We study the approximate message-passing decoder for sparse superposition coding on the additive white Gaussian noise channel and extend our preliminary work. We use heuristic statistical-physics-based tools, such as the cavity and the replica methods, for the statistical analysis of the scheme. While superposition codes asymptotically reach the Shannon capacity, we show that our iterative decoder is limited by a phase transition similar to the one that happens in low density parity check codes. We consider two solutions to this problem, that both allow to reach the Shannon capacity: 1) a power allocation strategy and 2) the use of spatial coupling, a novelty for these codes that appears to be promising. We present, in particular, simulations, suggesting that spatial coupling is more robust and allows for better reconstruction at finite code lengths. Finally, we show empirically that the use of a fast Hadamard-based operator allows for an efficient reconstruction, both in terms of computational time and memory, and the ability to deal with very large messages.
@cite_22: In a recent paper, the authors proposed a new class of low-complexity iterative thresholding algorithms for reconstructing sparse signals from a small set of linear measurements [1]. The new algorithms are broadly referred to as AMP, for approximate message passing. This is the first of two conference papers describing the derivation of these algorithms, connection with the related literature, extensions of the original framework, and new empirical evidence. In particular, the present paper outlines the derivation of AMP from standard sum-product belief propagation, and its extension in several directions. We also discuss relations with formal calculations based on statistical mechanics methods.
@cite_28: Abstract The Sherrmgton-Kirkpatrick model of a spin glass is solved by a mean field technique which is probably exact in the limit of infinite range interactions. At and above T c the solution is identical to that obtained by Sherrington and Kirkpatrick (1975) using the n → O replica method, but below T c the new result exhibits several differences and remains physical down to T = 0.
@cite_41: This paper surveys recent work in applying ideas from graphical models and message passing algorithms to solve large scale regularized regression problems. In particular, the focus is on compressed sensing reconstruction via 1 penalized least-squares (known as LASSO or BPDN). We discuss how to derive fast approximate message passing algorithms to solve this problem. Surprisingly, the analysis of such algorithms allows to prove exact high-dimensional limit results for the LASSO risk. This paper will appear as a chapter in a book on ‘Compressed Sensing’ edited by Yonina Eldar and Gitta Kutynok.
@cite_12: “Approximate message passing” (AMP) algorithms have proved to be effective in reconstructing sparse signals from a small number of incoherent linear measurements. Extensive numerical experiments further showed that their dynamics is accurately tracked by a simple one-dimensional iteration termed state evolution. In this paper, we provide rigorous foundation to state evolution. We prove that indeed it holds asymptotically in the large system limit for sensing matrices with independent and identically distributed Gaussian entries. While our focus is on message passing algorithms for compressed sensing, the analysis extends beyond this setting, to a general class of algorithms on dense graphs. In this context, state evolution plays the role that density evolution has for sparse graphs. The proof technique is fundamentally different from the standard approach to density evolution, in that it copes with a large number of short cycles in the underlying factor graph. It relies instead on a conditioning technique recently developed by Erwin Bolthausen in the context of spin glass theory.
@cite_17: We consider the estimation of a random vector observed through a linear transform followed by a componentwise probabilistic measurement channel. Although such linear mixing estimation problems are generally highly non-convex, Gaussian approximations of belief propagation (BP) have proven to be computationally attractive and highly effective in a range of applications. Recently, Bayati and Montanari have provided a rigorous and extremely general analysis of a large class of approximate message passing (AMP) algorithms that includes many Gaussian approximate BP methods. This paper extends their analysis to a larger class of algorithms to include what we call generalized AMP (G-AMP). G-AMP incorporates general (possibly non-AWGN) measurement channels. Similar to the AWGN output channel case, we show that the asymptotic behavior of the G-AMP algorithm under large i.i.d. Gaussian transform matrices is described by a simple set of state evolution (SE) equations. The general SE equations recover and extend several earlier results, including SE equations for approximate BP on general output channels by Guo and Wang.
The AMP algorithm, which stands at the roots of our approach, is a simple relaxation of loopy BP. While the principle behind AMP has been used for a long time in physics under the name of Thouless-Anderson-Palmer equations @cite_28 , the present form has been originally derived for compressed sensing @cite_22 @cite_17 @cite_41 and is naturally applied to sparse superposition codes as this scheme can be interpreted as a compressed sensing problem with structured sparsity. The state evolution technique @cite_12 is unfortunately not yet fully rigorous for the present AMP approach, due to the structured sparsity of the signal, but in spite of that, we conjecture that it is exact.