Fisher-Bingham-like normalizing flows on the sphere
This work addresses a specific challenge in probabilistic modeling for spherical data, particularly in astronomy, by providing a novel flow family that is incremental over existing methods.
The paper tackles the problem of constructing normalizing flows on the sphere that mimic Fisher-Bingham distributions, which are fundamental but not easily expressed as flows except in special cases. It introduces 'zoom-linear-project'-Fisher flows, which can handle conditional density estimation with varying scales, showing improved performance in astronomical applications.
A generic D-dimensional Gaussian can be conditioned or projected onto the D-1 unit sphere, thereby leading to the well-known Fisher-Bingham (FB) or Angular Gaussian (AG) distribution families, respectively. These are some of the most fundamental distributions on the sphere, yet cannot straightforwardly be written as a normalizing flow except in two special cases: the von-Mises Fisher in D=3 and the central angular Gaussian in any D. In this paper, we describe how to generalize these special cases to a family of normalizing flows that behave similarly to the full FB or AG family in any D. We call them "zoom-linear-project" (ZLP)-Fisher flows. Unlike a normal Fisher-Bingham distribution, their composition allows to gradually add complexity as needed. Furthermore, they can naturally handle conditional density estimation with target distributions that vary by orders of magnitude in scale - a setting that is important in astronomical applications but that existing flows often struggle with. A particularly useful member of the new family is the Kent analogue that can cheaply upgrade any flow in this situation to yield better performance.