Parametric dictionaries can increase the ability of sparse representations to meaningfully capture and interpret the underlying signal information, such as encountered in biomedical problems. Given a mapping function from the atom parameter space to the actual atoms, we propose a sparse Bayesian framework for learning the atom parameters, because of its ability to provide full posterior estimates, take uncertainty into account and generalize on unseen data. Inference is performed with Markov Chain Monte Carlo, that uses block sampling to generate the variables of the Bayesian problem. Since the parameterization of dictionary atoms results in posteriors that cannot be analytically computed, we use a Metropolis-Hastings-within-Gibbs framework, according to which variables with closed-form posteriors are generated with the Gibbs sampler, while the remaining ones with the Metropolis Hastings from appropriate candidate-generating densities. We further show that the corresponding Markov Chain is uniformly ergodic ensuring its convergence to a stationary distribution independently of the initial state. Results on synthetic data and real biomedical signals indicate that our approach offers advantages in terms of signal reconstruction compared to previously proposed Steepest Descent and Equiangular Tight Frame methods. This paper demonstrates the ability of Bayesian learning to generate parametric dictionaries that can reliably represent the exemplar data and provides the foundation towards inferring the entire variable set of the sparse approximation problem for signal denoising, adaptation, and other applications.

中文翻译：

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稀疏贝叶斯近似参数字典的马尔可夫链蒙特卡罗推理

参数字典可以提高稀疏表示的能力，以有意义地捕获和解释潜在的信号信息，例如在生物医学问题中遇到的信息。给定从原子参数空间到实际原子的映射函数，我们提出了一个用于学习原子参数的稀疏贝叶斯框架，因为它能够提供完整的后验估计、考虑不确定性并对看不见的数据进行泛化。使用马尔可夫链蒙特卡罗进行推理，它使用块采样来生成贝叶斯问题的变量。由于字典原子的参数化导致无法分析计算的后验，我们使用 Metropolis-Hastings-within-Gibbs 框架，根据该框架，使用 Gibbs 采样器生成具有封闭形式后验的变量，而其余的来自适当的候选生成密度的 Metropolis Hastings。我们进一步表明，相应的马尔可夫链是均匀遍历的，确保其收敛到与初始状态无关的平稳分布。合成数据和真实生物医学信号的结果表明，与先前提出的最速下降和等角紧框架方法相比，我们的方法在信号重建方面具有优势。本文展示了贝叶斯学习生成参数字典的能力，这些字典可以可靠地表示示例数据，并为推断稀疏逼近问题的整个变量集以用于信号去噪、自适应和其他应用奠定了基础。