Online EM for Functional Data

TL;DR

This paper introduces an online EM algorithm for functional data that efficiently extracts templates from noisy, deformed, and censored curves and images, improving computational efficiency over batch methods.

Contribution

It extends the Bayesian deformable template model with an online EM approach, enabling scalable and real-time template extraction from complex functional data.

Findings

More computationally efficient than batch algorithms

Effective in high-dimensional missing data scenarios

Successful application to curve registration and image template extraction

Abstract

A novel approach to perform unsupervised sequential learning for functional data is proposed. Our goal is to extract reference shapes (referred to as templates) from noisy, deformed and censored realizations of curves and images. Our model generalizes the Bayesian dense deformable template model (Allassonni\`ere et al., 2007), a hierarchical model in which the template is the function to be estimated and the deformation is a nuisance, assumed to be random with a known prior distribution. The templates are estimated using a Monte Carlo version of the online Expectation-Maximization algorithm, extending the work from Capp\'e and Moulines (2009). Our sequential inference framework is significantly more computationally efficient than equivalent batch learning algorithms, especially when the missing data is high-dimensional. Some numerical illustrations on curve registration problem and templates extraction from images are provided to support our findings.