2022

Neural Optimal Transport

Korotin, Alexander, Selikhanovych, Daniil, Burnaev, Evgeny

Understand

We present a novel neural-networks-based algorithm to compute optimal transport maps and plans for strong and weak transport costs.

To justify the usage of neural networks, we prove that they are universal approximators of transport plans between probability distributions.
We evaluate the performance of our optimal transport algorithm on toy examples and on the unpaired image-to-image translation.

Built on

On the translocation of masses
Leonid Kantorovitch · 1958
Earlier work this paper cites.
Characterization of the subdifferentials of convex functions
Ralph Rockafellar · 1966
Earlier work this paper cites.
Integral functionals, normal integrands and measurable selections
R Tyrrell Rockafellar · 1976
Earlier work this paper cites.
Foundations of modern probability , volume 2
Olav Kallenberg · 1997
Earlier work this paper cites.
Real analysis: modern techniques and their applications , volume 40
Gerald B Folland · 1999
Earlier work this paper cites.
Optimal transport: old and new , volume 338
Cédric Villani · 2008
Earlier work this paper cites.
Generative adversarial nets
Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio · 2014
Earlier work this paper cites.
Adam: A method for stochastic optimization
Original
Diederik P Kingma and Jimmy Ba · 2014
Earlier work this paper cites.
A survey of the schrödinger problem and some of its connections with optimal transport
Christian Léonard · 2014
Earlier work this paper cites.
Fine-grained visual comparisons with local learning
Aron Yu and Kristen Grauman · 2014
Earlier work this paper cites.
Learning deep features for scene recognition using places database
Bolei Zhou, Agata Lapedriza, Jianxiong Xiao, Antonio Torralba, and Aude Oliva · 2014
Earlier work this paper cites.
Deep learning face attributes in the wild
Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang · 2015
Earlier work this paper cites.
U-net: Convolutional networks for biomedical image segmentation
Olaf Ronneberger, Philipp Fischer, and Thomas Brox · 2015
Earlier work this paper cites.
Optimal transport for applied mathematicians
Filippo Santambrogio · 2015
Earlier work this paper cites.
Lsun: Construction of a large-scale image dataset using deep learning with humans in the loop
Original
Fisher Yu, Ari Seff, Yinda Zhang, Shuran Song, Thomas Funkhouser, and Jianxiong Xiao · 2015
Earlier work this paper cites.
Stochastic optimization for large-scale optimal transport
Aude Genevay, Marco Cuturi, Gabriel Peyré, and Francis Bach · 2016
Earlier work this paper cites.
Perceptual losses for real-time style transfer and super-resolution
Justin Johnson, Alexandre Alahi, and Li Fei-Fei · 2016
Earlier work this paper cites.
f-GAN: Training generative neural samplers using variational divergence minimization
Sebastian Nowozin, Botond Cseke, and Ryota Tomioka · 2016
Earlier work this paper cites.

Similar

Input convex neural networks
Brandon Amos, Lei Xu, and J Zico Kolter · 2017
Cited alongside, same era.
Wasserstein generative adversarial networks
Martin Arjovsky, Soumith Chintala, and Léon Bottou · 2017
Cited alongside, same era.
One-sided unsupervised domain mapping
Sagie Benaim and Lior Wolf · 2017
Cited alongside, same era.
Unbalanced optimal transport: Models, numerical methods, applications
Lenaic Chizat · 2017
Cited alongside, same era.
Kantorovich duality for general transport costs and applications
Nathael Gozlan, Cyril Roberto, Paul-Marie Samson, and Prasad Tetali · 2017
Cited alongside, same era.
Improved training of Wasserstein GANs
Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron C Courville · 2017
Cited alongside, same era.
GANs trained by a two time-scale update rule converge to a local nash equilibrium
Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter · 2017
Cited alongside, same era.
Learning to discover cross-domain relations with generative adversarial networks
Taeksoo Kim, Moonsu Cha, Hyunsoo Kim, Jung Kwon Lee, and Jiwon Kim · 2017
Cited alongside, same era.
Unsupervised image-to-image translation networks
Ming-Yu Liu, Thomas Breuel, and Jan Kautz · 2017
Cited alongside, same era.
On the regularization of wasserstein gans
Original
Henning Petzka, Asja Fischer, and Denis Lukovnicov · 2017
Cited alongside, same era.
Large scale optimal transport and mapping estimation
Vivien Seguy, Bharath Bhushan Damodaran, Remi Flamary, Nicolas Courty, Antoine Rolet, and Mathieu Blondel · 2017
Cited alongside, same era.
Augmented cyclegan: Learning many-to-many mappings from unpaired data
Amjad Almahairi, Sai Rajeshwar, Alessandro Sordoni, Philip Bachman, and Aaron Courville · 2018
Cited alongside, same era.

