The Overlap Gap Property in Principal Submatrix Recovery

We study support recovery for a $k \times k$ principal submatrix with elevated mean $\lambda/N$, hidden in an $N\times N$ symmetric mean zero Gaussian matrix. Here $\lambda>0$ is a universal constant, and we assume $k = N \rho$ for some constant $\rho \in (0,1)$. We establish that the MLE recovers a constant proportion of the hidden submatrix if and only if $\lambda \gtrsim \sqrt{\frac{1}{\rho} \log \frac{1}{\rho}}$. The MLE is computationally intractable in general, and in fact, for $\rho>0$ sufficiently small, this problem is conjectured to exhibit a statistical-computational gap. To provide rigorous evidence for this, we study the likelihood landscape for this problem, and establish that for some $\varepsilon>0$ and $\sqrt{\frac{1}{\rho} \log \frac{1}{\rho} } \ll \lambda \ll \frac{1}{\rho^{1/2 + \varepsilon}}$, the problem exhibits a variant of the Overlap-Gap-Property(OGP). As a direct consequence, we establish that a family of local MCMC based algorithms do not achieve optimal recovery. Finally, we establish that for $\lambda > 1/\rho$, a simple spectral method recovers a constant proportion of the hidden submatrix.