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.
This work elaborates on the important problem of (1) designing optimal randomized routing policies for reaching a target node t from a source note s on a weighted directed graph G and (2) defining distance measures between nodes interpolating between the least cost (based on optimal movements) and the commute-cost (based on a random walk on G), depending on a temperature parameter T. To this end, the randomized shortest path formalism (RSP, [2, 99, 124]) is rephrased in terms of Tsallis divergence regularization, instead of Kullback-Leibler divergence.
The framework assumes a unit flow injected into a source node and collected from a target node with flows minimizing the expected transportation cost, together with a relative entropy regularization term.
This work derives closed-form expressions computing the expectation of co-presence and of number of co-occurrences of nodes on paths sampled from a network according to general path weights (a bag of paths).
The present work extends the randomized shortest-paths framework (RSP), interpolating between shortest-path and random-walk routing in a network, in three directions.
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 volume of data generated by internet and social networks is increasing every day, and there is a clear need for efficient ways of extracting useful information from them.
A major issue of existing approaches is that they usually rely on a manual annotation in expressive styles, which tends to be rather subjective.
This probability captures a notion of relatedness between nodes of the graph: two nodes are considered as highly related when they are connected by many, preferably low-cost, paths.
In particular, we see that the results obtained with the free energy distance are among the best in all the experiments.