Spatially Coupled Sparse Regression Codes: Design and State Evolution Analysis

We consider the design and analysis of spatially coupled sparse regression codes (SC-SPARCs), which were recently introduced by Barbier et al. for efficient communication over the additive white Gaussian noise channel. SC-SPARCs can be efficiently decoded using an Approximate Message Passing (AMP) decoder, whose performance in each iteration can be predicted via a set of equations called state evolution. In this paper, we give an asymptotic characterization of the state evolution equations for SC-SPARCs. For any given base matrix (that defines the coupling structure of the SC-SPARC) and rate, this characterization can be used to predict whether AMP decoding will succeed in the large system limit. We then consider a simple base matrix defined by two parameters $(\omega, \Lambda)$, and show that AMP decoding succeeds in the large system limit for all rates $R < \mathcal{C}$. The asymptotic result also indicates how the parameters of the base matrix affect the decoding progression. Simulation results are presented to evaluate the performance of SC-SPARCs defined with the proposed base matrix.

[1]  Michael Lentmaier,et al.  Spatially Coupled LDPC Codes Constructed From Protographs , 2014, IEEE Transactions on Information Theory.

[2]  Nicolas Macris,et al.  Threshold Saturation for Spatially Coupled LDPC and LDGM Codes on BMS Channels , 2013, IEEE Transactions on Information Theory.

[3]  Ramji Venkataramanan,et al.  Techniques for Improving the Finite Length Performance of Sparse Superposition Codes , 2017, IEEE Transactions on Communications.

[4]  Nicolas Macris,et al.  Threshold saturation of spatially coupled sparse superposition codes for all memoryless channels , 2016, 2016 IEEE Information Theory Workshop (ITW).

[5]  Florent Krzakala,et al.  Probabilistic reconstruction in compressed sensing: algorithms, phase diagrams, and threshold achieving matrices , 2012, ArXiv.

[6]  Henry D. Pfister,et al.  A Simple Proof of Maxwell Saturation for Coupled Scalar Recursions , 2013, IEEE Transactions on Information Theory.

[7]  Nicolas Macris,et al.  Universal Sparse Superposition Codes With Spatial Coupling and GAMP Decoding , 2017, IEEE Transactions on Information Theory.

[8]  Li Ping,et al.  Clipping Can Improve the Performance of Spatially Coupled Sparse Superposition Codes , 2017, IEEE Communications Letters.

[9]  Andrea Montanari,et al.  The dynamics of message passing on dense graphs, with applications to compressed sensing , 2010, 2010 IEEE International Symposium on Information Theory.

[10]  Sundeep Rangan,et al.  Generalized approximate message passing for estimation with random linear mixing , 2010, 2011 IEEE International Symposium on Information Theory Proceedings.

[11]  Florent Krzakala,et al.  Approximate message-passing with spatially coupled structured operators, with applications to compressed sensing and sparse superposition codes , 2013, 1312.1740.

[12]  Adel Javanmard,et al.  Information-Theoretically Optimal Compressed Sensing via Spatial Coupling and Approximate Message Passing , 2011, IEEE Transactions on Information Theory.

[13]  Andrew R. Barron,et al.  Fast Sparse Superposition Codes Have Near Exponential Error Probability for $R<{\cal C}$ , 2014, IEEE Transactions on Information Theory.

[14]  Sanghee Cho,et al.  APPROXIMATE ITERATIVE BAYES OPTIMAL ESTIMATES FOR HIGH-RATE SPARSE SUPERPOSITION CODES , 2013 .

[15]  Ramji Venkataramanan,et al.  The error exponent of sparse regression codes with AMP decoding , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[16]  Andrew R. Barron,et al.  Least squares superposition codes of moderate dictionary size, reliable at rates up to capacity , 2010, 2010 IEEE International Symposium on Information Theory.

[17]  Nicolas Macris,et al.  Proof of threshold saturation for spatially coupled sparse superposition codes , 2016, 2016 IEEE International Symposium on Information Theory (ISIT).

[18]  Ramji Venkataramanan,et al.  Capacity-Achieving Sparse Superposition Codes via Approximate Message Passing Decoding , 2015, IEEE Transactions on Information Theory.

[19]  Florent Krzakala,et al.  Approximate Message-Passing Decoder and Capacity Achieving Sparse Superposition Codes , 2015, IEEE Transactions on Information Theory.

[20]  Andrea Montanari,et al.  Message-passing algorithms for compressed sensing , 2009, Proceedings of the National Academy of Sciences.