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WASSERSTEIN MIRROR GRADIENT FLOW AND THE LIMIT OF THE SINKHORN ALGORITHM

Abstract Number:

3266 

Submission Type:

Contributed Abstract 

Contributed Abstract Type:

Paper 

Participants:

Nabarun Deb (1), Young-Heon Kim (2), Soumik Pal (3), Geoffrey Schiebinger (2)

Institutions:

(1) University of Chicago, N/A, (2) University of British Columbia, N/A, (3) University of Washington, Seattle, N/A

Co-Author(s):

Young-Heon Kim  
University of British Columbia
Soumik Pal  
University of Washington, Seattle
Geoffrey Schiebinger  
University of British Columbia

First Author:

Nabarun Deb  
University of Chicago

Presenting Author:

Nabarun Deb  
N/A

Abstract Text:

We prove that the sequence of marginals obtained from the iterations of the Sinkhorn algorithm or the iterative proportional fitting procedure
(IPFP) on joint densities, converges to an absolutely continuous curve on the 2-Wasserstein space, as the regularization parameter ε goes to zero and the
number of iterations is scaled as 1/ε (and other technical assumptions). This limit, which we call the Sinkhorn flow, is an example of a Wasserstein mirror
gradient flow, a concept we introduce here inspired by the well-known Euclidean mirror gradient flows. In the case of Sinkhorn, the gradient is that of
the relative entropy functional with respect to one of the marginals and the mirror is half of the squared Wasserstein distance functional from the other
marginal. Interestingly, the norm of the velocity field of this flow can be interpreted as the metric derivative with respect to the linearized optimal transport
(LOT) distance. We derive conditions for exponential convergence for this limiting flow. We also construct a Mckean-Vlasov diffusion whose
marginal distributions follow the Sinkhorn flow.

Keywords:

Entropy regularized optimal transport|Mckean-Vlasov diffusion|mirror descent|parabolic Monge-Amp`ere|Sinkhorn algorithm|Wasserstein mirror gradient flow

Sponsors:

IMS

Tracks:

Foundations of Machine Learning

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