quantum computation, with emphasis on robust performance under finite precision and structured matrix classes.

Abstract

This document presents a series of open questions arising in matrix computations, i.e., the numerical solution of linear algebra problems. It is a result of working groups at the workshop \emph{Linear Systems and Eigenvalue Problems}, which was organized at the Simons Institute for the Theory of Computing program on \emph{Complexity and Linear Algebra} in Fall 2025. The complexity and numerical solution of linear algebra problems %in matrix computations and related fields is a crosscutting area between theoretical computer science and numerical analysis. The value of the particular problem formulations here is that they were produced via discussions between researchers from both groups. The open questions are organized in five categories: iterative solvers for linear systems, eigenvalue computation, low-rank approximation, randomized sketching, and other areas including tensors, quantum systems, and matrix functions.
Paper Structure (35 sections, 1 theorem, 62 equations)

This paper contains 35 sections, 1 theorem, 62 equations.

Table of Contents

  1. Introduction
  2. Linear algebraic systems
  3. Scalable iterative solvers for a set of parameterized benchmark problems
  4. Correctness of multigrid methods beyond the standard model problems
  5. Conjugate gradient versus sketch-and-project
  6. The effect of outlying singular values (or eigenvalues)
  7. Preconditioning and convergence of iterative solvers
  8. Symmetric Krylov methods in finite precision arithmetic
  9. The Forsythe conjecture
  10. Eigenvalue problems
  11. Deterministic pseudospectral shattering / Minami bound
  12. Remarks.
  13. Accuracy vs. precision in efficient diagonalization algorithms
  14. Remarks.
  15. Ritz values and approximate invariant subspaces
  16. ...and 20 more sections

Key Result

Theorem 5.4

A traditional SRHT is a subspace embedding with $\alpha,\beta = \mathrm{const}$ with embedding dimension $\mathcal{O}((r + \log n) \log r)$. Moreover, there is an $r$-dimensional subspace $\mathcal{V} \subseteq \mathbb{R}^{r^2}$ such that a traditional SRHT is not a subspace injection if $k = o(r \l

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Randomized coordinate descent
  • Definition 2.4
  • Definition 2.5
  • Definition 5.1: Subspace embeddings and injections
  • Definition 5.2: SparseStack KN12
  • Definition 5.3: Subsampled randomized Hadamard transform
  • Theorem 5.4: Subsampled randomized Hadamard transform
  • Definition 6.1