Bayesian Matrix Completion Under Geometric Constraints

TL;DR

This paper presents a hierarchical Bayesian method for Euclidean distance matrix completion that effectively incorporates geometric constraints, improving accuracy in sparse and noisy data scenarios.

Contribution

It introduces a novel hierarchical Bayesian framework with structured priors on latent points, enabling automatic regularization and robust noise handling for EDM completion.

Findings

Improved reconstruction accuracy over deterministic methods in sparse data regimes.

Effective handling of noise through hierarchical priors.

Demonstrated robustness and accuracy on synthetic datasets.

Abstract

The completion of a Euclidean distance matrix (EDM) from sparse and noisy observations is a fundamental challenge in signal processing, with applications in sensor network localization, acoustic room reconstruction, molecular conformation, and manifold learning. Traditional approaches, such as rank-constrained optimization and semidefinite programming, enforce geometric constraints but often struggle under sparse or noisy conditions. This paper introduces a hierarchical Bayesian framework that places structured priors directly on the latent point set generating the EDM, naturally embedding geometric constraints. By incorporating a hierarchical prior on latent point set, the model enables automatic regularization and robust noise handling. Posterior inference is performed using a Metropolis-Hastings within Gibbs sampler to handle coupled latent point posterior. Experiments on synthetic data demonstrate improved reconstruction accuracy compared to deterministic baselines in sparse regimes.