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Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance

Luise, G; Rudi, A; Pontil, M; Ciliberto, C; (2018) Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance. In: Bengio, S and Wallach, H and Larochelle, H and Grauman, K and CesaBianchi, N and Garnett, R, (eds.) Advances in Neural Information Processing Systems 31 (NIPS 2018). Neural Information Processing Systems Foundation, Inc.: Montréal, Canada. Green open access

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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.

Type: Proceedings paper
Title: Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance
Event: Advances in Neural Information Processing Systems 31 (NIPS 2018)
Location: Montreal, Canada
Dates: 2nd-8th December 2018
Open access status: An open access version is available from UCL Discovery
Publisher version: https://papers.nips.cc/paper/7827-differential-pro...
Language: English
Additional information: This version is the version of record. For information on re-use, please refer to the publisher's terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10073325
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