The overlap gap property in principal submatrix recovery

Abstract

Abstract We study support recovery for a

$$k \times k$$

k × k

principal submatrix with elevated mean

$$\lambda /N$$

λ / N

, hidden in an

$$N\times N$$

N × N

symmetric mean zero Gaussian matrix. Here

$$\lambda >0$$

λ > 0

is a universal constant, and we assume

$$k = N \rho $$

k = N ρ

for some constant

$$\rho \in (0,1)$$

ρ ∈ ( 0 , 1 )

. We establish that there exists a constant

$$C>0$$

C > 0

such that the MLE recovers a constant proportion of the hidden submatrix if

$$\lambda {\ge C} \sqrt{\frac{1}{\rho } \log \frac{1}{\rho }}$$

λ

≥ C

1 ρ

log

1 ρ

, while such recovery is information theoretically impossible if

$$\lambda = o( \sqrt{\frac{1}{\rho } \log \frac{1}{\rho }} )$$

λ = o (

1 ρ

log

1 ρ

)

. The MLE is computationally intractable in general, and in fact, for

$$\rho >0$$

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

$$\varepsilon >0$$

ε > 0

and

$$\sqrt{\frac{1}{\rho } \log \frac{1}{\rho } } \ll \lambda \ll \frac{1}{\rho ^{1/2 + \varepsilon }}$$

1 ρ

log

1 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

$$\lambda > 1/\rho $$

λ > 1 / ρ

, a simple spectral method recovers a constant proportion of the hidden submatrix.