In this paper, we develop solutions for sparse tensor signal recovery (SR) and tensor dictionary learning (DL) problems via the sparse Bayesian learning (SBL) approach. We consider a class of tensor system which has the special sparsity structure that a given (say the $i$ th) row of every unfolding matrices of the tensor involved is simultaneously zero or non-zero. For both problems, we propose a Kronecker-like prior distribution for the variables to be recovered in the framework of SBL to take advantage of this sparsity structure. For tensor DL, we consider a de-noising problem in which the clear version is recoverable from sparse coefficients and several separable dictionaries. Our prior distribution model for sparse coefficients entails that the same column of these separable dictionaries have a common prior distribution. We show that our SBL-based algorithms for solving the SR and DL problems require much lower complexity than that of the corresponding vector-matrix system by reducing the matrix inversion size. The proposed SBL-DL and SBL-SR algorithms are utilized, invoking a tensor virtual channel model, to estimate the channel response of a millimeter wave communication system which employs uniform planar arrays on both sides. The superiority of the resulting channel estimates is verified by computer simulations.

中文翻译：

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基于稀疏贝叶斯学习的张量词典学习和信号恢复及其在MIMO信道估计中的应用

在本文中，我们通过稀疏贝叶斯学习（SBL）方法开发了稀疏张量信号恢复（SR）和张量字典学习（DL）问题的解决方案。我们考虑一类张量系统，该张量系统具有给定的特殊稀疏结构（例如$ i $ th）所涉及的张量的每个展开矩阵的行同时为零或非零。对于这两个问题，我们为要在SBL框架中恢复的变量提出了一种类似于Kronecker的先验分布，以利用这种稀疏结构。对于张量DL，我们考虑了一个去噪问题，其中清晰的版本可以从稀疏系数和几个可分离的字典中恢复。我们稀疏系数的先验分布模型要求这些可分离词典的同一列具有共同的先验分布。我们表明，通过减少矩阵求逆大小，用于解决SR和DL问题的基于SBL的算法所需的复杂度比相应的矢量矩阵系统低得多。利用提出的SBL-DL和SBL-SR算法，调用张量虚拟通道模型，估计毫米波通信系统的信道响应，该系统在两侧均采用统一的平面阵列。所得信道估计的优越性已通过计算机仿真得到了验证。