Computational Barriers in Minimax Submatrix Detection
Computational Barriers in Minimax Submatrix Detection
复制标题
极小极大子矩阵检测中的计算障碍
DOI：
10.1214/14-aos1300
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Yihong Wu
中科院分区：
文献类型：
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作者：
Zongming Ma;Yihong Wu
本文研究了受加性高斯噪声污染的大矩阵中提高均值的小子阵的Minimax检测问题。为了从复杂性理论的角度研究统计性能和计算成本之间的权衡，我们考虑了一系列离散模型，这些模型渐近等价于高斯模型。假设当集团大小的阶数小于图大小的平方根时，种植集团检测问题不能在随机多项式时间内解决，则建立以下相变现象：当大矩阵$p的大小\to\infty$，如果子矩阵大小$k=\Theta（p^{\alpha}）$ for any $\alpha\in（0，{2}/{3}）$，计算复杂性约束可能会导致统计性能的严重损失，因为任何随机多项式时间测试都是极小极大次优的多项式因子;如果对于任意{2}/{3}，1}中的{\alpha}，k=\Theta（p^{\alpha}），则在线性时间内，在常数因子内，可以获得Minimax最优检测。使用Schatten范数损失作为一个代表性的例子，我们表明，达到极小极大估计率的硬度可以在很大程度上取决于损失函数。支持恢复的硬度的影响也得到。
This paper studies the minimax detection of a small submatrix of elevated mean in a large matrix contaminated by additive Gaussian noise. To investigate the tradeoff between statistical performance and computational cost from a complexity-theoretic perspective, we consider a sequence of discretized models which are asymptotically equivalent to the Gaussian model. Under the hypothesis that the planted clique detection problem cannot be solved in randomized polynomial time when the clique size is of smaller order than the square root of the graph size, the following phase transition phenomenon is established: when the size of the large matrix $p\to\infty$, if the submatrix size $k=\Theta(p^{\alpha})$ for any $\alpha\in(0,{2}/{3})$, computational complexity constraints can incur a severe penalty on the statistical performance in the sense that any randomized polynomial-time test is minimax suboptimal by a polynomial factor in $p$; if $k=\Theta(p^{\alpha})$ for any $\alpha\in({2}/{3},1)$, minimax optimal detection can be attained within constant factors in linear time. Using Schatten norm loss as a representative example, we show that the hardness of attaining the minimax estimation rate can crucially depend on the loss function. Implications on the hardness of support recovery are also obtained.
关于高斯随机矩阵中大平均和方差分析拟合子矩阵的最大尺寸。
DOI：
10.3150/11-bej394
发表时间：
2013
期刊：
Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability
影响因子：
--
作者：
Sun,Xing;Nobel,AndrewB
通讯作者：
Nobel,AndrewB