We study the maximum-average submatrix problem, in which given an $N \times N$ matrix $J$ one needs to find the $k \times k$ submatrix with the largest average of entries. We study the problem for random matrices $J$ whose entries are i.i.d. random variables by mapping it to a variant of the Sherrington-Kirkpatrick spin-glass model at fixed magnetization. We characterize analytically the phase diagram of the model as a function of the submatrix average and the size of the submatrix $k$ in the limit $N\to\infty$. We consider submatrices of size $k = m N$ with $0 < m < 1$. We find a rich phase diagram, including dynamical, static one-step replica symmetry breaking and full-step replica symmetry breaking. In the limit of $m \to 0$, we find a simpler phase diagram featuring a frozen 1-RSB phase, where the Gibbs measure is composed of exponentially many pure states each with zero entropy. We discover an interesting phenomenon, reminiscent of the phenomenology of the binary perceptron: there exist efficient algorithms that provably work in the frozen 1-RSB phase.

翻译：我们研究最大均值子矩阵问题：给定一个 $N \times N$ 矩阵 $J$，需要找出 $k \times k$ 子矩阵使其元素均值最大。我们通过将该问题映射至固定磁化强度下的Sherrington-Kirkpatrick自旋玻璃模型变体，研究元素为独立同分布随机变量的随机矩阵 $J$ 对应的该问题。我们解析地刻画了在 $N\to\infty$ 极限下，模型相图作为子矩阵均值及子矩阵大小 $k$ 的函数特征。考虑大小为 $k = m N$（$0 < m < 1$）的子矩阵，我们发现了一个丰富的相图，包括动态、静态一步复制对称性破缺及完全步复制对称性破缺。在 $m \to 0$ 极限下，我们得到一个更简单的相图，其特征为冻结的一步复制对称性破缺相，其中吉布斯测度由指数多个熵为零的纯态构成。我们观察到一种有趣现象，类似于二元感知器的现象学：存在可证明在该冻结一步复制对称性破缺相中有效工作的算法。