243 research outputs found
On the complexity of approximating Wasserstein barycenter
We study the complexity of approximating Wassertein barycenter of discrete measures, or histograms by contrasting two alternative approaches, both using entropic regularization. We provide a novel analysis for our approach based on the Iterative Bregman Projections (IBP) algorithm to approximate the original non-regularized barycenter. We also get the complexity bound for alternative accelerated-gradient-descent-based approach and compare it with the bound obtained for IBP. As a byproduct, we show that the regularization parameter in both approaches has to be proportional to ", which causes instability of both algorithms when the desired accuracy is high. To overcome this issue, we propose a novel proximal-IBP algorithm, which can be seen as a proximal gradient method, which uses IBP on each iteration to make a proximal step. We also consider the question of scalability of these algorithms using approaches from distributed optimization and show that the first algorithm can be implemented in a centralized distributed setting (master/slave), while the second one is amenable to a more general decentralized distributed setting with an arbitrary network topology
On the complexity of approximating Wasserstein barycenter
We study the complexity of approximating Wassertein barycenter of discrete measures, or
histograms by contrasting two alternative approaches, both using entropic regularization. We provide
a novel analysis for our approach based on the Iterative Bregman Projections (IBP) algorithm
to approximate the original non-regularized barycenter. We also get the complexity bound for alternative
accelerated-gradient-descent-based approach and compare it with the bound obtained
for IBP. As a byproduct, we show that the regularization parameter in both approaches has to
be proportional to ", which causes instability of both algorithms when the desired accuracy is
high. To overcome this issue, we propose a novel proximal-IBP algorithm, which can be seen as
a proximal gradient method, which uses IBP on each iteration to make a proximal step. We also
consider the question of scalability of these algorithms using approaches from distributed optimization
and show that the first algorithm can be implemented in a centralized distributed setting
(master/slave), while the second one is amenable to a more general decentralized distributed
setting with an arbitrary network topology
On the Complexity of Approximating Wasserstein Barycenter
We study the complexity of approximating Wassertein barycenter of
discrete measures, or histograms of size by contrasting two alternative
approaches, both using entropic regularization. The first approach is based on
the Iterative Bregman Projections (IBP) algorithm for which our novel analysis
gives a complexity bound proportional to to
approximate the original non-regularized barycenter. Using an alternative
accelerated-gradient-descent-based approach, we obtain a complexity
proportional to . As a byproduct, we show that
the regularization parameter in both approaches has to be proportional to
, which causes instability of both algorithms when the desired
accuracy is high. To overcome this issue, we propose a novel proximal-IBP
algorithm, which can be seen as a proximal gradient method, which uses IBP on
each iteration to make a proximal step. We also consider the question of
scalability of these algorithms using approaches from distributed optimization
and show that the first algorithm can be implemented in a centralized
distributed setting (master/slave), while the second one is amenable to a more
general decentralized distributed setting with an arbitrary network topology.Comment: Corrected misprints. Added a reference to accelerated Iterative
Bregman Projections introduced in arXiv:1906.0362
Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm
We present a novel algorithm to estimate the barycenter of arbitrary
probability distributions with respect to the Sinkhorn divergence. Based on a
Frank-Wolfe optimization strategy, our approach proceeds by populating the
support of the barycenter incrementally, without requiring any pre-allocation.
We consider discrete as well as continuous distributions, proving convergence
rates of the proposed algorithm in both settings. Key elements of our analysis
are a new result showing that the Sinkhorn divergence on compact domains has
Lipschitz continuous gradient with respect to the Total Variation and a
characterization of the sample complexity of Sinkhorn potentials. Experiments
validate the effectiveness of our method in practice.Comment: 46 pages, 8 figure
Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm
We study the fixed-support Wasserstein barycenter problem (FS-WBP), which
consists in computing the Wasserstein barycenter of discrete probability
measures supported on a finite metric space of size . We show first that the
constraint matrix arising from the standard linear programming (LP)
representation of the FS-WBP is \textit{not totally unimodular} when
and . This result resolves an open question pertaining to the
relationship between the FS-WBP and the minimum-cost flow (MCF) problem since
it proves that the FS-WBP in the standard LP form is not an MCF problem when and . We also develop a provably fast \textit{deterministic}
variant of the celebrated iterative Bregman projection (IBP) algorithm, named
\textsc{FastIBP}, with a complexity bound of
, where is the
desired tolerance. This complexity bound is better than the best known
complexity bound of for the IBP algorithm in
terms of , and that of from
accelerated alternating minimization algorithm or accelerated primal-dual
adaptive gradient algorithm in terms of . Finally, we conduct extensive
experiments with both synthetic data and real images and demonstrate the
favorable performance of the \textsc{FastIBP} algorithm in practice.Comment: Accepted by NeurIPS 2020; fix some confusing parts in the proof and
improve the empirical evaluatio
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