COMPUTATIONAL BARRIERS IN MINIMAX SUBMATRIX DETECTION

• 成果类型: Article
• 署名作者: Ma, Zongming; Wu, Yihong
• 署名单位: University of Pennsylvania; University of Illinois System; University of Illinois Urbana-Champaign
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/14-AOS1300
• 发表日期: 2015
• 页码: 1089-1116
• 关键词: large-average; cliques

摘要

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 -> infinity, if the submatrix size k = Theta(p(alpha)) for any alpha is an element of (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 is an element of (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.