High-dimensional structured data such as text and images is often poorly understood and misrepresented in statistical modelling. Typical approaches to modelling such data involve, either explicitly or implicitly, arbitrary geometric assumptions. In this chapter, we consider statistical modelling of non-Euclidean data whose geometry is obtained by embedding the data in a statistical manifold. The resulting models perform better than their Euclidean counterparts on real world data and draw an

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... resting connection betweenČencov and Campbell's axiomatic characterisation of the Fisher information and the recently proposed diffusion kernels and square root embedding.