Learning on manifolds

Finding the lowest bit of a krazy straw, from the inside

October 21, 2011 — January 26, 2022

functional analysis
Gaussian
geometry
Hilbert space
how do science
kernel tricks
machine learning
networks
PDEs
physics
regression
signal processing
spatial
statistics
stochastic processes
time series
uncertainty

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A placeholder for learning on curved spaces. Not discussed: learning OF the curvature of spaces.

AFAICT this usually boils down to defining an appropriate stochastic process on a manifold.

1 Learning on a given manifold

Learning where there is an a priori manifold seems to also be a usage here? For example, the manifold of positive definite matrices is treated in depth in Chikuse and 筑瀬 (2003).

See the work of, e.g.

Manifold optimisation implementations:

There are at least two textbooks online:

2 Information Geometry

The unholy offspring of Fisher information and differential geometry, about which I know little except that it sounds like it should be intuitive. It is probably synonymous with some of the other items on this page if I could sort out all this terminology. See information geometry.

3 Hamiltonian Monte Carlo

You can also discuss Hamiltonian Monte Carlo in this setting. I will not.

4 Langevin Monte Carlo

Girolami et al discuss Langevin Monte Carlo in this context.

5 Natural gradient

See natural gradients.

6 Homogeneous probability

Albert Tarantola’s framing, from his manuscript. How does it relate to information geometry? I don’t know yet. Haven’t had time to read. Also not a common phrasing, which is a danger sign.

7 Incoming

8 References

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Amari, Shunʼichi. 2001. “Information Geometry on Hierarchy of Probability Distributions.” IEEE Transactions on Information Theory.
Aswani, Bickel, and Tomlin. 2011. “Regression on Manifolds: Estimation of the Exterior Derivative.” The Annals of Statistics.
Azangulov, Smolensky, Terenin, et al. 2022. “Stationary Kernels and Gaussian Processes on Lie Groups and Their Homogeneous Spaces I: The Compact Case.”
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