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arXiv · 2608.02588

The Condition-Number Barrier in Sparse Least Squares

Honghao Lin·Vahab Mirrokni·David P. Woodruff
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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 $\gamma\in(0,1]$, there is no randomized polynomial-time algorithm that, with probability at least $2/3$, returns a vector $x$ such that, writing $s=\lVert x\rVert_0$, \[ \lVert Ax-b\rVert_2^2 \leq \min_{\lVert z\rVert_0\leq k}\lVert Az-b\rVert_2^2+\varepsilon \quad\text{and}\quad s=O\!\left(k\,\kappa_{s+k}^{\,1-\gamma}\right), \] where $\kappa_r$ 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.

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Honghao Lin, Vahab Mirrokni, David P. Woodruff. 2026-08-03. The Condition-Number Barrier in Sparse Least Squares. https://arxiv.org/abs/2608.02588

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