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Convolutional wasserstein distances

Justin Solomon, Fernando de Goes, Gabriel Peyré, Marco Cuturi, Adrian Butscher, Andy Nguyen, Tao Du, Leonidas Guibas
2015 ACM Transactions on Graphics  
Abstract This paper introduces a new class of algorithms for optimization problems involving optimal transportation over geometric domains.  ...  To this end, we approximate optimal transportation distances using entropic regularization.  ...  Regularized Optimal Transportation In this section, we present a modification of Wasserstein distances suitable for computation on geometric domains.  ... 
doi:10.1145/2766963 fatcat:ahfl5hkejjcgpk72ob27ixytdy

Geometric Dataset Distances via Optimal Transport [article]

David Alvarez-Melis, Nicolò Fusi
2020 arXiv   pre-print
This distance relies on optimal transport, which provides it with rich geometry awareness, interpretable correspondences and well-understood properties.  ...  Current methods to quantify it are often heuristic, make strong assumptions on the label sets across the tasks, and many are architecture-dependent, relying on task-specific optimal parameters (e.g., require  ...  Optimal Transport for Domain Adaptation Using label information to guide the optimal transport problem towards class-coherent matches has been explored before, e. g., by enforcing group-norm penalties  ... 
arXiv:2002.02923v1 fatcat:3en6i66g4rahlaomanqkhfxrwu

Augmented Sliced Wasserstein Distances [article]

Xiongjie Chen, Yongxin Yang, Yunpeng Li
2022 arXiv   pre-print
We show that the hypersurfaces can be optimized by gradient ascent efficiently.  ...  The sliced Wasserstein distance and its variants improve the computational efficiency through the random projection, yet they suffer from low accuracy if the number of projections is not sufficiently large  ...  We proposed a novel variant of the sliced Wasserstein distance, namely the augmented sliced Wasserstein distance (ASWD), which is flexible, has a high projection efficiency, and generalizes well.  ... 
arXiv:2006.08812v7 fatcat:vzfdrmfnrvbyhlyeslqff6fzvm

Image Generation Via Minimizing Fréchet Distance in Discriminator Feature Space [article]

Khoa D. Doan and Saurav Manchanda and Fengjiao Wang and Sathiya Keerthi and Avradeep Bhowmik and Chandan K. Reddy
2020 arXiv   pre-print
For distributional distance, we employ one of two choices: the Fréchet distance or direct optimal transport (OT); these respectively lead us to two new GAN methods: Fréchet-GAN and OT-GAN.  ...  representative primal and dual GAN approaches based on the Wasserstein distance.  ...  Alternatively, from the primal domain, [15] estimates the empirical Wasserstein distance directly from the samples by solving the Optimal Transport (OT) problem.  ... 
arXiv:2003.11774v2 fatcat:3u7icmhdcvcenpeirieyusdrt4

Diffusion Earth Mover's Distance and Distribution Embeddings [article]

Alexander Tong, Guillaume Huguet, Amine Natik, Kincaid MacDonald, Manik Kuchroo, Ronald Coifman, Guy Wolf, Smita Krishnaswamy
2021 arXiv   pre-print
Here, Diffusion EMD can derive distances between patients on the manifold of cells at least two orders of magnitude faster than equally accurate methods.  ...  We model the datasets as distributions supported on common data graph that is derived from the affinity matrix computed on the combined data.  ...  Convolutional wasserstein distances: Efficient optimal transportation on geometric domains. ACM Transactions on Graphics, 34 (4):1-11, 2015.  ... 
arXiv:2102.12833v2 fatcat:s663mkw2ine6fonnujgxcnabgy

LCS Graph Kernel Based on Wasserstein Distance in Longest Common Subsequence Metric Space [article]

Jianming Huang, Zhongxi Fang, Hiroyuki Kasai
2020 arXiv   pre-print
We also combine it with optimal transport theory to extract more in-depth features of graphs.  ...  Although this strategy provides an efficient means of extracting graph topological features, it brings excessive amounts of information that might greatly reduce its accuracy when dealing with large-scale  ...  The main motivation is that the Wasserstein distance based on optimal transport theory can assign less importance to such dissimilar paths between two graphs.  ... 
arXiv:2012.03612v1 fatcat:6i3n2urhuvdhzc374udvbia7iq

Wasserstein GANs Work Because They Fail (to Approximate the Wasserstein Distance) [article]

Jan Stanczuk, Christian Etmann, Lisa Maria Kreusser, Carola-Bibiane Schönlieb
2021 arXiv   pre-print
Wasserstein GANs are based on the idea of minimising the Wasserstein distance between a real and a generated distribution.  ...  to approximate the Wasserstein distance.  ...  distances (Deshpande et al., 2018) , (Paty & Cuturi, 2019) . • Replacing the L 2 -norm by a perceptually meaningful notion of distance: We intend to explore an optimal transport distance based on L  ... 
arXiv:2103.01678v4 fatcat:lzbp2t545jabzk2ategxvr6qhu

Computing Kantorovich-Wasserstein Distances on d-dimensional histograms using (d+1)-partite graphs [article]

