Sparse high-dimensional linear regression. Estimating squared error and a phase transition
稀疏高维线性回归：平方误差估计与相变AI 解读 ↓
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稀疏高维线性回归：平方误差估计与相变
- 研究问题
- 在高信噪比和次线性稀疏条件下，从高斯线性观测恢复二元稀疏向量时，统计可恢复性与算法可行性之间存在怎样的相变？
- 方法与思路
- 作者在高斯设计与高斯噪声模型中，分析最大似然恢复、重叠间隙性质及其相变，并研究相关算法的成功与失败。
- 材料陈述的贡献
- 摘要报告最大似然恢复的全有或全无现象、重叠间隙性质的相变及若干算法结果；具体阈值条件因文本残缺无法完整确认。
- 阅读关注点
- 应先查原PDF核对缺失的不等式，再检查渐近制度、相变阈值，以及重叠间隙性质与算法困难之间论证的准确强度。
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- OpenAlex 索引摘要版本未注明，且多处不等式与句段残缺；不能重构阈值范围或 LASSO 成败条件，需查阅原PDF。
原文摘要
We consider a sparse high-dimensional regression model where the goal is to recover a k-sparse unknown binary vector β∗ from n noisy linear observations of the form Y=Xβ∗+W∈Rn where X∈Rn×p has i.i.d. N(0,1) entries and W∈Rn has i.i.d. N(0,σ2) entries. In the high signal-to-noise ratio regime and sublinear sparsity regime, while the order of the sample size needed to recover the unknown vector information-theoretically is known to be n∗:=2klogp/log(k/σ2+1), no polynomial-time algorithm is known to succeed unless n>nalg:=(2k+σ2)logp. In this work, we offer a series of results investigating multiple computational and statistical aspects of the recovery task in the regime n∈[n∗,nalg]. First, we establish a novel information-theoretic property of the MLE of the problem happening around n=n∗ samples, which we coin as an “all-or-nothing behavior”: when n>n∗ it recovers almost perfectly the support of β∗, while if n 0 when n Cnalg OGP disappears. OGP is a geometric “disconnectivity” property, which initially appeared in the theory of spin glasses and is known to suggest algorithmic hardness when it occurs. Finally, using certain technical results obtained to establish the OGP phase transition, we additionally establish various novel positive and negative algorithmic results for the recovery task of interest, including the failure of LASSO with access to n Cnalg samples.