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Hessian transport gradient flows We derive new gradient flows of divergence functions in the probability space embedded with a class of Riemannian metrics. The Riemannian metric tensor is built from the transported Hessian operator of an entropy function. The new gradient flow is a generalized Fokker–Planck equation and is associated with a stochastic differential equation that depends on the reference measure. Several examples of Hessian transport gradient flows and the associated stochastic differential equations are presented, including the ones for the reverse Kullback–Leibler divergence, $$\alpha$$ α -divergence, Hellinger distance, Pearson divergence, and Jenson–Shannon divergence. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

, Volume 6 (4) – Oct 28, 2019
20 pages

Publisher
Springer Journals
Subject
Mathematics; Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis
eISSN
2197-9847
DOI
10.1007/s40687-019-0198-9
Publisher site
See Article on Publisher Site

### Abstract

We derive new gradient flows of divergence functions in the probability space embedded with a class of Riemannian metrics. The Riemannian metric tensor is built from the transported Hessian operator of an entropy function. The new gradient flow is a generalized Fokker–Planck equation and is associated with a stochastic differential equation that depends on the reference measure. Several examples of Hessian transport gradient flows and the associated stochastic differential equations are presented, including the ones for the reverse Kullback–Leibler divergence, $$\alpha$$ α -divergence, Hellinger distance, Pearson divergence, and Jenson–Shannon divergence.

### Journal

Research in the Mathematical SciencesSpringer Journals

Published: Oct 28, 2019

### References

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