The overlap gap property in principal submatrix recovery

• Main Authors: Gamarnik, David (Author), Jagannath, Aukosh (Author), Sen, Subhabrata (Author)
• Other Authors: Sloan School of Management (Contributor)
• Format: Article
• Language: English
• Published: Springer Berlin Heidelberg, 2022-01-07T21:44:58Z.
• Subjects: Article
• Online Access: Get fulltext

 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.