Gennaro Auricchio, Federico Bassetti, Stefano Gualandi, Marco Veneroni
2019 arXiv   pre-print
On these types of instances, our approach is competitive with state-of-the-art optimal transport algorithms.  ...  This paper presents a novel method to compute the exact Kantorovich-Wasserstein distance between a pair of d-dimensional histograms having n bins each.  ...  Acknowledgments We are deeply indebted to Giuseppe Savaré, for introducing us to optimal transport and for many stimulating discussions and suggestions.  ... 
arXiv:1805.07416v2 fatcat:z6ys7k3rc5h2hfyprpwhd5ylp4

Context Mover's Distance Barycenters: Optimal Transport of Contexts for Building Representations [article]

Sidak Pal Singh, Andreas Hug, Aymeric Dieuleveut, Martin Jaggi
2020 arXiv   pre-print
This enables us to consider representation learning from the perspective of Optimal Transport and take advantage of its tools such as Wasserstein distance and barycenters.  ...  transport between contexts and (d) easy applicability on top of existing point embedding methods.  ...  SPS is indebted to Marco Cuturi for teaching him about Optimal Transport and Honda Foundation for sponsoring that visit.  ... 
arXiv:1808.09663v6 fatcat:sojmvsoyo5d3rgcetfrloae6dm

Wasserstein Distance, Fourier Series and Applications [article]

Stefan Steinerberger
2020 arXiv   pre-print
We study the Wasserstein metric W_p, a notion of distance between two probability distributions, from the perspective of Fourier Analysis and discuss applications.  ...  Moreover, we show that for a Laplacian eigenfunction -Δ_g ϕ_λ = λϕ_λ on a compact Riemannian manifold W_p(max{ϕ_λ, 0}dx, max{-ϕ_λ, 0} dx) ≲_p √(logλ/λ)ϕ_λ_L^1^1/p which is at most a factor √(logλ) away  ...  Are the L p −norms on the right-hand side optimal? Is there a more elementary proof not passing through the notion of Wasserstein distance?  ... 
arXiv:1803.08011v3 fatcat:mw2lvnuqubfrdibmtskavks6de

Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance [article]

Giulia Luise, Alessandro Rudi, Massimiliano Pontil, Carlo Ciliberto
2018 arXiv   pre-print
On the other hand, the gradient formula allows us to efficiently solve learning and optimization problems in practice. Promising preliminary experiments complement our analysis.  ...  Applications of optimal transport have recently gained remarkable attention thanks to the computational advantages of entropic regularization.  ...  Optimal transport theory investigates how to compare probability measures over a domain X.  ... 
arXiv:1805.11897v1 fatcat:6w7wju3mhnbx7kcb26rg6iutdy

PSF field learning based on Optimal Transport Distances [article]

F. M. Ngolè Mboula, J.-L. Starck
2017 arXiv   pre-print
Contributions: we propose an interpolation framework based on Sliced Optimal Transport. A non-linear dimension reduction is first performed based on local pairwise approximated Wasserstein distances.  ...  Finally, the interpolated PSFs are calculated as approximated Wasserstein barycenters.  ...  We propose an intuitive framework for learning a PSFs set underlying geometry using Optimal Transport distances.  ... 
arXiv:1703.06066v1 fatcat:23z77eezpvfdrh3mq73i7bjqie

Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications [article]

Sloan Nietert, Ziv Goldfeld, Kengo Kato
2021 arXiv   pre-print
Our proof relies on controlling 𝖶_p^(σ) by a pth-order smooth Sobolev distance 𝖽_p^(σ) and deriving the limit distribution of √(n) 𝖽_p^(σ)(μ̂_n,μ), for all dimensions d.  ...  To combat the curse of dimensionality when estimating these distances from data, recent work has proposed smoothing out local irregularities in the measured distributions via convolution with a Gaussian  ...  The Wasserstein distance is an instance of the Kantorovich optimal transport problem, defined for Borel probability measures µ and ν on R d with cost c : R d × R d → R as inf π∈Π(µ,ν) R d c(x, y) dπ(x,  ... 
arXiv:2101.04039v3 fatcat:2llbhev6sbbulan35g56kmjrse

Faster Wasserstein Distance Estimation with the Sinkhorn Divergence [article]

Lenaic Chizat
2020 arXiv   pre-print
The squared Wasserstein distance is a natural quantity to compare probability distributions in a non-parametric setting.  ...  This quantity is usually estimated with the plug-in estimator, defined via a discrete optimal transport problem which can be solved to ϵ-accuracy by adding an entropic regularization of order ϵ and using  ...  On the rate of convergence in Wasserstein distance of the empirical measure.  ... 
arXiv:2006.08172v2 fatcat:5lo2o7whejazrkaakblb7ygc5i

Fused Gromov-Wasserstein Distance for Structured Objects

Titouan Vayer, Laetitia Chapel, Remi Flamary, Romain Tavenard, Nicolas Courty
2020 Algorithms  
The Kantorovitch formulation, leading to the Wasserstein distance, focuses on the features of the elements of the objects, but treats them independently, whereas the Gromov–Wasserstein distance focuses  ...  Optimal transport theory has recently found many applications in machine learning thanks to its capacity to meaningfully compare various machine learning objects that are viewed as distributions.  ...  This allows their comparison within the Optimal Transport (OT) framework that provides a meaningful way of comparing distributions by capturing the underlying geometric properties of the space through  ... 
doi:10.3390/a13090212 fatcat:mryu4dhkp5dsvmfmverkh3gpky
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