LPQP for MAP: Putting LP Solvers to Better Use

MAP inference for general energy functions remains a challenging problem. While most efforts are channeled towards improving the linear programming (LP) based relaxation, this work is motivated by the quadratic programming (QP) relaxation. We propose a novel MAP relaxation that penalizes the Kullback-Leibler divergence between the LP pairwise auxiliary variables, and QP equivalent terms given by the product of the unaries. We develop two efficient algorithms based on variants of this relaxation. The algorithms minimize the non-convex objective using belief propagation and dual decomposition as building blocks. Experiments on synthetic and real-world data show that the solutions returned by our algorithms substantially improve over the LP relaxation.Comment: ICML201

Image Labeling by Assignment

We introduce a novel geometric approach to the image labeling problem. Abstracting from specific labeling applications, a general objective function is defined on a manifold of stochastic matrices, whose elements assign prior data that are given in any metric space, to observed image measurements. The corresponding Riemannian gradient flow entails a set of replicator equations, one for each data point, that are spatially coupled by geometric averaging on the manifold. Starting from uniform assignments at the barycenter as natural initialization, the flow terminates at some global maximum, each of which corresponds to an image labeling that uniquely assigns the prior data. Our geometric variational approach constitutes a smooth non-convex inner approximation of the general image labeling problem, implemented with sparse interior-point numerics in terms of parallel multiplicative updates that converge efficiently

MAP-Inference for Highly-Connected Graphs with DC-Programming

Abstract. The design of inference algorithms for discrete-valued Markov Random Fields constitutes an ongoing research topic in computer vision. Large state-spaces, none-submodular energy-functions, and highlyconnected structures of the underlying graph render this problem particularly difficult. Established techniques that work well for sparsely connected grid-graphs used for image labeling, degrade for non-sparse models used for object recognition. In this context, we present a new class of mathematically sound algorithms that can be flexibly applied to this problem class with a guarantee to converge to a critical point of the objective function. The resulting iterative algorithms can be interpreted as simple message passing algorithms that converge by construction, in contrast to other message passing algorithms. Numerical experiments demonstrate its performance in comparison with established techniques.