  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 map 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, whilst still still enjoying
approximately the descriptive power of the Hellinger--Kantorovich metric.
