Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: a unified perspective

@article{Jain2018MeanfieldAC,
  title={Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: a unified perspective},
  author={Vishesh Jain and Frederic Koehler and Andrej Risteski},
  journal={Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing},
  year={2018}
}
The free energy is a key quantity of interest in Ising models, but unfortunately, computing it in general is computationally intractable. Two popular (variational) approximation schemes for estimating the free energy of general Ising models (in particular, even in regimes where correlation decay does not hold) are: (i) the mean-field approximation with roots in statistical physics, which estimates the free energy from below, and (ii) hierarchies of convex relaxations with roots in theoretical… 

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References

SHOWING 1-10 OF 45 REFERENCES

How to calculate partition functions using convex programming hierarchies: provable bounds for variational methods

This work considers dense and low threshold rank graphs, and interestingly, the reason the approach works on these types of graphs is because local correlations propagate to global correlations – completely the opposite of algorithms based on correlation decay.

The Computational Hardness of Counting in Two-Spin Models on d-Regular Graphs

  • A. SlyNike Sun
  • Mathematics
    2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
  • 2012
The proof establishes that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting ``free energy density'' which coincides with the (non-rigorous) Be the prediction of statistical physics.

Graphical Models, Exponential Families, and Variational Inference

The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.

Quick Approximation to Matrices and Applications

The matrix approximation is generalized to multi-dimensional arrays and from that derive approximation algorithms for all dense Max-SNP problems and the Regularity Lemma is derived.

Decomposition of mean-field Gibbs distributions into product measures

We show that under a low complexity condition on the gradient of a Hamiltonian, Gibbs distributions on the Boolean hypercube are approximate mixtures of product measures whose probability vectors are

Universality of the mean-field for the Potts model

We consider the Potts model with q colors on a sequence of weighted graphs with adjacency matrices $$A_n$$An, allowing for both positive and negative weights. Under a mild regularity condition on

Bethe States of Random Factor Graphs

It is verified that the Gibbs measure can be decomposed into a moderate number of Bethe states, subsets of the state space in which both short and long range correlations of the measure take a simple form.

Lasserre Hierarchy, Higher Eigenvalues, and Approximation Schemes for Graph Partitioning and Quadratic Integer Programming with PSD Objectives

  • V. GuruswamiA. Sinop
  • Computer Science, Mathematics
    2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
  • 2011
An approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semi definite objective functions and global linear constraints is presented, and an algorithm for independent sets in graphs that performs well when the Laplacian does not have too many eigenvalues bigger than $1+o(1).

Ising models on locally tree-like graphs

We consider Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that

The statistics of Curie-Weiss models

LetSn denote the random total magnetization of ann-site Curie-Weiss model, a collection ofn (spin) random variables with an equal interaction of strength 1/n between each pair of spins. The