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Bayesian Matrix Completion Under Geometric Constraints

Rohit Varma Chiluvuri, Santosh Nannuru

TL;DR

This work tackles Euclidean Distance Matrix completion from sparse and noisy observations by embedding geometric constraints directly into a Bayesian latent-point model. A hierarchical Gaussian prior on the latent points, coupled with Metropolis-Hastings within Gibbs inference, enables automatic regularization and principled uncertainty quantification for missing distances. The method, called BMC-GC, demonstrates superior reconstruction under high missingness and noise compared with deterministic baselines and yields posterior distributions for unobserved entries. By explicitly enforcing EDM geometry and quantifying uncertainty, it holds practical value for sensor localization, molecular conformation, and related distance-geometry applications. The framework also links to nuclear-norm based formulations via MAP equivalence while offering a fully Bayesian treatment with adaptive noise handling through a Normal–Wishart hyperprior.

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.

Bayesian Matrix Completion Under Geometric Constraints

TL;DR

This work tackles Euclidean Distance Matrix completion from sparse and noisy observations by embedding geometric constraints directly into a Bayesian latent-point model. A hierarchical Gaussian prior on the latent points, coupled with Metropolis-Hastings within Gibbs inference, enables automatic regularization and principled uncertainty quantification for missing distances. The method, called BMC-GC, demonstrates superior reconstruction under high missingness and noise compared with deterministic baselines and yields posterior distributions for unobserved entries. By explicitly enforcing EDM geometry and quantifying uncertainty, it holds practical value for sensor localization, molecular conformation, and related distance-geometry applications. The framework also links to nuclear-norm based formulations via MAP equivalence while offering a fully Bayesian treatment with adaptive noise handling through a Normal–Wishart hyperprior.

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.
Paper Structure (12 sections, 16 equations, 3 figures, 1 algorithm)

This paper contains 12 sections, 16 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Problem Formulation
  2. Background
  3. Nuclear norm minimization
  4. Probabilistic framework
  5. Bayesian framework for EDM
  6. Inference with Gaussian prior
  7. Inference with hierarchical Gaussian prior
  8. Hierarchical Prior
  9. Posterior Inference
  10. Simulation Results
  11. Results
  12. Conclusion

Figures (3)

  • Figure 1: Performance evaluation of BMC-GC on EDM denoising and completion. (a) Error vs fraction of observed entries under noisy conditions ($n=500$, SNR = 20 dB). (b) Convergence of the proposed sampler, showing relative reconstruction error over MCMC iterations with fraction of observed entries $0.2$. (c) Posterior distribution of 9 randomly selected missing entries of the EDM. Each subplot displays the posterior distribution of samples for a distinct unobserved pair $(i,j)$, along with the True Value (red) and Posterior Mean (blue). Results shown for $n=100$, fraction of observed entries $=0.5$, and $SNR=20$ dB.
  • Figure 2: Reconstruction Error in noiseless setup with $n=250$
  • Figure 3: Reconstruction Error in noiseless setup with $n=500$