Interpolating between Optimal Transport and MMD using Sinkhorn Divergences
By Jean Feydy, Thibault Séjourné, François-Xavier Vialard, Shun-ichi Amari, Alain Trouv'e, G. Peyr'e. International Conference on Artificial Intelligence and Statistics, 2018
706 citations recorded by the source corpus.
Comparing probability distributions is a fundamental problem in data sciences. Simple norms and divergences such as the total variation and the relative entropy only compare densities in a point-wise manner and fail to capture the geometric nature of the problem. In sharp contrast, Maximum Mean Discrepancies (MMD) and Optimal Transport distances (OT) are two classes of distances between measures that take into account the geometry of the underlying space and metrize the convergence in law. This paper studies the Sinkhorn divergences, a family of geometric divergences that interpolates between MMD and OT. Relying on a new notion of geometric entropy, we provide theoretical guarantees for these divergences: positivity, convexity and metrization of the convergence in law. On the practical side, we detail a numerical scheme that enables the large scale application of these divergences for machine learning: on the GPU, gradients of the Sinkhorn loss can be computed for batches of a million samples.
Subfields: Artificial Intelligence, Computational Theory and Mathematics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/dafb021ba04731669e9473477688345ef4962e63
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Citations in the corpus
- Wasserstein Barycenter Model Ensembling (2019, methodology): “The algorithm can be further accelerated by computing Sinkhorn divergences in batches as pointed in (Feydy et al., 2018).”
- A formula for the time derivative of the entropic cost and applications (2019, background): “Although its asymptotic regime has been the object of recent studies in connection with the ergodic behavior of Schrödinger bridges [3] and the limiting behavior of Sinkhorn divergence [12], very few results are available.”
- Generalized Energy Based Models (2020, methodology): “We then compared the performance of the GAN and GEBM using the Sinkhorn divergence Feydy et al. (2019) between each model and the data distribution.”
- Super-efficiency of automatic differentiation for functions defined as a minimum (2020, methodology): “It is used for instance in dictionary learning [Olshausen and Field, 1997, Mairal et al., 2010] or in optimal transport [Feydy et al., 2019].”
- PriorGrad: Improving Conditional Denoising Diffusion Models with Data-Dependent Adaptive Prior (2021, background): “…et al., 2020), Mel cepstral distortion (MCD) (Kubichek, 1993) and F 0 root mean square error (F 0 RMSE) and a debiased Sinkhorn divergence (Feydy et al., 2019) between the prior and the ground-truth S ( x T , x 0 ) , or the generated samples and the ground-truth S (˜ x 0 , x 0 ) ,…”
- Partially Does It: Towards Scene-Level FG-SBIR with Partial Input (2022, methodology): “Ours-Local-OT replace MMD with the more accurate [22] optimal transport to compute region-wise associativity between localised features.”
- Computational Optimal Transport (2018, positive): “It is proved in [Feydy et al., 2019] that if e−c/ε is a positive kernel, then a related corrected divergence (obtained by using LC in place of PC) is positive.”