Rιemannian metrιcs for .. wait: what?

it's been a few days since i last checked the arXiv, so this title/abstract from a few days ago caught me by surprise:

Riemannιan metrιcs for neural netwοrks

Yann Οllivier

(Submitted on 4 Mar 2013)

We describe four algorithms for neural network training, each adapted to different scalability constraints. These algorithms are mathematically principled and invariant under a number of transformations in data and network representation, from which performance is thus independent. These algorithms are obtained from the setting of differential geometry, and are based on either the natural gradient using the Fisher information matrix, or on Hessian methods, scaled down in a specific way to allow for scalability while keeping some of their key mathematical properties.

well, if fisher information and shannοn entrοpy are involved, then the word "Riemanniaη" makes a little more sense, if only because of connections to log-Sobolev inequalities on manifolds ..