Misspecified Maximum Likelihood Estimation for Non-Uniform Group Orbit Recovery

2025-09-26

AI Summary

This work addresses model misspecification in generalized orbit recovery—specifically, the bias arising when maximum likelihood estimation (MLE) incorrectly assumes a uniform group action distribution, as commonly encountered in projection-based settings such as single-particle cryo-electron microscopy. Theoretically, we show that without projections, the misspecified MLE remains consistent; however, projections induce systematic bias in signal recovery. To correct this, we propose a joint estimation framework that parametrizes the true group element distribution and incorporates it into the MLE objective. Leveraging a continuous Gaussian mixture model and optimization landscape theory, we rigorously characterize the bias magnitude and prove that, under mild conditions, the joint estimator eliminates bias and substantially improves recovery accuracy. This is the first systematic analysis of how group distribution misspecification affects orbit recovery, and it provides a provably convergent correction scheme with theoretical guarantees.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Large-scale security measurements

Abstract

We study maximum likelihood estimation (MLE) in the generalized group orbit recovery model, where each observation is generated by applying a random group action and a known, fixed linear operator to an unknown signal, followed by additive noise. This model is motivated by single-particle cryo-electron microscopy (cryo-EM) and can be viewed primarily as a structured continuous Gaussian mixture model. In practice, signal estimation is often performed by marginalizing over the group using a uniform distribution--an assumption that typically does not hold and renders the MLE misspecified. This raises a fundamental question: how does the misspecified MLE perform? We address this question from several angles. First, we show that in the absence of projection, the misspecified population log-likelihood has desired optimization landscape that leads to correct signal recovery. In contrast, when projections are present, the global optimizers of the misspecified likelihood deviate from the true signal, with the magnitude of the bias depending on the noise level. To address this issue, we propose a joint estimation approach tailored to the cryo-EM setting, which parameterizes the unknown distribution of the group elements and estimates both the signal and distribution parameters simultaneously.

Problem

• Analyzing performance of misspecified MLE in group orbit recovery models
• Studying bias in signal estimation when projections are present
• Proposing joint estimation method for cryo-EM signal and distribution parameters

Innovation

• Misspecified MLE analysis for group orbit recovery
• Joint estimation of signal and group distribution parameters
• Tailored approach for cryo-EM with unknown distributions