Randomized optimal transport on a graph: framework and new distance measures

Abstract

The recently developed bag-of-paths (BoP) framework consists in setting a Gibbs–Boltzmann distribution on all feasible paths of a graph. This probability distribution favors short paths over long ones, with a free parameter (the temperature T) controlling the entropic level of the distribution. This formalism enables the computation of new distances or dissimilarities, interpolating between the shortest-path and the resistance distance, which have been shown to perform well in clustering and classification tasks. In this work, the bag-of-paths formalism is extended by adding two independent equality constraints fixing starting and ending nodes distributions of paths (margins).When the temperature is low, this formalism is shown to be equivalent to a relaxation of the optimal transport problem on a network where paths carry a flow between two discrete distributions on nodes. The randomization is achieved by considering free energy minimization instead of traditional cost minimization. Algorithms computing the optimal free energy solution are developed for two types of paths: hitting (or absorbing) paths and non-hitting, regular, paths and require the inversion of an n × n matrix with n being the number of nodes. Interestingly, for regular paths on an undirected graph, the resulting optimal policy interpolates between the deterministic optimal transport policy (T → 0+) and the solution to the corresponding electrical circuit (T → ∞). Two distance measures between nodes and a dissimilarity between groups of nodes, both integrating weights on nodes, are derived from this framework.

Randomized Shortest Paths with Net Flows and Capacity Constraints

“…As already discussed in [44], the originality of the RSP model, in comparison with competing methods, lies in the fact that it adopts a path-based formalism with relative entropy regularization at the level of paths probabilities. That is, the quantities of interest are defined on the set of all possible paths (or walks) between two nodes of the network.…”

“…Concerning further work, we are interested in applications of the proposed models to operations research problems. For instance, it has been shown that solving optimal transport problems with entropic regularization can be significantly more efficient than using standard linear programming methods in some situations [24,42,44]. It would therefore be interesting to compare the RSP solution (with entropy regularization) to more standard algorithms solving minimum cost flow problems with capacity constraints [1,8,27,31,40,50,58].…”

Maximum likelihood estimation for randomized shortest paths with trajectory data

“…A similar model has been considered in the context of transportation research in [55]. An even more sophisticated approach would be the marginconstrained BoP model [22], where two distributions on the nodes of the network, k ∈ V , are given as input; one fixes the probability that k is a source node of a path, while the other fixes the probability that k is a target node of a path. The model then computes flows based on these fixed margins and the Gibbs-Boltzmann distribution over the set of all (hitting or regular) paths from the sources to the targets.…”

“…We consider the trajectories as extracted from hitting paths, where the hitting node is selected to be the first pixel within the summer range where the animal has been observed. As discussed in Section 3.1, more sophisticated methods could be used for determining the hitting target node, such as the margin-constrained RSP model [22], where the graph FIG. 7: Histogram of the MLE values for the 32 trajectories of reindeer.…”

Covariance and Correlation Kernels on a Graph in the Generalized Bag-of-Paths Formalism

“…When the temperature of the model is high, communication occurs through a random walk while, for low temperatures (close to zero), shorter paths are promoted. The model has been extended recently by adding a priori probabilities on the starting and ending nodes, thus allowing to weigh nodes [37,39].…”