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26
Compressed sensing of approximately-sparse signals: Phase transitions and optimal reconstruction
- in 50th Annual Allerton Conference on Communication, Control, and Computing
, 2012
"... Abstract—Compressed sensing is designed to measure sparse signals directly in a compressed form. However, most signals of interest are only “approximately sparse”, i.e. even though the signal contains only a small fraction of relevant (large) components the other components are not strictly equal to ..."
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Abstract—Compressed sensing is designed to measure sparse signals directly in a compressed form. However, most signals of interest are only “approximately sparse”, i.e. even though the signal contains only a small fraction of relevant (large) components the other components are not strictly equal to zero, but are only close to zero. In this paper we model the approximately sparse signal with a Gaussian distribution of small components, and we study its compressed sensing with dense random matrices. We use replica calculations to determine the mean-squared error of the Bayes-optimal reconstruction for such signals, as a function of the variance of the small components, the density of large components and the measurement rate. We then use the G-AMP algorithm and we quantify the region of parameters for which this algorithm achieves optimality (for large systems). Finally, we show that in the region where the G-AMP for the homogeneous measurement matrices is not optimal, a special “seeding ” design of a spatially-coupled measurement matrix allows to restore optimality. I.
Thresholds of spatially coupled systems via Lyapunov’s method
- in Proc. 2013 IEEE Inf. Theory Workshop
, 2013
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Performance improvement of iterative multiuser detection for large sparsely-spread CDMA systems by spatial coupling,” submitted to
- IEEE Trans. Inf. Theory, 2012, [Online]. Available
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Coupling Data Transmission for Capacity-Achieving Multiple-Access Communications
- IEEE TRAN. ON INFORM. THEORY
, 2012
"... We consider a signaling format where information is modulated via a superposition of independent data streams. Each data stream is formed by replication and permutation of encoded information bits. The relations between data bits and modulation symbols transmitted over the channel can be represented ..."
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We consider a signaling format where information is modulated via a superposition of independent data streams. Each data stream is formed by replication and permutation of encoded information bits. The relations between data bits and modulation symbols transmitted over the channel can be represented in the form of a sparse graph. The modulated streams are transmitted with a time offset enabling spatial coupling of the sparse modulation graphs. We prove that a two-stage demodulation/decoding method, in which iterative demodulation based on symbol estimation and interference cancellation is followed by parallel error correction decoding, achieves capacity on the additive white Gaussian noise (AWGN) channel.
Coupled Neural Associative Memories
"... We propose a novel architecture to design a neural associative memory that is capable of learning a large number of patterns and recalling them later in presence of noise. It is based on dividing the neurons into local clusters and parallel plains, very similar to the architecture of the visual cort ..."
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We propose a novel architecture to design a neural associative memory that is capable of learning a large number of patterns and recalling them later in presence of noise. It is based on dividing the neurons into local clusters and parallel plains, very similar to the architecture of the visual cortex of macaque brain. The common features of our proposed architecture with those of spatially-coupled codes enable us to show that the performance of such networks in eliminating noise is drastically better than the previous approaches while maintaining the ability of learning an exponentially large number of patterns. Previous work either failed in providing good performance during the recall phase or in offering large pattern retrieval (storage) capacities. We also present computational experiments that lend additional support to the theoretical analysis. I.
Replica analysis and approximate message passing decoder for superposition codes,” arXiv preprint arXiv:1403.8024
, 2014
"... Abstract—Superposition codes are efficient for the Additive White Gaussian Noise channel. We provide here a replica analysis of the performances of these codes for large signals. We also consider a Bayesian Approximate Message Passing decoder based on a belief-propagation approach, and discuss its p ..."
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Abstract—Superposition codes are efficient for the Additive White Gaussian Noise channel. We provide here a replica analysis of the performances of these codes for large signals. We also consider a Bayesian Approximate Message Passing decoder based on a belief-propagation approach, and discuss its performance using the density evolution technic. Our main findings are 1) for the sizes we can access, the message-passing decoder outperforms other decoders studied in the literature 2) its performance is limited by a sharp phase transition and 3) while these codes reach capacity as B (a crucial parameter in the code) increases, the performance of the message passing decoder worsen as the phase transition goes to lower rates. Superposition coding is a scheme for error-correction over the Additive White Gaussian Noise (AWGN) channel where a codeword Y ̃ is a sparse linear superposition of a random i.i.d
Properties of spatial coupling in compressed sensing
, 2014
"... In this paper we address a series of open questions about the construction of spatially coupled measurement matrices in compressed sensing. For hardware implementations one is forced to depart from the limiting regime of parameters in which the proofs of the so-called threshold saturation work. We ..."
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In this paper we address a series of open questions about the construction of spatially coupled measurement matrices in compressed sensing. For hardware implementations one is forced to depart from the limiting regime of parameters in which the proofs of the so-called threshold saturation work. We investigate quantitatively the behavior under finite coupling range, the dependence on the shape of the coupling interaction, and optimization of the so-called seed to minimize distance from optimality. Our analysis explains some of the properties observed empirically in previous works and provides new insight on spatially coupled compressed sensing.
