Faster Solves for Random Dense Linear Systems

Determine whether, with high probability over A, there exists an algorithm that solves Ax = b to high accuracy faster than O(n^ω) arithmetic operations when A is an n×n matrix with i.i.d. Gaussian (or Rademacher, or Bernoulli) entries; equivalently, establish superlinear speedups over classical dense direct methods (ω=3) for random inputs.

Background

Random dense matrices typically have linearly decaying singular values, so removing a small number of smallest singular values substantially improves conditioning; this suggests the possibility of faster algorithms than exact dense solves.

Despite progress on iterative and sketched methods, a general algorithm that beats O(nω) for random dense systems is not known.

References

Let $A$ be an $n\times n$ random matrix with i.i.d. Gaussian (or Rademacher, or Bernoulli) entries, and take any $n$-dimensional vector $b$. Is it possible (conditioned on a high-probability event) to solve the linear system $Ax=b$ to high accuracy faster than $O(n\omega)$? Note that this question is open (informally speaking) even if we restrict to classical matrix multiplication and let $\omega=3$.

— Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop  (2602.05394 - Amsel et al., 5 Feb 2026) in Subsection "The effect of outlying singular values (or eigenvalues)" (Section 2)