主子矩阵恢复中的重叠间隙性质
The Overlap Gap Property in Principal Submatrix Recovery
AI总结：
研究高斯矩阵中主子矩阵恢复的统计与计算极限，刻画MLE的相变阈值，证明似然景观的OGP性质导致局部MCMC算法非最优，并给出谱方法的恢复条件。
摘要：
我们研究了在 $N\times N$ 对称零均值高斯矩阵中隐藏的、具有升高均值 $\lambda/N$ 的 $k\times k$ 主子矩阵的支持恢复问题。这里 $\lambda>0$ 是一个普适常数，且我们假设 $k = N \rho$，其中 $\rho\in (0,1)$ 为常数。我们证明：存在常数 $C>0$，使得当 $\lambda \geq C \sqrt{\frac{1}{\rho} \log \frac{1}{\rho}}$ 时，极大似然估计（MLE）能恢复隐藏子矩阵的常数比例；而当 $\lambda = o\left( \sqrt{\frac{1}{\rho} \log \frac{1}{\rho}} \right)$ 时，这种恢复在信息论上是不可能的。MLE 通常计算上不可行，事实上，对于足够小的 $\rho>0$，该问题被推测存在“统计-计算间隙”。为提供严格证据，我们研究了该问题的似然景观，并证明：对于某个 $\varepsilon>0$ 且 $\sqrt{\frac{1}{\rho} \log \frac{1}{\rho}} \ll \lambda \ll \frac{1}{\rho^{1/2 + \varepsilon}}$，该问题表现出“重叠间隙性质（OGP）”的一种变体。作为直接推论，我们证明一族基于局部马尔可夫链蒙特卡洛（MCMC）的算法无法实现最优恢复。最后，我们证明当 $\lambda > 1/\rho$ 时，一种简单的谱方法能恢复隐藏子矩阵的常数比例。
英文摘要：
We study support recovery for a $k \times k$ principal submatrix with elevated mean $λ/N$, hidden in an $N\times N$ symmetric mean zero Gaussian matrix. Here $λ>0$ is a universal constant, and we assume $k = N ρ$ for some constant $ρ\in (0,1)$. We establish that {there exists a constant $C>0$ such that} the MLE recovers a constant proportion of the hidden submatrix if $λ{\geq C} \sqrt{\frac{1}ρ \log \frac{1}ρ}$, {while such recovery is information theoretically impossible if $λ= o( \sqrt{\frac{1}ρ \log \frac{1}ρ} )$}. The MLE is computationally intractable in general, and in fact, for $ρ>0$ sufficiently small, this problem is conjectured to exhibit a \emph{statistical-computational gap}. To provide rigorous evidence for this, we study the likelihood landscape for this problem, and establish that for some $\varepsilon>0$ and $\sqrt{\frac{1}ρ \log \frac{1}ρ } \ll λ\ll \frac{1}{ρ^{1/2 + \varepsilon}}$, the problem exhibits a variant of the \emph{Overlap-Gap-Property (OGP)}. As a direct consequence, we establish that a family of local MCMC based algorithms do not achieve optimal recovery. Finally, we establish that for $λ> 1/ρ$, a simple spectral method recovers a constant proportion of the hidden submatrix.
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