稀疏最小二乘中的条件数障碍

中文摘要 AI 辅助

在[AS21]中，Axiotis和Sviridenko猜想，稀疏凸优化中对受限条件数的线性依赖无法被多项式时间算法改进。我们基于Raghavendra、Steurer和Tulsiani[RST12]提出的加权正则图形式的随机精确体积小集扩张假设，为最小二乘目标建立了他们猜想的下界。具体而言，对于每个固定的γ∈(0,1]，不存在随机多项式时间算法能以至少2/3的概率返回向量x，其中s=∥x∥₀，满足∥Ax−b∥₂²≤min_{∥z∥₀≤k}∥Az−b∥₂²+ε且s=O(kκ_{s+k}^{1−γ})，其中κᵣ是稀疏度水平r下的受限条件数。该结果甚至在A为满列秩的有理实例上也成立。证明最初使用Google内部开发的基于Gemini的全自动智能体系统获得，作者已验证该证明并编辑以提高表述清晰度。

英文摘要

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.

发表机构

• Google Research（谷歌研究院）
• Carnegie Mellon University（卡内基梅隆大学）
• Texas A&M University（德克萨斯农工大学）

机构由 AI 辅助整理，请以论文原文为准。