Then

Wasserstein GAN with quadratic transport cost
Huidong Liu, Xianfeng Gu, and Dimitris Samaras · 2019
Later among the works it cites.
Guiding the one-to-one mapping in cyclegan via optimal transport
Guansong Lu, Zhiming Zhou, Yuxuan Song, Kan Ren, and Yong Yu · 2019
Later among the works it cites.
Computational optimal transport
Gabriel Peyré, Marco Cuturi, et al · 2019
Later among the works it cites.
2-Wasserstein approximation via restricted convex potentials with application to improved training for GANs
Original
Amirhossein Taghvaei and Amin Jalali · 2019
Later among the works it cites.
On scalable and efficient computation of large scale optimal transport
Yujia Xie, Minshuo Chen, Haoming Jiang, Tuo Zhao, and Hongyuan Zha · 2019
Later among the works it cites.
Scalable unbalanced optimal transport using generative adversarial networks
Karren D. Yang and Caroline Uhler · 2019
Later among the works it cites.
Stargan v2: Diverse image synthesis for multiple domains
Yunjey Choi, Youngjung Uh, Jaejun Yoo, and Jung-Woo Ha · 2020
Later among the works it cites.
On a mixture of brenier and strassen theorems
Nathael Gozlan and Nicolas Juillet · 2020
Later among the works it cites.
Denoising diffusion probabilistic models
Jonathan Ho, Ajay Jain, and Pieter Abbeel · 2020
Later among the works it cites.
Universal approximation with deep narrow networks
Patrick Kidger and Terry Lyons · 2020
Later among the works it cites.
Gmm-unit: Unsupervised multi-domain and multi-modal image-to-image translation via attribute gaussian mixture modeling
Original
Yahui Liu, Marco De Nadai, Jian Yao, Nicu Sebe, Bruno Lepri, and Xavier Alameda-Pineda · 2020
Later among the works it cites.
Large-scale optimal transport via adversarial training with cycle-consistency
Original
Guansong Lu, Zhiming Zhou, Jian Shen, Cheng Chen, Weinan Zhang, and Yong Yu · 2020
Later among the works it cites.
Optimal transport mapping via input convex neural networks
Ashok Makkuva, Amirhossein Taghvaei, Sewoong Oh, and Jason Lee · 2020
Later among the works it cites.
Coco-funit: Few-shot unsupervised image translation with a content conditioned style encoder
Kuniaki Saito, Kate Saenko, and Ming-Yu Liu · 2020
Later among the works it cites.
Score-based generative neural networks for large-scale optimal transport
Grady Daniels, Tyler Maunu, and Paul Hand · 2021
Later among the works it cites.
Scalable computation of monge maps with general costs
Jiaojiao Fan, Shu Liu, Shaojun Ma, Yongxin Chen, and Hao-Min Zhou · 2022
Closest in time.
Generative modeling with optimal transport maps
Litu Rout, Alexander Korotin, and Evgeny Burnaev · 2022
Closest in time.
Dual diffusion implicit bridges for image-to-image translation
Xuan Su, Jiaming Song, Chenlin Meng, and Stefano Ermon · 2023
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Reading the bibliography…