Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference

We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF inference problems. The core of our method is a very efficient bounding procedure, which combines scalable semidefinite programming (SDP) and a cutting-plane method for seeking violated constraints. In order to further speed up the computation, several strategies have been exploited, including model reduction, warm start and removal of inactive constraints. We analyze the performance of the proposed method under different settings, and demonstrate that our method either outperforms or performs on par with state-of-the-art approaches. Especially when the connectivities are dense or when the relative magnitudes of the unary costs are low, we achieve the best reported results. Experiments show that the proposed algorithm achieves better approximation than the state-of-the-art methods within a variety of time budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page

Contributions to Determining Exact Ground-States of Ising Spin-Glasses and to their Physics

In the last decades, much research has focused on a better understanding of so-called spin glasses (e.g., the alloys CuMn and AuFe.) Spin glasses are not yet fully understood. In order to be able to test the different theories that have been proposed for the nature of spin glasses we have to analyze numerically generated data. We consider spin glasses in the Ising model, where the spins (magnetic dipoles) have exactly two possibilities for aligning themselves. We are interested in the low-temperature states of the system, in which the spins are `frozen' and disordered. Determining a state of minimum energy, a ground state, amounts to calculating a maximum cut in a graph. The max-cut problem is a prominent NP-hard problem from combinatorial optimization. Maximum cuts can be determined exactly with a branch-and-cut algorithm. This thesis consists of two parts. In the first part we introduce the max-cut problem and a branch-and-cut algorithm for solving reasonably sized problems. We present several approaches for improving its performance for Ising spin-glass instances. In the second part of this work, we study the physics of spin glasses. We first discuss what is known in the literature. Then we present results for Bethe spin glasses. Finally we study the nature of short-range three-dimensional spin glasses. Results of the former were obtained in collaboration with M. Palassini and A.K. Hartmann, results of the latter in cooperation with M. Palassini and A. Peter Young

Fastest mixing Markov chain on graphs with symmetries

We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the second-largest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of large-scale instances that are otherwise computationally infeasible. We obtain analytic or semi-analytic results for particular classes of graphs, such as edge-transitive and distance-transitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and block-diagonalization, respectively. We also establish the connection between these two approaches.Comment: 39 pages, 15 figure

Cutting plane algorithms for variational inference in graphical models

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (leaves 65-66).In this thesis, we give a new class of outer bounds on the marginal polytope, and propose a cutting-plane algorithm for efficiently optimizing over these constraints. When combined with a concave upper bound on the entropy, this gives a new variational inference algorithm for probabilistic inference in discrete Markov Random Fields (MRFs). Valid constraints are derived for the marginal polytope through a series of projections onto the cut polytope. Projecting onto a larger model gives an efficient separation algorithm for a large class of valid inequalities arising from each of the original projections. As a result, we obtain tighter upper bounds on the logpartition function than possible with previous variational inference algorithms. We also show empirically that our approximations of the marginals are significantly more accurate. This algorithm can also be applied to the problem of finding the Maximum a Posteriori assignment in a MRF, which corresponds to a linear program over the marginal polytope. One of the main contributions of the thesis is to bring together two seemingly different fields, polyhedral combinatorics and probabilistic inference, showing how certain results in either field can carry over to the other.by David Alexander Sontag.S.M

Optimization Methods for Cluster Analysis in Network-based Data Mining

This dissertation focuses on two optimization problems that arise in network-based data mining, concerning identification of basic community structures (clusters) in graphs: the maximum edge weight clique and maximum induced cluster subgraph problems. We propose a continuous quadratic formulation for the maximum edge weight clique problem, and establish the correspondence between its local optima and maximal cliques in the graph. Subsequently, we present a combinatorial branch-and-bound algorithm for this problem that takes advantage of a polynomial-time solvable nonconvex relaxation of the proposed formulation. We also introduce a linear-time-computable analytic upper bound on the clique number of a graph, as well as a new method of upper-bounding the maximum edge weight clique problem, which leads to another exact algorithm for this problem. For the maximum induced cluster subgraph problem, we present the results of a comprehensive polyhedral analysis. We derive several families of facet-defining valid inequalities for the IUC polytope associated with a graph. We also provide a complete description of this polytope for some special classes of graphs. We establish computational complexity of the separation problems for most of the considered families of valid inequalities, and explore the effectiveness of employing the corresponding cutting planes in an integer (linear) programming framework for the maximum induced cluster subgraph problem