# Consistency issues in Gaussian Mixture Models reduction algorithms

arXiv: https://arxiv.org/abs/2104.12586 · HTML: https://mlsanity.com/paper/2104.12586/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 28 references that are in the map.
- First posted on arXiv 26 Apr 2021.

## Builds on (most cited first, 3 of 3)

- [A Wasserstein-type distance in the space of Gaussian Mixture Models](https://mlsanity.com/paper/1907.05254/) (2019)
- [Optimal transport for Gaussian mixture models](https://mlsanity.com/paper/1710.07876/) (2017)
- [Poisson multi-Bernoulli mixture filter: direct derivation and implementation](https://mlsanity.com/paper/1703.04264/) (2017)

## Built on by (most cited first, 2 of 2)

- [Learning with Noisy Labels Using Collaborative Sample Selection and Contrastive Semi-Supervised Learning](https://mlsanity.com/paper/2310.15533/) (2023)
- [Fixed-point iterations for several dissimilarity measure barycenters in the Gaussian case](https://mlsanity.com/paper/2205.04806/) (2022)

## Abstract (arXiv)

In many contexts Gaussian Mixtures (GM) are used to approximate probability distributions, possibly time-varying. In some applications the number of GM components exponentially increases over time, and reduction procedures are required to keep them reasonably limited. The GM reduction (GMR) problem can be formulated by choosing different measures of the dissimilarity of GMs before and after reduction, like the Kullback-Leibler Divergence (KLD) and the Integral Squared Error (ISE). Since in no case the solution is obtained in closed form, many approximate GMR algorithms have been proposed in the past three decades, although none of them provides optimality guarantees. In this work we discuss the importance of the choice of the dissimilarity measure and the issue of consistency of all steps of a reduction algorithm with the chosen measure. Indeed, most of the existing GMR algorithms are composed by several steps which are not consistent with a unique measure, and for this reason may produce reduced GMs far from optimality. In particular, the use of the KLD, of the ISE and normalized ISE is discussed and compared in this perspective.