# Estimating the Number of Components in Finite Mixture Models via the Group-Sort-Fuse Procedure

arXiv: https://arxiv.org/abs/2005.11641 · HTML: https://mlsanity.com/paper/2005.11641/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 May 2020.

## Built on by (most cited first, 8 of 8)

- [Learning latent causal graphs via mixture oracles](https://mlsanity.com/paper/2106.15563/) (2021)
- [Statistical Perspective of Top-K Sparse Softmax Gating Mixture of Experts](https://mlsanity.com/paper/2309.13850/) (2023)
- [Refined Convergence Rates for Maximum Likelihood Estimation under Finite Mixture Models](https://mlsanity.com/paper/2202.08786/) (2022)
- [Towards Convergence Rates for Parameter Estimation in Gaussian-gated Mixture of Experts](https://mlsanity.com/paper/2305.07572/) (2023)
- [Estimation and inference for the Wasserstein distance between mixing measures in topic models](https://mlsanity.com/paper/2206.12768/) (2022)
- [Beyond EM Algorithm on Over-specified Two-Component Location-Scale Gaussian Mixtures](https://mlsanity.com/paper/2205.11078/) (2022)
- [Bayesian nonparametric mixture inconsistency for the number of components: How worried should we be in practice?](https://mlsanity.com/paper/2207.14717/) (2022)
- [CavMerge: Merging K-means Based on Local Log-Concavity](https://mlsanity.com/paper/2604.04302/) (2026)

## Abstract (arXiv)

Estimation of the number of components (or order) of a finite mixture model is a long standing and challenging problem in statistics. We propose the Group-Sort-Fuse (GSF) procedure -- a new penalized likelihood approach for simultaneous estimation of the order and mixing measure in multidimensional finite mixture models. Unlike methods which fit and compare mixtures with varying orders using criteria involving model complexity, our approach directly penalizes a continuous function of the model parameters. More specifically, given a conservative upper bound on the order, the GSF groups and sorts mixture component parameters to fuse those which are redundant. For a wide range of finite mixture models, we show that the GSF is consistent in estimating the true mixture order and achieves the n-1/2 convergence rate for parameter estimation up to polylogarithmic factors. The GSF is implemented for several univariate and multivariate mixture models in the R package GroupSortFuse. Its finite sample performance is supported by a thorough simulation study, and its application is illustrated on two real data examples.