有界密度次高斯项的随机行子矩阵的极小奇异值
Extreme least singular values of random row submatrices with bounded-density subgaussian entries
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中文总结 AI 辅助
该研究确定了有界密度次高斯项随机行子矩阵极小奇异值的指数尺度，推广了实高斯结果，结合多种概率估计方法得到偏差概率界，在相位检索阈值处给出稳定性参数的指数基。
中文摘要 AI 辅助
设ξ为中心化实次高斯随机变量，具有正方差和有界勒贝格密度，令Aₘ∈ℝ^{Nₘ×m}的独立项服从ξ分布，其中Nₘ/m→γ>1。对每个满足|I|=m的集合I⊂[Nₘ]，记(Aₘ)ᵢ为以I为索引的行子矩阵，定义Mₘ(Aₘ):=min_{I⊂[Nₘ],|I|=m}σ_min((Aₘ)ᵢ)。我们确定其指数尺度：(1/m)log Mₘ(Aₘ)→^ℙ -h(γ)，其中h(γ):=γlogγ−(γ−1)log(γ−1)。这推广了相应的实高斯结果。主要新要素是避免对指数多的随机超平面进行一致控制的上尾论证，我们结合了非局域方向的密度级局部中心极限定理、超平面法向量的平均非局域估计、近平行对的指数界，以及利用线性数目的独立探测行的放大。对每个固定的ε∈(0,h(γ))，当m足够大时，ε偏差的概率至多为Cexp(-c√m)。在由单个无限独立同分布数组诱导的典范耦合下，该可和偏差估计在每个宽高比的紧范围上产生一致几乎必然指数律。特别地，在实相位检索阈值Nₘ=2m−1处，Balan–Wang稳定性参数在概率意义下具有指数基1/4，且在该耦合下几乎必然成立。
英文摘要
Let $ξ$ be a centered real subgaussian random variable with positive variance and a bounded Lebesgue density, and let $A_m\in\mathbb{R}^{N_m\times m}$ have independent entries distributed as $ξ$, where $N_m/m\toγ>1$. For each set $I\subset[N_m]$ with $|I|=m$, let $(A_m)_I$ denote the row submatrix indexed by $I$, and define $M_m(A_m):=\min_{I\subset[N_m],\,|I|=m}σ_{\min}((A_m)_I)$. We determine its exponential scale: $\frac{1}{m}\log M_m(A_m)\xrightarrow{\mathbb{P}}-h(γ)$, where $h(γ):=γ\logγ-(γ-1)\log(γ-1)$. This extends the corresponding real Gaussian result. The main new ingredient is an upper-tail argument that avoids uniform control over exponentially many random hyperplanes. We combine a density-level local central limit theorem for delocalized directions, an averaged delocalization estimate for hyperplane normals, an exponential bound for nearly parallel pairs, and amplification using a linear number of independent probe rows. For every fixed $\varepsilon\in(0,h(γ))$, the probability of an $\varepsilon$-deviation is at most $C\exp(-c\sqrt{m})$ for all sufficiently large $m$. Under the canonical coupling induced by a single infinite i.i.d. array, this summable deviation estimate yields a uniform almost-sure exponential law over every compact range of aspect ratios. In particular, at the real phase-retrieval threshold $N_m=2m-1$, the Balan--Wang stability parameter has exponential base $1/4$ in probability and, under this coupling, almost surely.
