Adaptive Euclidean maps for histograms: generalized Aitchison embeddings

Learning distances that are specifically designed to compare histograms in the probability simplex has recently attracted the attention of the machine learning community. Learning such distances is important because most machine learning problems involve bags of features rather than simple vectors. Ample empirical evidence suggests that the Euclidean distance in general and Mahalanobis metric learning in particular may not be suitable to quantify distances between points in the simplex. We propose in this paper a new contribution to address this problem by generalizing a family of embeddings proposed by Aitchison (J R Stat Soc 44:139–177, 1982) to map the probability simplex onto a suitable Euclidean space. We provide algorithms to estimate the parameters of such maps by building on previous work on metric learning approaches. The criterion we study is not convex, and we consider alternating optimization schemes as well as accelerated gradient descent approaches. These algorithms lead to representations that outperform alternative approaches to compare histograms in a variety of contexts.