The Linearized Hellinger-Kantorovich Distance

Tianji Cai, Junyi Cheng, Bernhard Schmitzer, Matthew Thorpe

Abstract: In this paper we study the local linearization of the Hellinger--Kantorovich distance via its Riemannian structure. We give explicit expressions for the logarithmic and exponential maps and identify a suitable notion of a Riemannian inner product. Samples can thus be represented as vectors in the tangent space of a suitable reference measure where the norm locally approximates the original metric. Working with the local linearization and the corresponding embeddings allows for the advantages of the Euclidean setting, such as faster computations and a plethora of data analysis tools, while still enjoying approximately the descriptive power of the Hellinger--Kantorovich metric.