Structural Properties, Cycloid Trajectories and Non-Asymptotic Guarantees of EM Algorithm for Mixed Linear Regression

• URL: http://arxiv.org/abs/2511.04937v1
• Date: Fri, 07 Nov 2025 02:34:14 GMT
• Title: Structural Properties, Cycloid Trajectories and Non-Asymptotic Guarantees of EM Algorithm for Mixed Linear Regression
• Authors: Zhankun Luo, Abolfazl Hashemi,
• Abstract summary: This work investigates the structural properties, cycloid trajectories, and non-asymptotic convergence guarantees of the Expectation-Maximization algorithm for two-component Mixed Linear Regression (2MLR)<n>Recent studies have established global convergence for 2MLR with known balanced weights and super-linear convergence in noiseless and high signal-to-noise ratio (SNR) regimes.
• Score: 11.223656960922478
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This work investigates the structural properties, cycloid trajectories, and non-asymptotic convergence guarantees of the Expectation-Maximization (EM) algorithm for two-component Mixed Linear Regression (2MLR) with unknown mixing weights and regression parameters. Recent studies have established global convergence for 2MLR with known balanced weights and super-linear convergence in noiseless and high signal-to-noise ratio (SNR) regimes. However, the theoretical behavior of EM in the fully unknown setting remains unclear, with its trajectory and convergence order not yet fully characterized. We derive explicit EM update expressions for 2MLR with unknown mixing weights and regression parameters across all SNR regimes and analyze their structural properties and cycloid trajectories. In the noiseless case, we prove that the trajectory of the regression parameters in EM iterations traces a cycloid by establishing a recurrence relation for the sub-optimality angle, while in high SNR regimes we quantify its discrepancy from the cycloid trajectory. The trajectory-based analysis reveals the order of convergence: linear when the EM estimate is nearly orthogonal to the ground truth, and quadratic when the angle between the estimate and ground truth is small at the population level. Our analysis establishes non-asymptotic guarantees by sharpening bounds on statistical errors between finite-sample and population EM updates, relating EM's statistical accuracy to the sub-optimality angle, and proving convergence with arbitrary initialization at the finite-sample level. This work provides a novel trajectory-based framework for analyzing EM in Mixed Linear Regression.

Related papers

• Learning Overspecified Gaussian Mixtures Exponentially Fast with the EM Algorithm [5.625796693054093]
  We investigate the convergence properties of the EM algorithm when applied to overspecified Gaussian mixture models.<n>We demonstrate that the population EM algorithm converges exponentially fast in terms of the Kullback-Leibler (KL) distance.
  arXiv  Detail & Related papers  (2025-06-13T14:57:57Z)
• Learning a Class of Mixed Linear Regressions: Global Convergence under General Data Conditions [1.9295130374196499]
  Mixed linear regression (MLR) has attracted increasing attention because of its great theoretical and practical importance in nonlinear relationships by utilizing a mixture of linear regression sub-models.<n>Although considerable efforts have been devoted to the learning problem of such systems, most existing investigations impose the strict independent and identically distributed (i.i.d.) or distributed PE conditions.
  arXiv  Detail & Related papers  (2025-03-24T09:57:39Z)
• Multivariate root-n-consistent smoothing parameter free matching estimators and estimators of inverse density weighted expectations [51.000851088730684]
  We develop novel modifications of nearest-neighbor and matching estimators which converge at the parametric $sqrt n $-rate.<n>We stress that our estimators do not involve nonparametric function estimators and in particular do not rely on sample-size dependent parameters smoothing.
  arXiv  Detail & Related papers  (2024-07-11T13:28:34Z)
• Unveiling the Cycloid Trajectory of EM Iterations in Mixed Linear Regression [5.883916678819683]
  We study the trajectory of iterations and the convergence rates of the Expectation-Maximization (EM) algorithm for two-component Mixed Linear Regression (2MLR) Recent results have established the super-linear convergence of EM for 2MLR in the noiseless and high SNR settings.
  arXiv  Detail & Related papers  (2024-05-28T14:46:20Z)
• Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation [52.73824786627612]
  This paper establishes new convergence results for textitgeodesic strongly monotone games.<n>Our key result shows that RGD attains last-iterate linear convergence in a textitgeometry-agnostic fashion.<n>Overall, this paper presents the first geometry-agnostic last-iterate convergence analysis for games beyond the Euclidean settings.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Sharp analysis of EM for learning mixtures of pairwise differences [14.01151780845689]
  We consider a symmetric mixture of linear regressions with random samples from the pairwise comparison design. We establish that the sequence converges linearly, providing an $ell_infty$-norm guarantee on the estimation error of the iterates. We show that the limit of the EM sequence achieves the sharp rate of estimation in the $ell$-norm, matching the information-theoretically optimal constant.
  arXiv  Detail & Related papers  (2023-02-20T16:13:19Z)
• A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Study [68.8204255655161]
  We investigate and compare the performance of several local and global smoothing techniques to a priori denoise the state measurements. We show that, in general, global methods, which use the entire measurement data set, outperform local methods, which employ a neighboring data subset around a local point.
  arXiv  Detail & Related papers  (2022-01-29T23:31:25Z)
• Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems [98.34292831923335]
  Motivated by the problem of online correlation analysis, we propose the emphStochastic Scaled-Gradient Descent (SSD) algorithm. We bring these ideas together in an application to online correlation analysis, deriving for the first time an optimal one-time-scale algorithm with an explicit rate of local convergence to normality.
  arXiv  Detail & Related papers  (2021-12-29T18:46:52Z)
• On the Double Descent of Random Features Models Trained with SGD [78.0918823643911]
  We study properties of random features (RF) regression in high dimensions optimized by gradient descent (SGD) We derive precise non-asymptotic error bounds of RF regression under both constant and adaptive step-size SGD setting. We observe the double descent phenomenon both theoretically and empirically.
  arXiv  Detail & Related papers  (2021-10-13T17:47:39Z)
• On the Convergence of Stochastic Extragradient for Bilinear Games with Restarted Iteration Averaging [96.13485146617322]
  We present an analysis of the ExtraGradient (SEG) method with constant step size, and present variations of the method that yield favorable convergence. We prove that when augmented with averaging, SEG provably converges to the Nash equilibrium, and such a rate is provably accelerated by incorporating a scheduled restarting procedure.
  arXiv  Detail & Related papers  (2021-06-30T17:51:36Z)
• Benign Overfitting of Constant-Stepsize SGD for Linear Regression [122.70478935214128]
  inductive biases are central in preventing overfitting empirically. This work considers this issue in arguably the most basic setting: constant-stepsize SGD for linear regression. We reflect on a number of notable differences between the algorithmic regularization afforded by (unregularized) SGD in comparison to ordinary least squares.
  arXiv  Detail & Related papers  (2021-03-23T17:15:53Z)
• Estimation, Confidence Intervals, and Large-Scale Hypotheses Testing for High-Dimensional Mixed Linear Regression [9.815103550891463]
  This paper studies the high-dimensional mixed linear regression (MLR) where the output variable comes from one of the two linear regression models with an unknown mixing proportion. We propose an iterative procedure for estimating the two regression vectors and establish their rates of convergence. A large-scale multiple testing procedure is proposed for testing the regression coefficients and is shown to control the false discovery rate (FDR) algorithmally.
  arXiv  Detail & Related papers  (2020-11-06T21:17:41Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.