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Fisher-Bingham-like normalizing flows on the sphere

Thorsten Glüsenkamp

TL;DR

This work tackles the lack of flexible normalizing flows on the sphere for general Fisher-Bingham and Angular Gaussian families. It introduces ZLP-Fisher flows, built from a Fisher-zoom (vMF) block and a linear-project (central angular Gaussian) block, enabling controllable complexity and stable conditional density estimation across scales, with a Kent-like variant that yields Gaussian behavior in the tangent space at high concentration. By ordering and constraining these blocks, the authors realize FB$_4$–FB$_8$-like distributions and generalize the vMF diffeomorphism to arbitrary dimension $D$, using efficient inversions in practice. Empirical tests on a 2-sphere conditional density task show competitive performance and demonstrate that Kent upgrades improve first- and second-moment modeling when target densities differ by orders of magnitude in scale, with potential applicability to astronomy and other directional-data problems. The approach offers a principled, scalable path to richer spherical NF models beyond the special-case flows previously known.

Abstract

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.

Fisher-Bingham-like normalizing flows on the sphere

TL;DR

This work tackles the lack of flexible normalizing flows on the sphere for general Fisher-Bingham and Angular Gaussian families. It introduces ZLP-Fisher flows, built from a Fisher-zoom (vMF) block and a linear-project (central angular Gaussian) block, enabling controllable complexity and stable conditional density estimation across scales, with a Kent-like variant that yields Gaussian behavior in the tangent space at high concentration. By ordering and constraining these blocks, the authors realize FB–FB-like distributions and generalize the vMF diffeomorphism to arbitrary dimension , using efficient inversions in practice. Empirical tests on a 2-sphere conditional density task show competitive performance and demonstrate that Kent upgrades improve first- and second-moment modeling when target densities differ by orders of magnitude in scale, with potential applicability to astronomy and other directional-data problems. The approach offers a principled, scalable path to richer spherical NF models beyond the special-case flows previously known.

Abstract

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.

Paper Structure

This paper contains 9 sections, 5 theorems, 16 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Fisher-Bingham and Angular Gaussian Families
  3. Normalizing flows on the sphere
  4. "Fisher zoom" - the von-Mises-Fisher distribution
  5. "linear-project" - the bimodal central angular Gaussian distribution
  6. ZLP - Combining "Z" and "LP" flows
  7. Tests on conditional density estimation
  8. Related Work
  9. Summary and Discussion

Key Result

Proposition 3.1

Let $\vec{x} = (x_1,\ldots,x_D)\in S^{D-1}$ be a point on the D-1 sphere. Let $h\colon [-1,1]\to\mathbb{R}$ be a diffeomorphism on the interval $[-1,1]$. Then $\Phi (\vec{x})=(x_1\cdot \sqrt{\frac{1-h(x_i)^2}{1-x_i^2}},\ldots,h(x_i),\ldots,x_D \cdot \sqrt{\frac{1-h(x_i)^2}{1-x_i^2}})$ is the corresp

Figures (3)

  • Figure 1: Results of specific 3-d Gaussians (contours illustrated by gray ellipsoids) conditioned or marginalized onto the surface of the 2-sphere (a-c). These yield members of the Fisher-Bingham (FB) or angular Gaussian (AG) family. For generic Gaussians (c), no normalizing-flow description exists. Picking the "Fisher zoom" from a) and the "linear-project" from b) with specific parameter constraints ($\Phi_\mathrm{LP,S_c}$, see section \ref{['section:combining_flows']}) leads to a dynamical normalizing-flow construction of a density with similar properties as the Kent distribution (d) - a bivariate unimodal distribution with a Gaussian limit in tangent space for large concentrations.
  • Figure 2: Visual depiction of test results for the letters "A" and "B" using ZLP-Fisher flow, the Exponential map flow with radial basis function (EXP-R) and rational quadratic splines/Möbius flow (RQS-M). A ZLP-Kent addition as final layer is indicated with "+ K". The first column shows a Mollweide projection of the whole sphere, the others use orthographic projection. In the orthographic projections, the patch size is roughly the same as shown in the column header. Ten samples (red) from the test letter shape are shown for reference aswell.
  • Figure 3: Improvement in test loss due to addition of a ZLP-Kent as the final flow layer. The x-axis shows PDF extents from 180 down to 0.2 degrees (zoom factors 1-1000). The upper plot shows a scaled test loss, where the entropy of a Gaussian of similar extent is subtracted to see the behavior more clearly.

Theorems & Definitions (5)

  • Proposition 3.1: Simple embedding-based density update
  • Theorem 3.2: Diffeomorphism to generate vMF flow in any dimension
  • Corollary 3.3: Finite sums for U and F
  • Corollary 3.4: Scaling behavior
  • Theorem 3.5