# The Overlap Gap Property in Principal Submatrix Recovery

We study support recovery for a k × k principal submatrix with elevated mean λ/N, hidden in an N× N symmetric mean zero Gaussian matrix. Here λ>0 is a universal constant, and we assume k = N ρ for some constant ρ∈ (0,1). We establish that the MLE recovers a constant proportion of the hidden submatrix if and only if λ≳√(1/ρlog1/ρ). The MLE is computationally intractable in general, and in fact, for ρ>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 ε>0 and √(1/ρlog1/ρ)≪λ≪1/ρ^1/2 + ε, 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 λ > 1/ρ, a simple spectral method recovers a constant proportion of the hidden submatrix.

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