Sharp minimax risks and phase transitions in sparse submatrix detection

In this paper, we study the minimax risk of detecting a sparse, elevated-mean Gaussian submatrix planted in a larger noisy matrix. When the planted submatrix has size $n\times n$ and the ambient matrix has size $N\times N$ with $N=n^{1+α}$, the seminal work of Butucea and Ingster (2013) identifies t