1,223 research outputs found
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
An Exponential Lower Bound on the Complexity of Regularization Paths
For a variety of regularized optimization problems in machine learning,
algorithms computing the entire solution path have been developed recently.
Most of these methods are quadratic programs that are parameterized by a single
parameter, as for example the Support Vector Machine (SVM). Solution path
algorithms do not only compute the solution for one particular value of the
regularization parameter but the entire path of solutions, making the selection
of an optimal parameter much easier.
It has been assumed that these piecewise linear solution paths have only
linear complexity, i.e. linearly many bends. We prove that for the support
vector machine this complexity can be exponential in the number of training
points in the worst case. More strongly, we construct a single instance of n
input points in d dimensions for an SVM such that at least \Theta(2^{n/2}) =
\Theta(2^d) many distinct subsets of support vectors occur as the
regularization parameter changes.Comment: Journal version, 28 Pages, 5 Figure
Regularized Decomposition of High-Dimensional Multistage Stochastic Programs with Markov Uncertainty
We develop a quadratic regularization approach for the solution of
high-dimensional multistage stochastic optimization problems characterized by a
potentially large number of time periods/stages (e.g. hundreds), a
high-dimensional resource state variable, and a Markov information process. The
resulting algorithms are shown to converge to an optimal policy after a finite
number of iterations under mild technical assumptions. Computational
experiments are conducted using the setting of optimizing energy storage over a
large transmission grid, which motivates both the spatial and temporal
dimensions of our problem. Our numerical results indicate that the proposed
methods exhibit significantly faster convergence than their classical
counterparts, with greater gains observed for higher-dimensional problems
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