TLDR: Researchers introduce “zoom-linear-project” (ZLP)-Fisher flows, a new family of normalizing flows that can effectively model complex probability distributions on spheres. These flows combine “Fisher zoom” and “linear-project” components, allowing for flexible complexity and robust handling of conditional density estimation where target distributions vary widely in scale, which is crucial for astronomical applications. A “Kent analogue” within this family can significantly improve the performance of existing flows.
Researchers have developed a new family of normalizing flows, called “zoom-linear-project” (ZLP)-Fisher flows, designed to model complex probability distributions on the surface of a sphere. This advancement addresses a significant challenge in statistical inference, particularly in fields like astronomy, where data often involves directional information and can vary widely in scale.
Normalizing flows are powerful tools in statistical inference, used for tasks like variational inference and neural posterior estimation. When dealing with directional data, especially on a 2-sphere, it’s crucial to use distributions that naturally fit this manifold. However, some of the most fundamental distributions on the sphere, such as the Fisher-Bingham (FB) and Angular Gaussian (AG) families, have been difficult to represent as normalizing flows, except for a few specific cases like the von-Mises Fisher distribution in three dimensions or the central angular Gaussian in any dimension.
The new ZLP-Fisher flows generalize these special cases, offering a flexible way to create distributions that behave similarly to the full FB or AG families across various dimensions. Unlike traditional Fisher-Bingham distributions, these flows can be composed to gradually increase complexity as needed. A key advantage is their ability to handle conditional density estimation where the target distributions can differ by orders of magnitude in scale. This is particularly important in astronomical applications, such as gravitational-wave or neutrino astronomy, where existing flows often struggle with such varying scales.
The ZLP-Fisher flows are built from two primary components: the “Fisher zoom” and the “linear-project” flow. The “Fisher zoom” is derived from the von-Mises-Fisher distribution and effectively “zooms in” on a specific region of the sphere. The “linear-project” component, based on the central angular Gaussian, allows for the introduction of covariance structure through a linear transformation followed by a projection onto the sphere. The researchers found that the specific order and combination of these “zoom” and “linear-project” steps, along with rotations, can recreate qualitatively all the distributions within the FB and AG families, including the most general FB8 type.
A particularly useful member of this new family is the “Kent analogue.” This Kent-like flow can efficiently enhance existing normalizing flows, improving their performance, especially in situations where conditional target distributions exhibit large variations in scale. The research demonstrates that this Kent version of the ZLP-Fisher flow approaches a multivariate Gaussian in tangent space as its concentration parameter increases, mirroring properties of the standard Kent distribution.
The effectiveness of ZLP-Fisher flows was tested in conditional density estimation tasks, using samples drawn from rotated and scaled alphabet letter shapes on the upper half-sphere. These tests simulated scenarios where posterior scales could vary by three orders of magnitude, from covering the entire sphere down to arcminutes. The ZLP-Fisher flow showed stable performance, outperforming or significantly improving other established flows like rational-quadratic splines with Möbius flows and exponential map flows with radial basis functions, particularly when augmented with the Kent-like upgrade. Existing flows often face numerical instabilities near poles or have limitations in describing arbitrarily localized regions, issues that ZLP-Fisher flows mitigate.
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This work provides a robust and flexible framework for constructing normalizing flows on spheres, offering a starting point to build increasingly complex distributions. It addresses a critical need for stable and efficient density estimation in high-stakes scientific applications. For more technical details, you can refer to the full research paper: Fisher-Bingham-like normalizing flows on the sphere.
