Expectation-Maximization as a Spectrally Governed Relaxation Flow

Statistics > Machine Learning [Submitted on 8 May 2026 (v1), last revised 31 May 2026 (this version, v2)] Title:Expectation-Maximization as a Spectrally Governed Relaxation Flow View PDF HTML (experimental)Abstract:The expectation--maximization (EM) algorithm combines global monotonicity, local linear convergence, and strong practical robustness, but these features are usually analyzed separately. Global descent is nonlinear, whereas local convergence is governed by the spectrum of the linearized EM map. How these two levels fit into a single dynamical picture has remained less transparent. We make explicit the latent-variable operator that connects them. Along the EM trajectory, the likelihood increment admits a global energy decomposition in terms of posterior-relative entropy. Linearization at a nondegenerate maximizer $\theta^\ast$ reveals the local operator \[ \mathcal G_{\theta^\ast}=I-DT(\theta^\ast), \] which coincides with both the missing-information ratio and the information-geometric Hessian of the observed likelihood. From this operator we derive two acceleration strategies. The \textbf{G-Accelerator} uses the spectral gap to obtain an optimal Nesterov-type momentum $\beta^* = (1-\sqrt{\lambda_*})/(1+\sqrt{\lambda_*})$. The \textbf{Geo-Adaptive} accelerator extends the geometric EM framework of Zhou, Alexander \& Lange by replacing their fixed correction strength $\gamma=8$ with the adaptive rule $\gamma_k = 1/\hat\lambda_k$, where $\hat\lambda_k$ is estimated online from the parameter trajectory. Both methods are parameter-free; Geo-Adaptive achieves dramatic acceleration precisely when the spectral gap is smallest. Numerical experiments on Gaussian mixtures demonstrate that both accelerators consistently outperform standard EM and fixed-$\gamma$ DCC-EM, with Geo-Adaptive attaining speedups exceeding $8\times$ in the most challenging regimes. Submission history From: Qiao Wang [view email][v1] Fri, 8 May 2026 14:49:24 UTC (68 KB) [v2] Sun, 31 May 2026 05:50:28 UTC (96 KB) Current browse context: stat.ML References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.