超出高斯宽度的受限特征值：重尾分布下的阈值占据

中文摘要 AI 辅助

受限特征值（RE）界控制着范数正则化估计器的稳定恢复。对于各向同性次高斯测量，基准样本量为 $1+w(A)^2$，其中 $w(A)$ 是归一化下降锥的高斯宽度。COLT 2015 开放问题笔记（Banerjee 等人，2015）提出，仅基于均匀小球条件，重尾设计是否遵循相同规律。我们对该一般问题给出明确且系统的否定答案：所提出的规律在其全维度自由、任意集合形式下不成立，缺失的阻碍是同时发生的阈值占据。具有固定小球常数的常宽度多面体下降锥，在达到环境维度一半的每条样本路径上都具有零经验 RE。更一般地，每个有限范围空间都能在任意窄的球冠中实现精确阈值编码，并提升为完整多面体下降锥截面。对于每个固定阈值 VC 维 $d$，当 $\beta\to0$ 时，最坏情况样本复杂度的精确值为 $\beta^{-1}[d\text{log}(1/\beta)+\text{log}(1/\delta)]$。在精确各向同性和所有有限矩下，这种差异依然存在：在同一常宽度锥上，高斯测量成功需要 $O(1+\text{log}(1/\delta))$ 个样本，而各向同性重尾设计在 $n\lesssim\sqrt{p/\text{log}p}$ 时逐路径失败。高斯平滑产生处处为正的 $C^\infty$ 密度，同时保留任意差的 RE。在各向同性下，由仿射维度乘以平方包围半径控制的无分布 fallback 在该族上是精确的。

英文摘要

Restricted eigenvalue (RE) bounds govern stable recovery by norm-regularized estimators. For isotropic sub-Gaussian measurements, the benchmark sample size is $1+w(A)^2$, where $w(A)$ is the Gaussian width of the normalized descent cone. The COLT 2015 open-problem note (Banerjee et al., 2015) asked whether the same law follows for heavy-tailed designs from a uniform small-ball condition alone. We give an explicit and systematic negative answer to the general question as formulated there: the proposed law fails in its full dimension-free, arbitrary-set form, and the missing obstruction is simultaneous threshold occupancy. A constant-width polyhedral descent cone with fixed small-ball constants has zero empirical RE on every sample path up to half the ambient dimension. More generally, every finite range space admits exact threshold encoding in an arbitrarily narrow spherical cap and a lift to a full polyhedral descent-cone section. For every fixed threshold VC dimension $d$, as $β\downarrow0$, the sharp worst-case sample complexity is $Θ(β^{-1}[d\log(1/β)+\log(1/δ)])$. The separation persists under exact isotropy and all finite moments: on the same constant-width cone, Gaussian measurements succeed with $O(1+\log(1/δ))$ samples, whereas an isotropic heavy-tailed design fails pathwise for $n\lesssim\sqrt{p/\log p}$. Gaussian smoothing yields an everywhere-positive $C^\infty$ density while retaining arbitrarily poor RE. Under isotropy, a distribution-free fallback governed by affine dimension times squared enclosing radius is sharp on this family.

发表机构

• Nanyang Technological University（南洋理工大学）

机构由 AI 辅助整理，请以论文原文为准。