The Square Root Normal Field Distance and Unbalanced Optimal Transport

Bauer, Martin; Hartman, Emmanuel; Klassen, Eric

doi:10.48550/arxiv.2105.06510

Cited by 2 publications

(3 citation statements)

References 39 publications

(80 reference statements)

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“…We now introduce a static formulation called semi-couplings introduced in [31] and used in [6] to derive an explicit algorithm. Semi-couplings are couples of nonnegative Radon measures γ, π on X × X such that γ 1 = α and π 2 = β.…”

Section: Dynamic Formulationmentioning

confidence: 99%

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Unbalanced Optimal Transport, from Theory to Numerics

Séjourné¹,

Peyré²,

Vialard³

2022

Preprint

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Optimal Transport (OT) has recently emerged as a central tool in data sciences to compare in a geometrically faithful way point clouds and more generally probability distributions. The wide adoption of OT into existing data analysis and machine learning pipelines is however plagued by several shortcomings. This includes its lack of robustness to outliers, its high computational costs, the need for a large number of samples in high dimension and the difficulty to handle data in distinct spaces. In this review, we detail several recently proposed approaches to mitigate these issues. We insist in particular on unbalanced OT, which compares arbitrary positive measures, not restricted to probability distributions (i.e. their total mass can vary). This generalization of OT makes it robust to outliers and missing data. The second workhorse of modern computational OT is entropic regularization, which leads to scalable algorithms while lowering the sample complexity in high dimension. The last point presented in this review is the Gromov-Wasserstein (GW) distance, which extends OT to cope with distributions belonging to different metric spaces. The main motivation for this review is to explain how unbalanced OT, entropic regularization and GW can work hand-in-hand to turn OT into efficient geometric loss functions for data sciences.

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Section: Dynamic Formulationmentioning

confidence: 99%

“…This is the main reason why this formulation is frequently used in applications, see Section 4 for more details. The semicoupling formulation, mentioned Section 3.2, can be useful in the WFR/KL setting to derive an algorithm [6].…”

Section: Computational Complexitymentioning

confidence: 99%

Unbalanced Optimal Transport, from Theory to Numerics

Séjourné¹,

Peyré²,

Vialard³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Some previous works directly adress the unregularized case ε = 0 and some specific entropy functionals. An alternate minimization scheme is proposed in [Bauer et al, 2021] for the special case of KL divergences, but without quantitative rates of convergence. In the case of a quadratic divergence ϕ(x) = ρ(x − 1) 2 , it is possible to compute the whole path of solutions for varying ρ using LARS algorithm [Chapel et al, 2021].…”

Section: Introductionmentioning

confidence: 99%

Faster Unbalanced Optimal Transport: Translation invariant Sinkhorn and 1-D Frank-Wolfe

Séjourné¹,

Vialard²,

Peyré³

2022

Preprint

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Unbalanced optimal transport (UOT) extends optimal transport (OT) to take into account mass variations to compare distributions. This is crucial to make OT successful in ML applications, making it robust to data normalization and outliers. The baseline algorithm is Sinkhorn, but its convergence speed might be significantly slower for UOT than for OT. In this work, we identify the cause for this deficiency, namely the lack of a global normalization of the iterates, which equivalently corresponds to a translation of the dual OT potentials. Our first contribution leverages this idea to develop a provably accelerated Sinkhorn algorithm (coined "translation invariant Sinkhorn") for UOT, bridging the computational gap with OT. Our second contribution focusses on 1-D UOT and proposes a Frank-Wolfe solver applied to this translation invariant formulation. The linear oracle of each steps amounts to solving a 1-D OT problems, resulting in a linear time complexity per iteration. Our last contribution extends this method to the computation of UOT barycenter of 1-D measures. Numerical simulations showcase the convergence speed improvement brought by these three approaches.

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The Square Root Normal Field Distance and Unbalanced Optimal Transport

Cited by 2 publications

References 39 publications

Unbalanced Optimal Transport, from Theory to Numerics

Unbalanced Optimal Transport, from Theory to Numerics

Faster Unbalanced Optimal Transport: Translation invariant Sinkhorn and 1-D Frank-Wolfe

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