Prior Support Knowledge-Aided Sparse Bayesian Learning with Partly Erroneous Support Information

It has been shown both experimentally and theoretically that sparse signal recovery can be significantly improved given that part of the signal's support is known \emph{a priori}. In practice, however, such prior knowledge is usually inaccurate and contains errors. Using such knowledge may result in severe performance degradation or even recovery failure. In this paper, we study the problem of sparse signal recovery when partial but partly erroneous prior knowledge of the signal's support is available. Based on the conventional sparse Bayesian learning framework, we propose a modified two-layer Gaussian-inverse Gamma hierarchical prior model and, moreover, an improved three-layer hierarchical prior model. The modified two-layer model employs an individual parameter $b_i$ for each sparsity-controlling hyperparameter $\alpha_i$, and has the ability to place non-sparsity-encouraging priors to those coefficients that are believed in the support set. The three-layer hierarchical model is built on the modified two-layer prior model, with a prior placed on the parameters $\{b_i\}$ in the third layer. Such a model enables to automatically learn the true support from partly erroneous information through learning the values of the parameters $\{b_i\}$. Variational Bayesian algorithms are developed based on the proposed hierarchical prior models. Numerical results are provided to illustrate the performance of the proposed algorithms.