# Elliptical belief propagation

Generalized least generalized squares

August 22, 2022 — August 23, 2022

We can generalize Gaussian belief propagation to use general elliptical laws by using Mahalanobis distance without presuming the Gaussian distribution (Agarwal et al. 2013; Davison and Ortiz 2019), making it into a kind of elliptical belief propagation.

## 1 Robust

If we use a robust Huber loss instead of a Gaussian log-likelihood, then the resulting algorithm is usually referred to as a *robust factor* or as *dynamic covariance scaling* (Agarwal et al. 2013; Davison and Ortiz 2019). The nice thing here is that we can imagine the transition from quadratic to linear losses gives us an estimate of which observations are outliers.

## 2 Student-\(t\)

Surely this is around? Certainly there is a special case in the t-process. It is mentioned, I think, in Lan et al. (2006) and possibly Proudler et al. (2007) although the latter seems to be something more *ad hoc*.

## 3 Gaussian mixture

Surely? TBD.

## 4 Generic

There seem to be generic update rules (Aste 2021; Bånkestad et al. 2020) which could be used to construct a generic elliptical belief propagation algorithm.

## 5 References

*2013 IEEE International Conference on Robotics and Automation*.

*Journal of Multivariate Analysis*.

*Journal of Economic Theory*.

*arXiv:1910.14139 [Cs]*.

*arXiv:1310.7320 [Cs, Math, Stat]*.

*J. Multivar. Anal.*

*Journal of Guidance, Control, and Dynamics*.

*Journal of Multivariate Analysis*.

*Computer Vision – ECCV 2006*.

*arXiv:2107.02308 [Cs]*.

*2007 15th International Conference on Digital Signal Processing*.

*Image and Vision Computing*.