Bayesian Matrix Completion Under Geometric Constraints

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.

💡 Research Summary

The paper tackles the problem of completing a Euclidean Distance Matrix (EDM) from sparse and noisy observations, a task that appears in sensor network localization, acoustic room reconstruction, molecular conformation, and manifold learning. Traditional deterministic approaches—rank‑constrained optimization, semidefinite programming, or nuclear‑norm minimization—enforce the geometric constraints of EDMs (rank ≤ d + 2, PSD Gram matrix) but require manual regularization and cannot quantify uncertainty, especially when observations are extremely sparse or heavily corrupted.

The authors propose a fully probabilistic, hierarchical Bayesian framework that places priors directly on the latent point set P ∈ ℝⁿˣᵈ that generates the EDM. Two prior configurations are examined. The first is a simple independent zero‑mean Gaussian prior on each point, p(P|σ²_p)=∏ₙ𝒩(P_i;0,σ²_p I_d). Under a maximum‑a‑posteriori (MAP) criterion this reduces exactly to the nuclear‑norm formulation used in deterministic methods, establishing a clear link between Bayesian regularization and classical convex optimization. However, this approach still suffers from a fixed regularization strength and provides no posterior uncertainty.

To overcome these limitations, the second configuration introduces a Normal‑Wishart hierarchical prior: each point is drawn from a Gaussian with unknown mean u_p and precision matrix Δ_p, while (u_p, Δ_p) themselves follow a conjugate Normal‑Wishart hyperprior. This hierarchy allows the model to learn the appropriate level of regularization from the data, automatically adapting to different noise levels and sampling densities. A Gamma hyperprior on the noise precision α = σ⁻² further enables joint learning of the observation noise.

The observation model assumes that only a subset Ω of squared distances is observed, each corrupted by additive Gaussian noise: D_ij = ‖P_i − P_j‖² + ε_ij, ε_ij ∼ 𝒩(0,σ²). The resulting likelihood is non‑linear in P, making direct Gibbs sampling impossible for the latent points. The authors therefore employ a hybrid Markov chain Monte Carlo (MCMC) scheme: Gibbs updates for the hyperparameters (u_p, Δ_p, α) are analytically tractable thanks to conjugacy, while each point P_i is updated via a Metropolis‑Hastings (MH) step with a Gaussian proposal centered at the current location. The acceptance probability incorporates both the prior term (quadratic in P_i) and the likelihood term (involving the squared‑distance residuals). The full algorithm (Algorithm 1) iterates hyperparameter updates, pointwise MH proposals, and recomputation of the EDM estimate ˆD_ij = ‖P_i − P_j‖².

Experiments are conducted on synthetic data where n = 100, 250, 500 points are drawn from a standard normal distribution in d = 3 dimensions. Observation fractions range from 2 % to 10 % and signal‑to‑noise ratios (SNR) from 20 dB to 25 dB. The proposed Bayesian Matrix Completion under Geometric Constraints (BMC‑GC) is compared against three state‑of‑the‑art deterministic baselines: OptSpace, Alternating Descent, and an Augmented Lagrangian method. Performance is measured by relative reconstruction error E = ‖Ď − D‖_F / ‖D‖_F. BMC‑GC consistently outperforms the baselines, especially in the most sparse regimes (2 %–6 % observed entries), achieving error reductions of 10 %–30 % over the best deterministic competitor. Convergence diagnostics show that after a conservative burn‑in of 1200 iterations (out of 1500 total), the chain stabilizes; posterior samples are thinned every 10 iterations, and the final EDM estimate is the average of 30 samples. Moreover, the authors visualize posterior distributions for nine randomly selected missing entries, demonstrating that the posterior means are close to the true values and that credible intervals correctly capture the ground truth, thereby validating the model’s uncertainty quantification capability.

In summary, the paper makes several notable contributions: (1) it formulates EDM completion as a hierarchical Bayesian inference problem that directly respects Euclidean geometry; (2) it establishes an equivalence between MAP estimation with a simple Gaussian prior and nuclear‑norm minimization, bridging probabilistic and convex‑optimization perspectives; (3) it introduces a Normal‑Wishart hierarchical prior that learns regularization strength and noise precision from data, eliminating the need for hand‑tuned hyperparameters; (4) it designs an efficient Metropolis‑within‑Gibbs sampler that handles the non‑linear likelihood while preserving conjugacy for hyperparameters; and (5) it empirically demonstrates superior reconstruction accuracy and meaningful uncertainty estimates compared to leading deterministic methods. The work opens avenues for applying Bayesian EDM completion to real‑world sensor networks, acoustic imaging, and molecular structure determination, and suggests future extensions such as variational approximations for scalability, incorporation of additional geometric constraints (e.g., anchor points), and application to large‑scale, high‑dimensional distance data.