12 research outputs found
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
Ensemble Riemannian data assimilation: towards large-scale dynamical systems
This paper presents the results of the ensemble
Riemannian data assimilation for relatively highdimensional
nonlinear dynamical systems, focusing on the
chaotic Lorenz-96 model and a two-layer quasi-geostrophic
(QG) model of atmospheric circulation. The analysis state
in this approach is inferred from a joint distribution that optimally
couples the background probability distribution and
the likelihood function, enabling formal treatment of systematic
biases without any Gaussian assumptions. Despite the
risk of the curse of dimensionality in the computation of the
coupling distribution, comparisons with the classic implementation
of the particle filter and the stochastic ensemble
Kalman filter demonstrate that, with the same ensemble size,
the presented methodology could improve the predictability
of dynamical systems. In particular, under systematic errors,
the root mean squared error of the analysis state can be reduced
by 20% (30 %) in the Lorenz-96 (QG) model
Optimal transport: discretization and algorithms
This chapter describes techniques for the numerical resolution of optimal
transport problems. We will consider several discretizations of these problems,
and we will put a strong focus on the mathematical analysis of the algorithms
to solve the discretized problems. We will describe in detail the following
discretizations and corresponding algorithms: the assignment problem and
Bertsekas auction's algorithm; the entropic regularization and Sinkhorn-Knopp's
algorithm; semi-discrete optimal transport and Oliker-Prussner or damped
Newton's algorithm, and finally semi-discrete entropic regularization. Our
presentation highlights the similarity between these algorithms and their
connection with the theory of Kantorovich duality
Optimal transport: discretization and algorithms
This chapter describes techniques for the numerical resolution of optimal transport problems. We will consider several discretizations of these problems, and we will put a strong focus on the mathematical analysis of the algorithms to solve the discretized problems. We will describe in detail the following discretizations and corresponding algorithms: the assignment problem and Bertsekas auction's algorithm; the entropic regularization and Sinkhorn-Knopp's algorithm; semi-discrete optimal transport and Oliker-Prussner or damped Newton's algorithm, and finally semi-discrete entropic regularization. Our presentation highlights the similarity between these algorithms and their connection with the theory of Kantorovich duality