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Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance

Abstract : Applications of optimal transport have recently gained remarkable attention thanks to the computational advantages of entropic regularization. However, in most situations the Sinkhorn approximation of the Wasserstein distance is replaced by a regularized version that is less accurate but easy to differentiate. In this work we characterize the differential properties of the original Sinkhorn distance, proving that it enjoys the same smoothness as its regularized version and we explicitly provide an efficient algorithm to compute its gradient. We show that this result benefits both theory and applications: on one hand, high order smoothness confers statistical guarantees to learning with Wasserstein approximations. On the other hand, the gradient formula allows us to efficiently solve learning and optimization problems in practice. Promising preliminary experiments complement our analysis.
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https://hal.inria.fr/hal-01958887
Contributor : Alessandro Rudi <>
Submitted on : Wednesday, December 19, 2018 - 12:10:15 AM
Last modification on : Tuesday, September 22, 2020 - 3:47:44 AM
Long-term archiving on: : Wednesday, March 20, 2019 - 3:29:03 PM

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  • HAL Id : hal-01958887, version 1
  • ARXIV : 1805.11897

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Citation

Giulia Luise, Alessandro Rudi, Massimiliano Pontil, Carlo Ciliberto. Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance. NIPS 2018 - Advances in Neural Information Processing Systems, Dec 2018, Montreal, Canada. pp.5864-5874. ⟨hal-01958887⟩

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