Relative Entropy-Regularized Optimal Transport on a Graph: a new algorithm and an experimental comparison

Abstract: Following [21, 23], the present work investigates a new relative entropy-regularized algorithm for solving the optimal transport on a graph problem within the randomized shortest paths formalism. More precisely, a unit flow is injected into a set of input nodes and collected from a set of output nodes while minimizing the expected transportation cost together with a paths relative entropy regularization term, providing a randomized routing policy. The main advantage of this new formulation is the fact that it can easily accommodate edge flow capacity constraints which commonly occur in real-world problems. The resulting optimal routing policy, i.e., the probability distribution of following an edge in each node, is Markovian and is computed by constraining the input and output flows to the prescribed marginal probabilities thanks to a variant of the algorithm developed in [8]. In addition, experimental comparisons with other recently developed techniques show that the distance measure between nodes derived from the introduced model provides competitive results on semi-supervised classification tasks.