Probabilistic graph and hypergraph matching

We consider the problem of finding a matching between two sets of features, given complex relations among them, going beyond pairwise. Each feature set is modeled by a hypergraph where the complex relations are represented by hyper-edges. A match between the feature sets is then modeled as a hypergraph matching problem. We derive the hyper-graph matching problem in a probabilistic setting represented by a convex optimization. First, we formalize a soft matching criterion that emerges from a probabilistic interpretation of the problem input and output, as opposed to previous methods that treat soft matching as a mere relaxation of the hard matching problem. Second, the model induces an algebraic relation between the hyper-edge weight matrix and the desired vertex-to-vertex probabilistic matching. Third, the model explains some of the graph matching normalization proposed in the past on a heuristic basis such as doubly stochastic normalizations of the edge weights. A key benefit of the model is that the global optimum of the matching criteria can be found via an iterative successive projection algorithm. The algorithm reduces to the well known Sinkhorn [15] row/column matrix normalization procedure in the special case when the two graphs have the same number of vertices and a complete matching is desired. Another benefit of our model is the straightforward scalability from graphs to hyper-graphs.