Dimensionality reduction in machine learning
Information Geometry is a rigorous framework which combines statistical inference, information theory and differential geometry.

Several open problems in statistics (and geometry) have found a new formulation and solution.
Specificaly the concept of dual connections and dualy-flat spaces. It has been shown that various statistical inference methods (e.g maximum likelihood, EM algorithm etc.) can be formulated in terms of dualy-flat spaces in a statistical manifold (reference "Methods of Information Geometry", Amari, Nagaoka). The new formulation allows a unified view of previously un-related methods and algorithms.

The problem of dimensionality reduction and PCA has been formulated information-geometricaly as follows (with emphasis on information):


"Information-Geometric Dimensionality Reduction"
"An Information Geometric Framework for Dimensionality Reduction"
"The e-PCA and m-PCA: Dimension Reduction of Parameters by Information Geometry"