Pattern Learning and Recognition on Statistical Manifolds: An Information-Geometric Review

We review the information-geometric framework for statistical pattern recognition: First, we explain the role of statistical similarity measures and distances in fundamental statistical pattern recognition problems. We then concisely review the main statistical distances and report a novel versatile family of divergences. Depending on their intrinsic complexity, the statistical patterns are learned by either atomic parametric distributions, semi-parametric finite mixtures, or non-parametric kernel density distributions. Those statistical patterns are interpreted and handled geometrically in statistical manifolds either as single points, weighted sparse point sets or non-weighted dense point sets. We explain the construction of the two prominent families of statistical manifolds: The Rao Riemannian manifolds with geodesic metric distances, and the Amari-Chentsov manifolds with dual asymmetric non-metric divergences. For the latter manifolds, when considering atomic distributions from the same exponential families (including the ubiquitous Gaussian and multinomial families), we end up with dually flat exponential family manifolds that play a crucial role in many applications. We compare the advantages and disadvantages of these two approaches from the algorithmic point of view. Finally, we conclude with further perspectives on how "geometric thinking" may spur novel pattern modeling and processing paradigms.