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 zero Gaussian matrix. Here >0$$ is universal constant, and we assume = N \rho $$ some constant $$\rho \in (0,1)$$ . establish that there exists $$C>0$$ such the MLE recovers proportion of if {\ge C} \sqrt{\frac{1}{\rho } \log \frac{1}{\rho }}$$ while information theoretically impossible o( }} )$$ The computationally intractable general, fact, sufficiently small, this problem conjectured to exhibit statistical-computational gap. To provide rigorous evidence this, likelihood landscape problem, $$\varepsilon $$\sqrt{\frac{1}{\rho \ll \lambda ^{1/2 + \varepsilon exhibits variant Overlap-Gap-Property (OGP). As direct consequence, family local MCMC based algorithms do not achieve optimal recovery. Finally, > 1/\rho simple spectral method submatrix.