4,387 research outputs found
Developments in the theory of randomized shortest paths with a comparison of graph node distances
There have lately been several suggestions for parametrized distances on a
graph that generalize the shortest path distance and the commute time or
resistance distance. The need for developing such distances has risen from the
observation that the above-mentioned common distances in many situations fail
to take into account the global structure of the graph. In this article, we
develop the theory of one family of graph node distances, known as the
randomized shortest path dissimilarity, which has its foundation in statistical
physics. We show that the randomized shortest path dissimilarity can be easily
computed in closed form for all pairs of nodes of a graph. Moreover, we come up
with a new definition of a distance measure that we call the free energy
distance. The free energy distance can be seen as an upgrade of the randomized
shortest path dissimilarity as it defines a metric, in addition to which it
satisfies the graph-geodetic property. The derivation and computation of the
free energy distance are also straightforward. We then make a comparison
between a set of generalized distances that interpolate between the shortest
path distance and the commute time, or resistance distance. This comparison
focuses on the applicability of the distances in graph node clustering and
classification. The comparison, in general, shows that the parametrized
distances perform well in the tasks. In particular, we see that the results
obtained with the free energy distance are among the best in all the
experiments.Comment: 30 pages, 4 figures, 3 table
Expectation Optimization with Probabilistic Guarantees in POMDPs with Discounted-sum Objectives
Partially-observable Markov decision processes (POMDPs) with discounted-sum
payoff are a standard framework to model a wide range of problems related to
decision making under uncertainty. Traditionally, the goal has been to obtain
policies that optimize the expectation of the discounted-sum payoff. A key
drawback of the expectation measure is that even low probability events with
extreme payoff can significantly affect the expectation, and thus the obtained
policies are not necessarily risk-averse. An alternate approach is to optimize
the probability that the payoff is above a certain threshold, which allows
obtaining risk-averse policies, but ignores optimization of the expectation. We
consider the expectation optimization with probabilistic guarantee (EOPG)
problem, where the goal is to optimize the expectation ensuring that the payoff
is above a given threshold with at least a specified probability. We present
several results on the EOPG problem, including the first algorithm to solve it.Comment: Full version of a paper published at IJCAI/ECAI 201
Randomized Optimal Transport on a Graph: framework and new distance measures
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 ) 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 matrix with
being the number of nodes. Interestingly, for regular paths on an undirected
graph, the resulting optimal policy interpolates between the deterministic
optimal transport policy () and the solution to the
corresponding electrical circuit (). Two distance
measures between nodes and a dissimilarity between groups of nodes, both
integrating weights on nodes, are derived from this framework.Comment: Preprint paper to appear in Network Science journal, Cambridge
University Pres
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