4,969 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
Structured Near-Optimal Channel-Adapted Quantum Error Correction
We present a class of numerical algorithms which adapt a quantum error
correction scheme to a channel model. Given an encoding and a channel model, it
was previously shown that the quantum operation that maximizes the average
entanglement fidelity may be calculated by a semidefinite program (SDP), which
is a convex optimization. While optimal, this recovery operation is
computationally difficult for long codes. Furthermore, the optimal recovery
operation has no structure beyond the completely positive trace preserving
(CPTP) constraint. We derive methods to generate structured channel-adapted
error recovery operations. Specifically, each recovery operation begins with a
projective error syndrome measurement. The algorithms to compute the structured
recovery operations are more scalable than the SDP and yield recovery
operations with an intuitive physical form. Using Lagrange duality, we derive
performance bounds to certify near-optimality.Comment: 18 pages, 13 figures Update: typos corrected in Appendi
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