38 research outputs found
Convergence rate of Tsallis entropic regularized optimal transport
In this paper, we consider Tsallis entropic regularized optimal transport and
discuss the convergence rate as the regularization parameter goes
to . In particular, we establish the convergence rate of the Tsallis
entropic regularized optimal transport using the quantization and shadow
arguments developed by Eckstein--Nutz. We compare this to the convergence rate
of the entropic regularized optimal transport with Kullback--Leibler (KL)
divergence and show that KL is the fastest convergence rate in terms of Tsallis
relative entropy.Comment: 21 page
Regularized Optimal Transport and the Rot Mover's Distance
This paper presents a unified framework for smooth convex regularization of
discrete optimal transport problems. In this context, the regularized optimal
transport turns out to be equivalent to a matrix nearness problem with respect
to Bregman divergences. Our framework thus naturally generalizes a previously
proposed regularization based on the Boltzmann-Shannon entropy related to the
Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We
call the regularized optimal transport distance the rot mover's distance in
reference to the classical earth mover's distance. We develop two generic
schemes that we respectively call the alternate scaling algorithm and the
non-negative alternate scaling algorithm, to compute efficiently the
regularized optimal plans depending on whether the domain of the regularizer
lies within the non-negative orthant or not. These schemes are based on
Dykstra's algorithm with alternate Bregman projections, and further exploit the
Newton-Raphson method when applied to separable divergences. We enhance the
separable case with a sparse extension to deal with high data dimensions. We
also instantiate our proposed framework and discuss the inherent specificities
for well-known regularizers and statistical divergences in the machine learning
and information geometry communities. Finally, we demonstrate the merits of our
methods with experiments using synthetic data to illustrate the effect of
different regularizers and penalties on the solutions, as well as real-world
data for a pattern recognition application to audio scene classification
Sparse Randomized Shortest Paths Routing with Tsallis Divergence Regularization
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 main consequence of this change is that the
resulting routing policy (local transition probabilities) becomes sparser when
T decreases, therefore inducing a sparse random walk on G converging to the
least-cost directed acyclic graph when T tends to 0. Experimental comparisons
on node clustering and semi-supervised classification tasks show that the
derived dissimilarity measures based on expected routing costs provide
state-of-the-art results. The sparse RSP is therefore a promising model of
movements on a graph, balancing sparse exploitation and exploration in an
optimal way