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Informationgeometric dimensionality reduction
 Signal Processing Magazine, IEEE
, 2011
"... We consider the problem of dimensionality reduction and manifold learning when the domain of interest is a set of probability distributions instead of a set of Euclidean data vectors. In this problem, one seeks to discover a low dimensional representation, called an embedding, that preserves certain ..."
Abstract

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We consider the problem of dimensionality reduction and manifold learning when the domain of interest is a set of probability distributions instead of a set of Euclidean data vectors. In this problem, one seeks to discover a low dimensional representation, called an embedding, that preserves certain properties such as distance between measured distributions or separation between classes of distributions. Such representations are useful for data visualization and clustering. While a standard Euclidean dimension reduction method like PCA, ISOMAP, or Laplacian Eigenmaps can easily be applied to distributional data – e.g. by quantization and vectorization of the distributions – this may not provide the best lowdimensional embedding. This is because the most natural measure of dissimilarity between probability distributions is the information divergence and not the standard Euclidean distance. If the information divergence is adopted then the space of probability distributions becomes a nonEuclidean space called an information geometry. This article presents methods that are specifically designed for the lowdimensional embedding of informationgeometric data, and we illustrate these methods for visualization in flow cytometry and demography analysis. Index Terms Information geometry, dimensionality reduction, statistical manifold, classification