708 research outputs found
A semidefinite program for unbalanced multisection in the stochastic block model
We propose a semidefinite programming (SDP) algorithm for community detection
in the stochastic block model, a popular model for networks with latent
community structure. We prove that our algorithm achieves exact recovery of the
latent communities, up to the information-theoretic limits determined by Abbe
and Sandon (2015). Our result extends prior SDP approaches by allowing for many
communities of different sizes. By virtue of a semidefinite approach, our
algorithms succeed against a semirandom variant of the stochastic block model,
guaranteeing a form of robustness and generalization. We further explore how
semirandom models can lend insight into both the strengths and limitations of
SDPs in this setting.Comment: 29 page
The power of sum-of-squares for detecting hidden structures
We study planted problems---finding hidden structures in random noisy
inputs---through the lens of the sum-of-squares semidefinite programming
hierarchy (SoS). This family of powerful semidefinite programs has recently
yielded many new algorithms for planted problems, often achieving the best
known polynomial-time guarantees in terms of accuracy of recovered solutions
and robustness to noise. One theme in recent work is the design of spectral
algorithms which match the guarantees of SoS algorithms for planted problems.
Classical spectral algorithms are often unable to accomplish this: the twist in
these new spectral algorithms is the use of spectral structure of matrices
whose entries are low-degree polynomials of the input variables. We prove that
for a wide class of planted problems, including refuting random constraint
satisfaction problems, tensor and sparse PCA, densest-k-subgraph, community
detection in stochastic block models, planted clique, and others, eigenvalues
of degree-d matrix polynomials are as powerful as SoS semidefinite programs of
roughly degree d. For such problems it is therefore always possible to match
the guarantees of SoS without solving a large semidefinite program. Using
related ideas on SoS algorithms and low-degree matrix polynomials (and inspired
by recent work on SoS and the planted clique problem by Barak et al.), we prove
new nearly-tight SoS lower bounds for the tensor and sparse principal component
analysis problems. Our lower bounds for sparse principal component analysis are
the first to suggest that going beyond existing algorithms for this problem may
require sub-exponential time
Community detection in sparse networks via Grothendieck's inequality
We present a simple and flexible method to prove consistency of semidefinite
optimization problems on random graphs. The method is based on Grothendieck's
inequality. Unlike the previous uses of this inequality that lead to constant
relative accuracy, we achieve any given relative accuracy by leveraging
randomness. We illustrate the method with the problem of community detection in
sparse networks, those with bounded average degrees. We demonstrate that even
in this regime, various simple and natural semidefinite programs can be used to
recover the community structure up to an arbitrarily small fraction of
misclassified vertices. The method is general; it can be applied to a variety
of stochastic models of networks and semidefinite programs.Comment: This is the final version, incorporating the referee's comment
Phase Transitions in Semidefinite Relaxations
Statistical inference problems arising within signal processing, data mining,
and machine learning naturally give rise to hard combinatorial optimization
problems. These problems become intractable when the dimensionality of the data
is large, as is often the case for modern datasets. A popular idea is to
construct convex relaxations of these combinatorial problems, which can be
solved efficiently for large scale datasets.
Semidefinite programming (SDP) relaxations are among the most powerful
methods in this family, and are surprisingly well-suited for a broad range of
problems where data take the form of matrices or graphs. It has been observed
several times that, when the `statistical noise' is small enough, SDP
relaxations correctly detect the underlying combinatorial structures.
In this paper we develop asymptotic predictions for several `detection
thresholds,' as well as for the estimation error above these thresholds. We
study some classical SDP relaxations for statistical problems motivated by
graph synchronization and community detection in networks. We map these
optimization problems to statistical mechanics models with vector spins, and
use non-rigorous techniques from statistical mechanics to characterize the
corresponding phase transitions. Our results clarify the effectiveness of SDP
relaxations in solving high-dimensional statistical problems.Comment: 71 pages, 24 pdf figure
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