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The problem of finding large average submatrices of a real-valued matrixarises in the exploratory analysis of data from a variety of disciplines,ranging from genomics to social sciences. In this paper we provide a detailedasymptotic analysis of large average submatrices of an $n \times n$ Gaussianrandom matrix. The first part of the paper addresses global maxima. For fixed$k$ we identify the average and the joint distribution of the $k \times k$submatrix having largest average value. As a dual result, we establish that thesize of the largest square sub-matrix with average bigger than a fixed positiveconstant is, with high probability, equal to one of two consecutive integersthat depend on the threshold and the matrix dimension $n$. The second part ofthe paper addresses local maxima. Specifically we consider submatrices withdominant row and column sums that arise as the local optima of iterative searchprocedures for large average submatrices. For fixed $k$, we identify thelimiting average value and joint distribution of a $k \times k$ submatrixconditioned to be a local maxima. In order to understand the density of suchlocal optima and explain the quick convergence of such iterative procedures, weanalyze the number $L_n(k)$ of local maxima, beginning with exact asymptoticexpressions for the mean and fluctuation behavior of $L_n(k)$. For fixed $k$,the mean of $L_{n}(k)$ is $\Theta(n^{k}/(\log{n})^{(k-1)/2})$ while thestandard deviation is $\Theta(n^{2k^2/(k+1)}/(\log{n})^{k^2/(k+1)})$. Ourprincipal result is a Gaussian central limit theorem for $L_n(k)$ that is basedon a new variant of Stein's method.

Energy Landscape for large average submatrix detection problems in Gaussian random matrices