The overlap gap property in principal submatrix recovery

Abstract

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 there exists a constant $$C>0$$ such that the MLE recovers a constant proportion of the hidden submatrix if $$\lambda {\ge C} \sqrt{\frac{1}{\rho } \log \frac{1}{\rho }}$$ , while such recovery is information theoretically impossible if $$\lambda = o( \sqrt{\frac{1}{\rho } \log \frac{1}{\rho }} )$$ . 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.