Learning on manifolds

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

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.

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:

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.

Hamiltonian Monte Carlo

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

Langevin Monte Carlo

Girolami et al discuss Langevin Monte Carlo in this context.

Natural gradient

See natural gradients.

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.


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