Elliptical belief propagation

Generalized least generalized squares

We can generalize Gaussian belief propagation to use general elliptical laws by using Mahalanobis distance without presuming the Gaussian distribution , making it into a kind of elliptical belief propagation.

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 . 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.

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.

Surely? TBD.

Generic

There seem to be generic update rules which could be used to construct a generic elliptical belief propagation algorithm.

References

Agarwal, Pratik, Gian Diego Tipaldi, Luciano Spinello, Cyrill Stachniss, and Wolfram Burgard. 2013. In 2013 IEEE International Conference on Robotics and Automation, 62–69.
Aste, Tomaso. 2021. arXiv.
Bånkestad, Maria, Jens Sjölund, Jalil Taghia, and Thomas Schön. 2020. arXiv.
Davison, Andrew J., and Joseph Ortiz. 2019. arXiv:1910.14139 [Cs], October.
Donoho, David L., and Andrea Montanari. 2013. arXiv:1310.7320 [Cs, Math, Stat], October.
Karlgaard, Christopher D., and Hanspeter Schaub. 2011. Journal of Guidance, Control, and Dynamics 34 (2): 388–402.
Lan, Xiangyang, Stefan Roth, Daniel Huttenlocher, and Michael J. Black. 2006. In Computer Vision – ECCV 2006, edited by Aleš Leonardis, Horst Bischof, and Axel Pinz, 3952:269–82. Berlin, Heidelberg: Springer Berlin Heidelberg.
Ortiz, Joseph, Talfan Evans, and Andrew J. Davison. 2021. arXiv:2107.02308 [Cs], July.
Proudler, I., S. Roberts, S. Reece, and I. Rezek. 2007. In 2007 15th International Conference on Digital Signal Processing, 355–58.

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