Multi Terminal Probabilistic Compressed Sensing

In this paper, the `Approximate Message Passing' (AMP) algorithm, initially developed for compressed sensing of signals under i.i.d. Gaussian measurement matrices, has been extended to a multi-terminal setting (MAMP algorithm). It has been shown that similar to its single terminal counterpart, the behavior of MAMP algorithm is fully characterized by a `State Evolution' (SE) equation for large block-lengths. This equation has been used to obtain the rate-distortion curve of a multi-terminal memoryless source. It is observed that by spatially coupling the measurement matrices, the rate-distortion curve of MAMP algorithm undergoes a phase transition, where the measurement rate region corresponding to a low distortion (approximately zero distortion) regime is fully characterized by the joint and conditional Renyi information dimension (RID) of the multi-terminal source. This measurement rate region is very similar to the rate region of the Slepian-Wolf distributed source coding problem where the RID plays a role similar to the discrete entropy. Simulations have been done to investigate the empirical behavior of MAMP algorithm. It is observed that simulation results match very well with predictions of SE equation for reasonably large block-lengths.Comment: 11 pages, 13 figures. arXiv admin note: text overlap with arXiv:1112.0708 by other author

Polarization of the Renyi Information Dimension with Applications to Compressed Sensing

In this paper, we show that the Hadamard matrix acts as an extractor over the reals of the Renyi information dimension (RID), in an analogous way to how it acts as an extractor of the discrete entropy over finite fields. More precisely, we prove that the RID of an i.i.d. sequence of mixture random variables polarizes to the extremal values of 0 and 1 (corresponding to discrete and continuous distributions) when transformed by a Hadamard matrix. Further, we prove that the polarization pattern of the RID admits a closed form expression and follows exactly the Binary Erasure Channel (BEC) polarization pattern in the discrete setting. We also extend the results from the single- to the multi-terminal setting, obtaining a Slepian-Wolf counterpart of the RID polarization. We discuss applications of the RID polarization to Compressed Sensing of i.i.d. sources. In particular, we use the RID polarization to construct a family of deterministic $\pm 1$-valued sensing matrices for Compressed Sensing. We run numerical simulations to compare the performance of the resulting matrices with that of random Gaussian and random Hadamard matrices. The results indicate that the proposed matrices afford competitive performances while being explicitly constructed.Comment: 12 pages, 2 figure

Achieving the Fundamental Limit of Lossless Analog Compression via Polarization

In this paper, we study the lossless analog compression for i.i.d. nonsingular signals via the polarization-based framework. We prove that for nonsingular source, the error probability of maximum a posteriori (MAP) estimation polarizes under the Hadamard transform, which extends the polarization phenomenon to analog domain. Building on this insight, we propose partial Hadamard compression and develop the corresponding analog successive cancellation (SC) decoder. The proposed scheme consists of deterministic measurement matrices and non-iterative reconstruction algorithm, providing benefits in both space and computational complexity. Using the polarization of error probability, we prove that our approach achieves the information-theoretical limit for lossless analog compression developed by Wu and Verdu.Comment: 48 pages, 5 figures. This work was presented in part at the 2023 IEEE Global Communications Conferenc

Compressed Sensing of Memoryless Sources:A Deterministic Hadamard Construction

Compressed sensing is a new trend in signal processing for efficient sampling and signal acquisition. The idea is that most real-world signals have a sparse representation in an appropriate basis and this can be exploited to capture the sparse signal by taking only a few linear projections. The recovery is possible by running appropriate low-complexity algorithms that exploit the sparsity (prior information) to reconstruct the signal from the linear projections (posterior information). The main benefit is that the required number of measurements is much smaller than the dimension of the signal. This results in a huge gain in sensor cost (in measurement devices) or a dramatic saving in data acquisition time. However, some difficulties naturally arise in applying the compressed sensing to real-world applications such as robustness issues in taking the linear projections and computational complexity of the recovery algorithm. In this thesis, we design structured matrices for compressed sensing. In particular, we claim that some of the practical difficulties can be reasonably solved by imposing some structure on the measurement matrices. The thesis evolves around the Hadamard matrices which are $\{+1,-1\}$-valued matrices with many applications in signal processing, coding, optics and mathematics. As the title of the thesis implies, there are two main ingredients to this thesis. First, we use a memoryless assumption for the source, i.e., we assume that the nonzero components of the sparse signal are independently generated by a given probability distribution and their position is completely random. This allows us to use tools from probability, information theory and coding theory to rigorously assess the achievable performance. Second, using the mathematical properties of the Hadamard matrices, we design measurement matrices by selecting specific rows of a Hadamard matrix according to a deterministic criterion. We call the resulting matrices ``partial Hadamard matrices''. We design partial Hadamard matrices for three signal models: memoryless discrete signals and sparse signals with linear or sub-linear sparsity. A signal has linear sparsity if the number $k$ of its nonzero components is proportional to $n$, the dimension of signal, whereas it has a sub-linear sparsity if $k$ scales like $O(n^\alpha)$ for some $\alpha \in (0,1)$. We develop tools to rigorously analyze the performance of the proposed constructions by borrowing ideas from information theory and coding theory. We also extend our construction to distributed (multi-terminal) signals. Distributed compressed sensing is a ubiquitous problem in distributed data acquisition systems such as ad-hoc sensor networks. From both a theoretical and an engineering point of view, it is important to know how many measurement per dimension are necessary from different terminals in order to have a reliable estimate of the distributed data. We theoretically analyze this problem for a very simple setup where the components of the distributed signal are generated by a joint probability distribution which captures the spatial correlation among different terminals. We give an information-theoretic characterization of the measurements-rate region that results in a negligible recovery distortion. We also propose a low-complexity distributed message passing algorithm to achieve the theoretical limits