Research And Applications Of Structured Signal Processing Algorithm Based On Approximate Message Passing

The era of big data brings new needs and challenges for increasingly effi-cient ways of both acquisiting and processing high-order and high-dimensional data. Recently, a promising trend for addressing these demands is to effectively reduce the intrinsic dimensionality of data by exploiting its sparse or low-rank representation or other structured constraints, while keeping the notable features of the data. Then, the dramatically dimensionality reduction can be leveraged to enable reliable inferences to be drawn in scenarios where it is infeasible to perform inference using classical techniques.To date, most works for taking advantage of the low intrinsic dimension-ality inherent in data have focused on identifying concise representations (i.e.,sparse and low-rank) of the data, seeking to represent the data using only a few of "significant" elements. While powerful in their own right, such meth-ods make no additional assumptions regarding possible structured constraints within the significant elements. In this paper, applying Bayesian approximate message passing (AMP) methodology to the generic affine rank minimization(ARM) and tensor CANDECOMP/PARAFAC (CP) decomposition problems,we propose two corresponding generalized approaches and investigate ways to incorporate knowledge of various structured constraints into them, so that leads to a superior performance gains. In addition, considering the difficulty to solve complicated structured signal processing problems, a turbo AMP frame-work can be leveraged to decouple the global problem into several subprob-lems which enable approximate inferences to be made using classical message passing-based methods.The contributions are divided into the following three parts:1) By extending the canonical turbo AMP framework to more general-ized case which may consist of multiple linear inverse problems, a generalized turbo AMP framework is proposed. Then, two turbo AMP-based algorithms are proposed to address the simultaneously low-rank and joint-sparse (L&S) com-pressive sensing problem and the parallel matrix decomposition based tensor completion problem, respectively, and are then applied to compressive hyper-spectral imaging and seismic data interpolation, respectively. The computa-tional complexity of both algorithms is also analyzed.2) Using probabilistic low-rank factorization, we derive a generalized AMP-based approach for the generic ARM problem, and analyze its computational complexity. By jointly representing the linear affine transformation and bilinear matrix factorization in factor graph, the proposed approach, called ARM-AMP,adopts efficient approximations and the "Onsager" corrections to the formal belief propagation recursion over the factor graph, leading to effective recon-struction of structured low-rank signal matrix. Then, we demonstrate how the ARM-AMP approach can be instantiated to solve various structured-matrix es-timation problems, including L&S compressive sensing and compressed robust principal component analysis (compressed RPCA). The resulting algorithms are applied to compressive hyperspectral imaging and compressed video surveil-lance, respectively.3) Using a scalar CP decomposition, we derive a generalized AMP-based approach for the generic CP-based tensor decomposition problem, and analyze its computational complexity. The proposed approach, called CP-AMP, explic-itly represents the probabilistic CP factorization in the factor graph, then adopts AMP approximations to the belief propagation recursion over the multi-linear factor graph. Then, we demonstrate how this CP-AMP approach is instantiated to solve the CP-based tensor completion problem. Applications are considered in facial image synthesis.Simulation results with both synthetic and real data demonstrate that the proposed algorithms yield the state-of-the-art reconstruction performances while maintaining low computational complexity.