The Condition-Number Barrier in Sparse Least Squares

2026-08-03

AI Summary

This work investigates whether the linear dependence on the restricted condition number κ in sparse least-squares optimization can be overcome in polynomial time. Under the randomized exact Small-Set Expansion hypothesis, the authors establish—via a complexity-theoretic reduction coupled with a refined analysis of the restricted condition number—the first proof of the Axiotis–Sviridenko conjecture in this setting: for any fixed γ ∈ (0,1], no polynomial-time randomized algorithm can output, with probability exceeding 2/3, a solution of sparsity O(k·κ^{1−γ}) that meets the target error, even when the input instance consists of rational entries and has full column rank. A key component of the argument was derived with the aid of an AI-driven automated theorem-proving system, thereby establishing a theoretical lower bound on the condition number barrier.

Abstract

In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm. We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis in the weighted regular-graph formulation of Raghavendra, Steurer, and Tulsiani [RST12]. Concretely, for every fixed $γ\in(0,1]$, there is no randomized polynomial-time algorithm that, with probability at least $2/3$, returns a vector $x$ such that, writing $s=\lVert x\rVert_0$, \[ \lVert Ax-b\rVert_2^2 \leq \min_{\lVert z\rVert_0\leq k}\lVert Az-b\rVert_2^2+\varepsilon \quad\text{and}\quad s=O\!\left(k\,κ_{s+k}^{\,1-γ}\right), \] where $κ_r$ is the restricted condition number at sparsity level $r$. The result holds even on rational instances with $A$ of full column rank. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.

Problem

• sparse least squares
• restricted condition number
• polynomial-time algorithm
• lower bound
• Small-Set Expansion Hypothesis

Innovation

• sparse least squares
• restricted condition number
• computational lower bound
• Small-Set Expansion Hypothesis
• AI-assisted proof