Multimodal Bayesian Registration of Noisy Functions using Hamiltonian Monte Carlo

TL;DR

This paper introduces a Bayesian functional data registration method that handles noisy data and multiple optimal alignments by modifying Hamiltonian Monte Carlo for infinite-dimensional spaces, demonstrated on simulated and real datasets.

Contribution

It presents a novel Bayesian approach using modified Hamiltonian Monte Carlo to effectively register noisy functions and identify multiple optimal alignments without pre-smoothing.

Findings

Efficiently handles noisy functional data.

Successfully captures multiple optimal alignments.

Provides uncertainty quantification for warping functions.

Abstract

Functional data registration is a necessary processing step for many applications. The observed data can be inherently noisy, often due to measurement error or natural process uncertainty, which most functional alignment methods cannot handle. A pair of functions can also have multiple optimal alignment solutions, which is not addressed in current literature. In this paper, a flexible Bayesian approach to functional alignment is presented, which appropriately accounts for noise in the data without any pre-smoothing required. Additionally, by running parallel MCMC chains, the method can account for multiple optimal alignments via the multi-modal posterior distribution of the warping functions. To most efficiently sample the warping functions, the approach relies on a modification of the standard Hamiltonian Monte Carlo to be well-defined on the infinite-dimensional Hilbert space. This flexible Bayesian alignment method is applied to both simulated data and real data sets to show its efficiency in handling noisy functions and successfully accounting for multiple optimal alignments in the posterior; characterizing the uncertainty surrounding the warping functions.