Pairwise dissimilarity representations are frequently used as an alternative to feature vectors in pattern recognition.One of the problems encountered in the analysis of such data, is that the dissimilarities are rarely Euclidean, while statistical learning algorithms often rely on Euclidean distances. Such non-Euclidean dissimilarities are often corrected or imposed geometry via embedding. This talk reviews and and extends the field of analysing non-Euclidean dissimilarity data.

In this chapter, we compare and contrast two approaches to the problem of embedding non-Euclidean data, namely geometric and structure preserving embedding. Under the first heading, we explore how spherical embedding can be used to embed data onto the surface of sphere of optimal radius. Here we explore both elliptic and hyperbolic geometries, i. e., positive and negative curvatures. Our results on synthetic and real data show that the elliptic embedding performs well under noisy conditions and can deliver low-distortion embeddings for a wide variety of datasets. Hyperbolic data seems to be much less common (at least in our datasets) and is more difficult to accurately embed. Under the second heading, we show how the Ihara zeta function can be used to embed hypergraphs in a manner which reflects their underlying relational structure. Specifically, we show how a polynomial characterization derived from the Ihara zeta function leads to an embedding which captures the prime cycle structure of the hypergraphs.