大型噪声矩阵子矩阵局部化的计算和统计有界性
Computational and Statistical Boundaries for Submatrix Localization in a Large Noisy Matrix
摘要：
在高维推理中，计算效率和统计精度之间的相互作用引起了越来越多的关注。在本文中，我们研究了计算和统计边界的子矩阵本地化。给定被噪声矩阵（大小为$m \times n$）污染的（一个或多个非重叠的）信号子矩阵（大小为$λ$和大小为$k_m \times k_n$）的一个观测，我们根据$m$、$n$、$k_m$和$k_n$建立信噪比$λ/σ$的两个转换阈值。第一阈值$\sf SNR_c$对应于计算边界。低于这个阈值，它表明，没有多项式时间算法可以成功地识别的子矩阵，在\textit{hidden clique hypothesis}。我们介绍了自适应线性时间谱算法，识别子矩阵的概率高时，信号强度高于阈值$\sf SNR_c$。第二个阈值，$\sf SNR_s$，捕获统计边界，低于该边界，在极大极小意义下，没有方法可以成功地概率为1。穷举搜索方法成功地找到了这个阈值以上的子矩阵。结果表明，一个有趣的现象，$\sf SNR_c$总是显着大于$\sf SNR_s$，这意味着统计最优性和计算效率的子矩阵本地化之间的本质差距。
英文摘要：
The interplay between computational efficiency and statistical accuracy in high-dimensional inference has drawn increasing attention in the literature. In this paper, we study computational and statistical boundaries for submatrix localization. Given one observation of (one or multiple non-overlapping) signal submatrix (of magnitude $λ$ and size $k_m \times k_n$) contaminated with a noise matrix (of size $m \times n$), we establish two transition thresholds for the signal to noise $λ/σ$ ratio in terms of $m$, $n$, $k_m$, and $k_n$. The first threshold, $\sf SNR_c$, corresponds to the computational boundary. Below this threshold, it is shown that no polynomial time algorithm can succeed in identifying the submatrix, under the \textit{hidden clique hypothesis}. We introduce adaptive linear time spectral algorithms that identify the submatrix with high probability when the signal strength is above the threshold $\sf SNR_c$. The second threshold, $\sf SNR_s$, captures the statistical boundary, below which no method can succeed with probability going to one in the minimax sense. The exhaustive search method successfully finds the submatrix above this threshold. The results show an interesting phenomenon that $\sf SNR_c$ is always significantly larger than $\sf SNR_s$, which implies an essential gap between statistical optimality and computational efficiency for submatrix localization.
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