小球边际不通过欧几里得高斯宽度控制受限特征值
Small-Ball Marginals Do Not Control Restricted Eigenvalues by Euclidean Gaussian Width
- University of California, San Diego（加利福尼亚大学圣迭戈分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
通过构造重尾分布反例，证明均匀小球条件无法由欧几里得高斯宽度控制受限特征值下界，否定COLT 2015问题的无分布形式。
AI中文摘要：
Banerjee、Chen和Sivakumar在COLT 2015会议上提出一个问题：随机设计矩阵行的均匀小球条件是否强制一个受限特征值下界，其样本复杂度由任意球面子集的普通欧几里得高斯宽度控制。我们对该问题的自然无分布形式给出否定答案。对于每个样本量$n$，我们构造一个中心化、真正重尾的行分布，维度$p=4^n+1$，以及一个集合$A=C\cap\mathbb{S}^{p-1}$，其中$C$是闭多面体凸锥，使得\\[ \inf_{v\ne 0}\mathbb{P}\\!\left( \left|\left\langle Z,v\right\rangle\right| \ge \frac{\left\lVert v\right\rVert_2}{\sqrt{2}} \right)\ge \frac{1}{12} \quad\text{且}\quad w(A)<2. \\] 然而，对于以$Z$的$n$个独立副本为行的矩阵$X$，\\[ \mathbb{P}\\!\left( \inf_{u\in A}\left\lVert Xu\right\rVert_2^2=0 \right) \ge 1-\exp(-2^n). \\] 因此，仅依赖于固定小球常数的正常数不能以高概率产生所提议形式$c_1n-c_2w(A)^2$的下界。该构造隔离了障碍：边际小球下界控制每个固定方向，但不控制搜索许多方向时依赖于分布的复杂度。我们明确陈述量词，并讨论为何各向同性或上尾假设会导致一个不同但仍然有意义的问题。
英文摘要：
Banerjee, Chen, and Sivakumar asked at COLT 2015 whether a uniform small-ball condition on the rows of a random design matrix forces a restricted-eigenvalue lower bound whose sample complexity is governed by the ordinary Euclidean Gaussian width of an arbitrary spherical subset. We give a negative answer to the natural distribution-free formulation of that question. For every sample size $n$, we construct a centered, genuinely heavy-tailed row distribution in dimension $p=4^n+1$ and a set $A=C\cap\mathbb{S}^{p-1}$, where $C$ is a closed polyhedral convex cone, such that \[ \inf_{v\ne 0}\mathbb{P}\!\left( \left|\left\langle Z,v\right\rangle\right| \ge \frac{\left\lVert v\right\rVert_2}{\sqrt{2}} \right)\ge \frac{1}{12} \quad\text{and}\quad w(A)<2. \] Nevertheless, for the matrix $X$ with $n$ independent copies of $Z$ as rows, \[ \mathbb{P}\!\left( \inf_{u\in A}\left\lVert Xu\right\rVert_2^2=0 \right) \ge 1-\exp(-2^n). \] Thus no positive constants depending only on the fixed small-ball parameters can yield a lower bound of the proposed form $c_1n-c_2w(A)^2$ with high probability. The construction isolates the obstruction: a marginal small-ball lower bound controls every fixed direction, but does not control the distribution-dependent complexity of searching over many directions. We state the quantifiers explicitly and discuss why isotropic or upper-tail assumptions lead to a different, still meaningful problem.
