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The overlap gap property in principal submatrix recovery
Name
440_2021_1089_ReferencePDF.pdf
Size
840.09 KB
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Unknown
Checksum (MD5)
29f72dfd603c62a97f4620a3c6a03031
Author(s)
Gamarnik, David
Date Issued
September 27, 2021
Journal
Probability theory and related fields
Publisher
Springer Berlin Heidelberg
Citation
Gamarnik, David, Jagannath, Aukosh and Sen, Subhabrata. 2021. "The overlap gap property in principal submatrix recovery."
Version
Author's final manuscript
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.
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s00440-021-01089-7
