Hessian metric via transport information geometry

Wuchen Li UCLA

Differential Geometry Information Theory Mathematical Physics mathscidoc:2004.22001

arXiv:2003.10526, 2020.3
We propose to study the Hessian metric of given functional in the space of probability space embedded with L^2–Wasserstein (optimal transport) metric. We name it transport Hessian metric, which contains and extends the classical L^2–Wasserstein metric. We formulate several dynamical systems associated with transport Hessian metrics. Several connections between transport Hessian metrics and math physics equations are discovered. E.g., the transport Hessian gradient flow, including Newton’s flow, formulates a mean-field kernel Stein variational gradient flow; The transport Hessian Hamiltonian flow of negative Boltzmann-Shannon entropy forms the Shallow water’s equation; The transport Hessian gradient flow of Fisher information forms the heat equation. Several examples and closed-form solutions of finite-dimensional transport Hessian metrics and dynamics are presented.
Optimal transport; Information geometry; Geometric analysis; Machine learning
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@inproceedings{wuchen2020hessian,
  title={Hessian metric via transport information geometry},
  author={Wuchen Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200427013354574759642},
  booktitle={arXiv:2003.10526},
  year={2020},
}
Wuchen Li. Hessian metric via transport information geometry. 2020. In arXiv:2003.10526. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200427013354574759642.
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