Tree-Sliced Variants of Wasserstein Distances

Le, Tam; Yamada, Makoto; Fukumizu, Kenji; Cuturi, Marco

doi:10.48550/arxiv.1902.00342

Cited by 3 publications

(3 citation statements)

References 34 publications

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“…As shown in Bachoc [46] many eigenvalues of the Wasserstein induced covariance matrix are negative. Three recent papers [47][48][49] have analysed the specific conditions in which the exponentiation of WST yields a positive definite kernel.…”

Section: Wasserstein Induced Kernelmentioning

confidence: 99%

On the use of Wasserstein distance in the distributional analysis of human decision making under uncertainty

Candelieri

Ponti

Giordani³

et al. 2022

Ann Math Artif Intell

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The key contribution of this paper is a theoretical framework to analyse humans’ decision-making strategies under uncertainty, and more specifically how human subjects manage the trade-off between information gathering (exploration) and reward seeking (exploitation) in particular active learning in a black-box optimization task. Humans’ decisions making according to these two objectives can be modelled in terms of Pareto rationality. If a decision set contains a Pareto efficient (dominant) strategy, a rational decision maker should always select the dominant strategy over its dominated alternatives. A distance from the Pareto frontier determines whether a choice is (Pareto) rational. The key element in the proposed analytical framework is the representation of behavioural patterns of human learners as a discrete probability distribution, specifically a histogram considered as a non-parametric estimate of discrete probability density function on the real line. Thus, the similarity between users can be captured by a distance between their associated histograms. This maps the problem of the characterization of humans’ behaviour into a space, whose elements are probability distributions, structured by a distance between histograms, namely the optimal transport-based Wasserstein distance. The distributional analysis gives new insights into human behaviour in search tasks and their deviations from Pareto rationality. Since the uncertainty is one of the two objectives defining the Pareto frontier, the analysis has been performed for three different uncertainty quantification measures to identify which better explains the Pareto compliant behavioural patterns. Beside the analysis of individual patterns Wasserstein has also enabled a global analysis computing the WST barycenters and performing k-means Wasserstein clustering.

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Section: Wasserstein Induced Kernelmentioning

confidence: 99%

On the use of Wasserstein distance in the distributional analysis of human decision making under uncertainty

Candelieri

Ponti

Giordani³

et al. 2022

Ann Math Artif Intell

View full text Add to dashboard Cite

show abstract

“…In what follows, to avoid confusion, we refer to the standard 1-Wasserstein distance between point sets as the optimal transport (OT) distance. Starting from [2] and [3], there has been a long line of work to approximate the OT-distance for Euclidean point sets using randomly shifted quadtrees (e.g, [7,19,15,22,28,1]). In particular, we consider two such approaches, the L 1 -embedding approach by [15], and the flowtree approach by [1].…”

Section: Introductionmentioning

confidence: 99%

Approximation algorithms for 1-Wasserstein distance between persistence diagrams

Chen¹,

Wang²

2021

Preprint

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Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially complex input object (be it a graph, an image, or a point set and so on) to a unified type of feature summary, called the persistence diagrams. One can then carry out downstream data analysis tasks using such persistence diagram representations.A key problem is to compute the distance between two persistence diagrams efficiently. In particular, a persistence diagram is essentially a multiset of points in the plane, and one popular distance is the so-called 1-Wasserstein distance between persistence diagrams. In this paper, we present two algorithms to approximate the 1-Wasserstein distance for persistence diagrams in near-linear time. These algorithms primarily follow the same ideas as two existing algorithms to approximate optimal transport between two finite point-sets in Euclidean spaces via randomly shifted quadtrees. We show how these algorithms can be effectively adapted for the case of persistence diagrams. Our algorithms are much more efficient than previous exact and approximate algorithms, both in theory and in practice, and we demonstrate its efficiency via extensive experiments. They are conceptually simple and easy to implement, and the code is publicly available in github.

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“…The Sliced Wasserstein distance is one of many approximations of the Wasserstein distance based on one-dimensional projections. We mention here the Generalized Sliced (Kolouri et al, 2019), Tree-Sliced (Le et al, 2019), and Subspace Robust (Paty and Cuturi, 2019) Wasserstein distances.…”

Section: Introductionmentioning

confidence: 99%

Minimax Confidence Intervals for the Sliced Wasserstein Distance

Manole,

Balakrishnan,

Wasserman

2019

Preprint

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The Wasserstein distance has risen in popularity in the statistics and machine learning communities as a useful metric for comparing probability distributions. We study the problem of uncertainty quantification for the Sliced Wasserstein distance-an easily computable approximation of the Wasserstein distance. Specifically, we construct confidence intervals for the Sliced Wasserstein distance which have finite-sample validity under no assumptions or mild moment assumptions, and are adaptive in length to the smoothness of the underlying distributions. We also bound the minimax risk of estimating the Sliced Wasserstein distance, and show that the length of our proposed confidence intervals is minimax optimal over appropriate distribution classes. To motivate the choice of these classes, we also study minimax rates of estimating a distribution under the Sliced Wasserstein distance. These theoretical findings are complemented with a simulation study.

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Tree-Sliced Variants of Wasserstein Distances

Cited by 3 publications

References 34 publications

On the use of Wasserstein distance in the distributional analysis of human decision making under uncertainty

On the use of Wasserstein distance in the distributional analysis of human decision making under uncertainty

Approximation algorithms for 1-Wasserstein distance between persistence diagrams

Minimax Confidence Intervals for the Sliced Wasserstein Distance

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