A placeholder for learning on curved spaces. Not discussed: learning OF curved spaces.
Also: learning where there is an a priori manifold seems to also be a usage here? See the work of, e.g. Nina Miolane and collaborators on the Geomstats project.
Girolami et al discuss Langevin Monte Carlo in this context.
The below headings may one day be filled in.
The unholy offspring of Fisher information and differential geometry, about which I know little except that it sounds like it should be intuitive. See also information criteria. I also know that even though this sounds intuitive, it is not mainstream and it has also not been especially useful to me even in places where it seemed that it should, at least not beyond the basic delta method.
Hamiltonian Monte Carlo
You can also discuss Hamiltonian Monte Carlo in this setting. I will not.
See natural gradients.
Albert Tarantola’s framing, from his maybe forthcoming manuscript. How does it relate to information geometry? I don’t know yet. Haven’t had time to read. Also not a very common phrasing, which is a danger sign.
- Divergence in everything: Cramér-Rao from data processing
- Azimuth’s Information Geometry Series plus the overview
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Amari, Shun-ichi. 1998. “Natural Gradient Works Efficiently in Learning.” Neural Computation 10 (2): 251–76. https://doi.org/10.1162/089976698300017746.
Amari, Shunʼichi. 1987. “Differential Geometrical Theory of Statistics.” In Differential Geometry in Statistical Inference, 19–94.
———. 2001. “Information Geometry on Hierarchy of Probability Distributions.” IEEE Transactions on Information Theory 47: 1701–11. https://doi.org/10.1109/18.930911.
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Barndorff-Nielsen, O E. 1987. “Differential and Integral Geometry in Statistical Inference.” In Differential Geometry in Statistical Inference. Sn Aarhus.
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Ge, Rong, and Tengyu Ma. 2017. “On the Optimization Landscape of Tensor Decompositions.” In Advances in Neural Information Processing Systems. http://arxiv.org/abs/1706.05598.
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