Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions

Alain Durmus; Antoine Liutkus; Fabian-Robert St\"oter; Szymon Majewski; Umut \c{S}im\c{s}ekli

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Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions

Antoine Liutkus, Umut Şimşekli, Szymon Majewski, Alain Durmus, Fabian-Robert Stöter

Machine Learning · Submitted 21 Jun 2018

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The authors' main claim Our experimental results support our theory and show that our algorithm is able to successfully capture the structure of different types of data distributions.

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arXiv ID1806.08141 (version 2)

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CitationAlain Durmus, Antoine Liutkus, Fabian-Robert St\"oter, Szymon Majewski, Umut \c{S}im\c{s}ekli. "Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions." arXiv:1806.08141 (2018). https://pith.science/paper/6H7LYKFJ

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What the abstract itself states, as probabilities: claims first 0.94 · states limitation 0.08 · human participants 0.03 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.16 · compares with prior results 0.51 · mentions code or data availability 0.02.

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By building upon the recent theory that established the connection between implicit generative modeling (IGM) and optimal transport, in this study, we propose a novel parameter-free algorithm for learning the underlying distributions of complicated datasets and sampling from them. The proposed algorithm is based on a functional optimization problem, which aims at finding a measure that is close to the data distribution as much as possible and also expressive enough for generative modeling purposes. We formulate the problem as a gradient flow in the space of probability measures. The connections between gradient flows and stochastic differential equations let us develop a computationally efficient algorithm for solving the optimization problem. We provide formal theoretical analysis where we prove finite-time error guarantees for the proposed algorithm. To the best of our knowledge, the proposed algorithm is the first nonparametric IGM algorithm with explicit theoretical guarantees. Our experimental results support our theory and show that our algorithm is able to successfully capture the structure of different types of data distributions.

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s0	By building upon the recent theory that established the connection between implicit generative modeling (IGM) and optimal transport, in this study, we propose a novel parameter-free algorithm for learning the underlying distributions of complicated datasets and sampling from them.	contribution 1.00	0.89	0.33	asserted 1.00
s1	The proposed algorithm is based on a functional optimization problem, which aims at finding a measure that is close to the data distribution as much as possible and also expressive enough for generative modeling purposes.	method 0.87	0.80	0.08	asserted 1.00
s2	We formulate the problem as a gradient flow in the space of probability measures.	method 0.95	0.88	0.05	asserted 1.00
s3	The connections between gradient flows and stochastic differential equations let us develop a computationally efficient algorithm for solving the optimization problem.	method 0.96	0.82	0.05	asserted 0.97
s4	We provide formal theoretical analysis where we prove finite-time error guarantees for the proposed algorithm.	contribution 0.88	0.96	0.12	asserted 1.00
s5	To the best of our knowledge, the proposed algorithm is the first nonparametric IGM algorithm with explicit theoretical guarantees.	contribution 0.78	0.52	0.36	asserted 0.95
s6	Our experimental results support our theory and show that our algorithm is able to successfully capture the structure of different types of data distributions.	result 0.94	0.97	0.40	asserted 0.99

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Original abstract

By building upon the recent theory that established the connection between implicit generative modeling (IGM) and optimal transport, in this study, we propose a novel parameter-free algorithm for learning the underlying distributions of complicated datasets and sampling from them. The proposed algorithm is based on a functional optimization problem, which aims at finding a measure that is close to the data distribution as much as possible and also expressive enough for generative modeling purposes. We formulate the problem as a gradient flow in the space of probability measures. The connections between gradient flows and stochastic differential equations let us develop a computationally efficient algorithm for solving the optimization problem. We provide formal theoretical analysis where we prove finite-time error guarantees for the proposed algorithm. To the best of our knowledge, the proposed algorithm is the first nonparametric IGM algorithm with explicit theoretical guarantees. Our experimental results support our theory and show that our algorithm is able to successfully capture the structure of different types of data distributions.

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8 arXiv papers cite this in 16 sentences

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Theorem [ref] is a generalization of Theorem 2 of [ref] and Theorem 5.6.1 of [ref] , which we retrieve when $σ→0$.

Differentially Private Gradient Flow based on the Sliced Wasserstein Distance · Gradient Flow of the Smoothed Sliced Wasserstein Distance · 0.85

Uses its method or tool 1 paper

The experiments performed on the MNIST and FashionMNIST datasets use the same autoencoder architecture as the framework proposed by [ref] .

Differentially Private Gradient Flow based on the Sliced Wasserstein Distance · Settings and Datasets · 1.00

Cites it as support 1 paper

However, the linear nature of these projections does not guarantee an efficient evaluation of the sliced-Wasserstein distance: in very high-dimensional settings, the data often lives in a thin manifold and the number of randomly chosen linear projections required to capture the structure of the data distribution grows very quickly [ref] .

Generalized Sliced Wasserstein Distances · Introduction · 0.98

Cites it as background 4 papers

In a non-DP scenario, gradient flows are commonly employed to address minimization problems like the one described in <ref> [ref] and are a viable alternative to other generative models [ref] .

Differentially Private Gradient Flow based on the Sliced Wasserstein Distance · Introduction · 0.87
Recently, Optimal Transport-based adversarial divergences have attracted significant attention from the GANs literature, such as the Wasserstein distance [ref] , the Sliced-Wasserestein distance [ref] or the Sinkhorn divergence [ref] .

Generalization Properties of Optimal Transport GANs with Latent Distribution Learning · Background · 0.99
Recently new promising classes of algorithms based on the theory of gradient flows takes the form of McKean-Vlasov ODEs or SDEs [ref] .

New particle representations for ergodic McKean-Vlasov SDEs · Introduction · 0.88

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Compares with 2 papers

These results are obtained by running different IGM approaches on the MNIST dataset, namely GAN [ref] , Wasserstein GAN (W-GAN) [ref] and the Sliced-Wasserstein Generator (SWG) [ref] .

Generative Modeling using the Sliced Wasserstein Distance · Experiments on real data · 0.96
This aspect is reminiscent of the compressive learning methodology [ref] , except we exploit quantiles of random projections here, instead of random generalized moments as done there.

Compressive Statistical Learning with Random Feature Moments · Approximate Euler-Maruyama discretization · 0.93

Cites as background 2 papers

Implicit generative modeling (IGM) [ref] has become very popular recently and has proven successful in various fields; variational auto-encoders (VAE) [ref] and generative adversarial networks (GAN) [ref] being its two well-known examples.

Auto-Encoding Variational Bayes · Introduction · 1.00
Implicit generative modeling (IGM) [ref] has become very popular recently and has proven successful in various fields; variational auto-encoders (VAE) [ref] and generative adversarial networks (GAN) [ref] being its two well-known examples.

Learning in Implicit Generative Models · Introduction · 1.00

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