# Learning a Class of Mixed Linear Regressions: Global Convergence under General Data Conditions

arXiv: https://arxiv.org/abs/2503.18500 · HTML: https://mlsanity.com/paper/2503.18500/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 31 references that are in the map.
- First posted on arXiv 24 Mar 2025.

## Builds on (most cited first, 6 of 6)

- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval](https://mlsanity.com/paper/1803.07726/) (2018)
- [EM Converges for a Mixture of Many Linear Regressions](https://mlsanity.com/paper/1905.12106/) (2019)
- [Imbalanced Mixed Linear Regression](https://mlsanity.com/paper/2301.12559/) (2023)
- [Alternating Minimization for Mixed Linear Regression](https://mlsanity.com/paper/1310.3745/) (2013)
- [Global Convergence of Online Identification for Mixed Linear Regression](https://mlsanity.com/paper/2311.18506/) (2023)

## Abstract (arXiv)

Mixed linear regression (MLR) has attracted increasing attention because of its great theoretical and practical importance in capturing nonlinear relationships by utilizing a mixture of linear regression sub-models. Although considerable efforts have been devoted to the learning problem of such systems, i.e., estimating data labels and identifying model parameters, most existing investigations employ the offline algorithm, impose the strict independent and identically distributed (i.i.d.) or persistent excitation (PE) conditions on the regressor data, and provide local convergence results only. In this paper, we investigate the recursive estimation and data clustering problems for a class of stochastic MLRs with two components. To address this inherently nonconvex optimization problem, we propose a novel two-step recursive identification algorithm to estimate the true parameters, where the direction vector and the scaling coefficient of the unknown parameters are estimated by the least squares and the expectation-maximization (EM) principles, respectively. Under a general data condition, which is much weaker than the traditional i.i.d. and PE conditions, we establish the global convergence and the convergence rate of the proposed identification algorithm for the first time. Furthermore, we prove that, without any excitation condition on the regressor data, the data clustering performance including the cumulative mis-classification error and the within-cluster error can be optimal asymptotically. Finally, we provide a numerical example to illustrate the performance of the proposed learning algorithm.