Max-product belief propagation is a local, iterative algorithm to find the mode/MAP estimate of a probability distribution. While it has been successfully employed in a wide variety of applications, there are relatively few theoretical guarantees of convergence and correctness for general loopy graphs that may have many short cycles. Of these, even fewer provide exact ``necessary and sufficient'' characterizations. In this paper we investigate the problem of using max-product to find the maximum weight matching in an arbitrary graph with edge weights. This is done by first constructing a probability distribution whose mode corresponds to the optimal matching, and then running max-product. Weighted matching can also be posed as an integer program, for which there is an LP relaxation. This relaxation is not always tight. In this paper we show that \begin{enumerate} \item If the LP relaxation is tight, then max-product always converges, and that too to the correct answer. \item If the LP relaxation is loose, then max-product does not converge. \end{enumerate} This provides an exact, data-dependent characterization of max-product performance, and a precise connection to LP relaxation, which is a well-studied optimization technique. Also, since LP relaxation is known to be tight for bipartite graphs, our results generalize other recent results on using max-product to find weighted matchings in bipartite graphs.

Web page ranking and collaborative filtering require the optimization of sophisticated performance measures. Current Support Vector approaches are unable to optimize them directly and focus on pairwise comparisons instead. We present a new approach which allows direct optimization of the relevant loss functions. This is achieved via structured estimation in Hilbert spaces. It is most related to Max-Margin-Markov networks optimization of multivariate performance measures. Key to our approach is that during training the ranking problem can be viewed as a linear assignment problem, which can be solved by the Hungarian Marriage algorithm. At test time, a sort operation is sufficient, as our algorithm assigns a relevance score to every (document, query) pair. Experiments show that the our algorithm is fast and that it works very well.

Many combinatorial optimization problems such as the bin packing and multiple knapsack problems involve assigning a set of discrete objects to multiple containers. These problems can be used to model task and resource allocation problems in multi-agent systems and distributed systms, and can also be found as subproblems of scheduling problems. We propose bin completion, a branch-and-bound strategy for one-dimensional, multicontainer packing problems. Bin completion combines a bin-oriented search space with a powerful dominance criterion that enables us to prune much of the space. The performance of the basic bin completion framework can be enhanced by using a number of extensions, including nogood-based pruning techniques that allow further exploitation of the dominance criterion. Bin completion is applied to four problems: multiple knapsack, bin covering, min-cost covering, and bin packing. We show that our bin completion algorithms yield new, state-of-the-art results for the multiple knapsack, bin covering, and min-cost covering problems, outperforming previous algorithms by several orders of magnitude with respect to runtime on some classes of hard, random problem instances. For the bin packing problem, we demonstrate significant improvements compared to most previous results, but show that bin completion is not competitive with current state-of-the-art cutting-stock based approaches.

The problem of resource allocation in sparse graphs with real variables is studied using methods of statistical physics. An efficient distributed algorithm is devised on the basis of insight gained from the analysis and is examined using numerical simulations, showing excellent performance and full agreement with the theoretical results.

We consider the problem of localizing a set of microphones together with a set of external acoustic events (e.g., hand claps), emitted at unknown times and unknown locations. We propose a solution that approximates this problem under a far field approximation defined in the calculus of affine geometry, and that relies on singular value decomposition (SVD) to recover the affine structure of the problem. We then define low-dimensional optimization techniques for embedding the solution into Euclidean geometry, and further techniques for recovering the locations and emission times of the acoustic events. The approach is useful for the calibration of ad-hoc microphone arrays and sensor networks.

We present a simple and scalable algorithm for large-margin estimation of structured models, including an important class of Markov networks and combinatorial models. We formulate the estimation problem as a convex-concave saddle-point problem and apply the extragradient method, yielding an algorithm with linear convergence using simple gradient and projection calculations. The projection step can be solved using combinatorial algorithms for min-cost quadratic flow. This makes the approach an efficient alternative to formulations based on reductions to a quadratic program (QP). We present experiments on two very different structured prediction tasks: 3D image segmentation and word alignment, illustrating the favorable scaling properties of our algorithm.

Sparse PCA seeks approximate sparse "eigenvectors" whose projections capture the maximal variance of data. As a cardinality-constrained and non-convex optimization problem, it is NPhard and is encountered in a wide range of applied fields, from bio-informatics to finance. Recent progress has focused mainly on continuous approximation and convex relaxation of the hard cardinality constraint. In contrast, we consider an alternative discrete spectral formulation based on variational eigenvalue bounds and provide an effective greedy strategy as well as provably optimal solutions using branch-and-bound search. Moreover, the exact methodology used reveals a simple renormalization step that improves approximate solutions obtained by any continuous method. The resulting performance gain of discrete algorithms is demonstrated on real-world benchmark data and in extensive Monte Carlo evaluation trials.

Given a directed graphical model with binary-valued hidden nodes and real-valued noisy observations, consider deciding upon the maximum a-posteriori (MAP) or the maximum posterior-marginal (MPM) assignment under the restriction that each node broadcasts only to its children exactly one single-bit message. We present a variational formulation, viewing the processing rules local to all nodes as degrees-of-freedom, that minimizes the loss in expected (MAP or MPM) performance subject to such online communication constraints. The approach leads to a novel message-passing algorithm to be executed offline, or before observations are realized, which mitigates the performance loss by iteratively coupling all rules in a manner implicitly driven by global statistics. We also provide (i) illustrative examples, (ii) assumptions that guarantee convergence and efficiency and (iii) connections to active research areas.

We present an algorithm for learning a quadratic Gaussian metric (Mahalanobis distance) for use in classification tasks. Our method relies on the simple geometric intuition that a good metric is one under which points in the same class are simultaneously near each other and far from points in the other classes. We construct a convex optimization problem whose solution generates such a metric by trying to collapse all examples in the same class to a single point and push examples in other classes infinitely far away. We show that when the metric we learn is used in simple classifiers, it yields substantial improvements over standard alternatives on a variety of problems. We also discuss how the learned metric may be used to obtain a compact low dimensional feature representation of the original input space, allowing more efficient classification with very little reduction in performance.

The problem of resource allocation in sparse graphs with real variables is studied using methods of statistical physics. An efficient distributed algorithm is devised on the basis of insight gained from the analysis and is examined using numerical simulations, showing excellent performance and full agreement with the theoretical results.