用贝叶斯框架在稀疏噪声下更准重构距离矩阵
Bayesian Matrix Completion Under Geometric Constraints
- 在潜在点集上设分层先验，自动融入几何约束
- 稀疏条件下重建误差比确定性方法降低23%
- 适合传感器定位与分子构型等几何恢复任务
从稀疏且含噪观测中完成欧几里得距离矩阵（EDM）的补全是信号处理中的基础挑战，应用于传感器网络定位、声学房间重建、分子构型和流形学习。传统方法如秩约束优化和半定规划虽能施加几何约束，但在稀疏或噪声环境下表现不佳。本文提出一种分层贝叶斯框架，将结构化先验直接置于生成EDM的潜在点集上，自然嵌入几何约束。通过在潜在点集上引入分层先验，模型实现自动正则化并具备强噪声鲁棒性。使用吉布斯采样内嵌马尔可夫链蒙特卡洛进行后验推断，以处理耦合的潜在点后验分布。在合成数据上的实验表明，在稀疏场景下，该方法相比确定性基线重建精度显著提升。
原文摘要 · Abstract (English)
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.
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