Finding a Large Submatrix of a Gaussian Random Matrix

We consider the problem of finding a $k\times k$ submatrix of an $n\times n$ matrix with i.i.d. standard Gaussian entries, which has a large average entry. It was shown earlier by Bhamidi et al. that the largest average value of such a matrix is $2\sqrt{\log n/k}$ with high probability. In the same paper an evidence was provided that a natural greedy algorithm called Largest Average Submatrix ($\LAS$) should produce a matrix with average entry approximately $\sqrt{2}$ smaller. In this paper we show that the matrix produced by the $\LAS$ algorithm is indeed $\sqrt{2\log n/k}$ w.h.p. Then by drawing an analogy with the problem of finding cliques in random graphs, we propose a simple greedy algorithm which produces a $k\times k$ matrix with asymptotically the same average value. Since the greedy algorithm is the best known algorithm for finding cliques in random graphs, it is tempting to believe that beating the factor $\sqrt{2}$ performance gap suffered by both algorithms might be very challenging. Surprisingly, we show the existence of a very simple algorithm which produces a matrix with average value $(4/3)\sqrt{2\log n/k}$. To get an insight into the algorithmic hardness of this problem, and motivated by methods originating in the theory of spin glasses, we conduct the so-called expected overlap analysis of matrices with average value asymptotically $\alpha\sqrt{2\log n/k}$. The overlap corresponds to the number of common rows and common columns for pairs of matrices achieving this value. We discover numerically an intriguing phase transition at $\alpha^*\approx 1.3608..$: when $\alpha<\alpha^*$ the space of overlaps is a continuous subset of $[0,1]^2$, whereas $\alpha=\alpha^*$ marks the onset of discontinuity, and the model exhibits the Overlap Gap Property when $\alpha>\alpha^*$. We conjecture that $\alpha>\alpha^*$ marks the onset of the algorithmic hardness.