Fréchet Means and Procrustes Analysis in Wasserstein Space

Zemel, Yoav; Panaretos, Victor M.

doi:10.48550/arxiv.1701.06876

Cited by 3 publications

(13 citation statements)

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“…Uniqueness in fact holds at the population level as well: the condition is that the random covariance operator be injective with positive probability. On R d this was observed by Bigot and Klein [10] in a parametric setting, and extended to the nonparametric setting by Zemel and Panaretos [60]; the analytical idea dates back to Álvarez-Esteban et al [3].…”

Section: Existence and Uniqueness Of Fréchet Meansmentioning

confidence: 84%

“…When N " 2, multicouplings are simply couplings and finding an optimal multicoupling is the optimal transport problem. On R d , multicouplings were studied by Gangbo and Swiech [25] (also see Zemel and Panaretos [60]). In analogy with the optimal transport problem, an optimal multicoupling always exists, and if µ 1 is regular an optimal multicoupling takes the form pI , S 2 , .…”

Section: Existence and Uniqueness Of Fréchet Meansmentioning

confidence: 99%

“…The Wasserstein formalism allows one to construct an alternative algorithm, that is also similar in spirit to generalised Procrustes analysis: instead of averaging pairwise SVD matchings, one averages pairwise optimal transport maps. This algorithm was proposed independently and concurrently by Zemel and Panaretos [60] and Álvarez-Esteban et al [4] in a finite dimensional setting. Though it can be applied to any finite collection of measures in Wasserstein space, we shall outline it here in the setup of covariance operators only (i.e., for collections of centred Gaussian measures).…”

Section: Computation Of Fréchet Means: Procrustes Algorithms and Grad...mentioning

confidence: 99%

“…In finite dimensions 6 , the Wasserstein-inspired algorithm is shown [60,4] to converge to the unique Fréchet mean Σ of Σ 1 , . .…”

Section: Computation Of Fréchet Means: Procrustes Algorithms and Grad...mentioning

confidence: 99%

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Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

2018

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Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen-Loève expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on covariance operators, and the intrinsic infinitedimensionality of these operators. In this paper, we describe the manifold geometry of the space of trace-class infinite-dimensional covariance operators and associated key statistical properties, under the recently proposed infinite-dimensional version of the Procrustes metric. We identify this space with that of centred Gaussian processes equipped with the Wasserstein metric of optimal transportation. The identification allows us to provide a complete description of those aspects of this manifold geometry that are important in terms of statistical inference, and establish key properties of the Fréchet mean of a random sample of covariances, as well as generative models that are canonical for such metrics and link with the problem of registration of functional data.

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Section: Existence and Uniqueness Of Fréchet Meansmentioning

confidence: 84%

Section: Existence and Uniqueness Of Fréchet Meansmentioning

confidence: 99%

Section: Computation Of Fréchet Means: Procrustes Algorithms and Grad...mentioning

confidence: 99%

“…In finite dimensions 6 , the Wasserstein-inspired algorithm is shown [60,4] to converge to the unique Fréchet mean Σ of Σ 1 , . .…”

Section: Computation Of Fréchet Means: Procrustes Algorithms and Grad...mentioning

confidence: 99%

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Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

2018

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“…Applications for optimization of mass transport for multiple marginals arise in a variety of fields: probabilistic Fréchet means in statistics [20,29], team matching in game theory [8,9,22], option prices and price equilibria in economics [3,10], and electron correlations in matter physics [11], to name a few. The variety of applications has led to considerable activity on the topic; a search for 'optimal mass transport problems for several marginals' on GoogleScholar returns about 19,600 hits, including a seminal book by Villani [27], which has over 2,800 citations at the time of this writing.…”

Section: Introductionmentioning

confidence: 99%

Improved Linear Programs for Discrete Barycenters

Borgwardt

Patterson

2020

INFORMS Journal on Optimization

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Discrete barycenters are the optimal solutions to mass transport problems for a set of discrete measures. Such transport problems arise in many applications of operations research and statistics. The best known algorithms for exact barycenters are based on linear programming, but these programs scale exponentially in the number of measures, making them prohibitive for practical purposes.In this paper, we improve on these algorithms. First, by using the optimality conditions to restrict the search space, we provide a reduced linear program that contains dramatically fewer variables compared to previous formulations. Second, we recall a proof from the literature, which lends itself to a linear program that has not been considered for computations. We show that this second formulation is the best model for data in general position. Third, we combine the two programs into a single hybrid model that retains the best properties of both formulations for partially structured data. We study these models through an analysis of their scaling in size, the hardness of the required preprocessing, and computational experiments. In doing so, we show that each of the improved linear programs becomes the best model for different types of data.

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An LP-based, Strongly-Polynomial 2-Approximation Algorithm for Sparse Wasserstein Barycenters

Borgwardt¹

2017

Preprint

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Wasserstein barycenters correspond to optimal solutions of transportation problems for several marginals, which arise in a wide range of fields. In many applications, data is given as a set of probability measures with finite support. The discrete barycenters in this setting exhibit favorable properties: All barycenters have finite support, and there always is one with a provably sparse support. Further, each barycenter allows a non-mass splitting optimal transport to each of the marginals. It is open whether the computation of a discrete barycenter is possible in polynomial time. The best known exact algorithms are based on linear programming, but the sizes of these programs scale exponentially. In this paper, we prove that there is a strongly polynomial, tight 2-approximation, based on restricting the possible support of an approximate barycenter to the union of supports of the measures. The resulting measure is sparse, but an optimal transport will generally split mass. We then exhibit a strongly polynomial algorithm to improve this measure to another one, for which there exists a non-mass splitting transport of lower cost. Finally, we present an iterative scheme that alternates between these two algorithms. It terminates with a 2-approximation that has a sparse support and an associated non-mass splitting optimal transport. We conclude with some practical computations.

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Fréchet Means and Procrustes Analysis in Wasserstein Space

Cited by 3 publications

References 0 publications

Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes

Improved Linear Programs for Discrete Barycenters

An LP-based, Strongly-Polynomial 2-Approximation Algorithm for Sparse Wasserstein Barycenters

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