Applications of Coding Theory to Massive Multiple Access and Big Data Problems

The broad theme of this dissertation is design of schemes that admit iterative algorithms with low computational complexity to some new problems arising in massive multiple access and big data. Although bipartite Tanner graphs and low-complexity iterative algorithms such as peeling and message passing decoders are very popular in the channel coding literature they are not as widely used in the respective areas of study and this dissertation serves as an important step in that direction to bridge that gap. The contributions of this dissertation can be categorized into the following three parts. In the first part of this dissertation, a timely and interesting multiple access problem for a massive number of uncoordinated devices is considered wherein the base station is interested only in recovering the list of messages without regard to the identity of the respective sources. A coding scheme with polynomial encoding and decoding complexities is proposed for this problem, the two main features of which are (i) design of a close-to-optimal coding scheme for the T-user Gaussian multiple access channel and (ii) successive interference cancellation decoder. The proposed coding scheme not only improves on the performance of the previously best known coding scheme by ≈ 13 dB but is only ≈ 6 dB away from the random Gaussian coding information rate. In the second part construction-D lattices are constructed where the underlying linear codes are nested binary spatially-coupled low-density parity-check codes (SCLDPC) codes with uniform left and right degrees. It is shown that the proposed lattices achieve the Poltyrev limit under multistage belief propagation decoding. Leveraging this result lattice codes constructed from these lattices are applied to the three user symmetric interference channel. For channel gains within 0.39 dB from the very strong interference regime, the proposed lattice coding scheme with the iterative belief propagation decoder, for target error rates of ≈ 10^-5, is only 2:6 dB away the Shannon limit. The third part focuses on support recovery in compressed sensing and the nonadaptive group testing (GT) problems. Prior to this work, sensing schemes based on left-regular sparse bipartite graphs and iterative recovery algorithms based on peeling decoder were proposed for the above problems. These schemes require O(K logN) and Ω(K logK logN) measurements respectively to recover the sparse signal with high probability (w.h.p), where N, K denote the dimension and sparsity of the signal respectively (K (double backward arrow) N). Also the number of measurements required to recover at least (1 - €) fraction of defective items w.h.p (approximate GT) is shown to be cv€_K logN/K. In this dissertation, instead of the left-regular bipartite graphs, left-and- right regular bipartite graph based sensing schemes are analyzed. It is shown that this design strategy enables to achieve superior and sharper results. For the support recovery problem, the number of measurements is reduced to the optimal lower bound of Ω (K log N/K). Similarly for the approximate GT, proposed scheme only requires c€_K log N/ K measurements. For the probabilistic GT, proposed scheme requires (K logK log vN/ K) measurements which is only log K factor away from the best known lower bound of Ω (K log N/ K). Apart from the asymptotic regime, the proposed schemes also demonstrate significant improvement in the required number of measurements for finite values of K, N

Applications of Coding Theory to Massive Multiple Access and Big Data Problems

The broad theme of this dissertation is design of schemes that admit iterative algorithms with low computational complexity to some new problems arising in massive multiple access and big data. Although bipartite Tanner graphs and low-complexity iterative algorithms such as peeling and message passing decoders are very popular in the channel coding literature they are not as widely used in the respective areas of study and this dissertation serves as an important step in that direction to bridge that gap. The contributions of this dissertation can be categorized into the following three parts. In the first part of this dissertation, a timely and interesting multiple access problem for a massive number of uncoordinated devices is considered wherein the base station is interested only in recovering the list of messages without regard to the identity of the respective sources. A coding scheme with polynomial encoding and decoding complexities is proposed for this problem, the two main features of which are (i) design of a close-to-optimal coding scheme for the T-user Gaussian multiple access channel and (ii) successive interference cancellation decoder. The proposed coding scheme not only improves on the performance of the previously best known coding scheme by ≈ 13 dB but is only ≈ 6 dB away from the random Gaussian coding information rate. In the second part construction-D lattices are constructed where the underlying linear codes are nested binary spatially-coupled low-density parity-check codes (SCLDPC) codes with uniform left and right degrees. It is shown that the proposed lattices achieve the Poltyrev limit under multistage belief propagation decoding. Leveraging this result lattice codes constructed from these lattices are applied to the three user symmetric interference channel. For channel gains within 0.39 dB from the very strong interference regime, the proposed lattice coding scheme with the iterative belief propagation decoder, for target error rates of ≈ 10^-5, is only 2:6 dB away the Shannon limit. The third part focuses on support recovery in compressed sensing and the nonadaptive group testing (GT) problems. Prior to this work, sensing schemes based on left-regular sparse bipartite graphs and iterative recovery algorithms based on peeling decoder were proposed for the above problems. These schemes require O(K logN) and Ω(K logK logN) measurements respectively to recover the sparse signal with high probability (w.h.p), where N, K denote the dimension and sparsity of the signal respectively (K (double backward arrow) N). Also the number of measurements required to recover at least (1 - €) fraction of defective items w.h.p (approximate GT) is shown to be cv€_K logN/K. In this dissertation, instead of the left-regular bipartite graphs, left-and- right regular bipartite graph based sensing schemes are analyzed. It is shown that this design strategy enables to achieve superior and sharper results. For the support recovery problem, the number of measurements is reduced to the optimal lower bound of Ω (K log N/K). Similarly for the approximate GT, proposed scheme only requires c€_K log N/ K measurements. For the probabilistic GT, proposed scheme requires (K logK log vN/ K) measurements which is only log K factor away from the best known lower bound of Ω (K log N/ K). Apart from the asymptotic regime, the proposed schemes also demonstrate significant improvement in the required number of measurements for finite values of K, N

Computing a k-sparse n-length Discrete Fourier Transform using at most 4k samples and O(k log k) complexity

Given an $n$-length input signal \mbf{x}, it is well known that its Discrete Fourier Transform (DFT), \mbf{X}, can be computed in $O(n \log n)$ complexity using a Fast Fourier Transform (FFT). If the spectrum \mbf{X} is exactly $k$-sparse (where $k<<n$), can we do better? We show that asymptotically in $k$ and $n$, when $k$ is sub-linear in $n$ (precisely, $k \propto n^{\delta}$ where $0 < \delta <1$), and the support of the non-zero DFT coefficients is uniformly random, we can exploit this sparsity in two fundamental ways (i) {\bf {sample complexity}}: we need only $M=rk$ deterministically chosen samples of the input signal \mbf{x} (where $r < 4$ when $0 < \delta < 0.99$); and (ii) {\bf {computational complexity}}: we can reliably compute the DFT \mbf{X} using $O(k \log k)$ operations, where the constants in the big Oh are small and are related to the constants involved in computing a small number of DFTs of length approximately equal to the sparsity parameter $k$. Our algorithm succeeds with high probability, with the probability of failure vanishing to zero asymptotically in the number of samples acquired, $M$.Comment: 36 pages, 15 figures. To be presented at ISIT-2013, Istanbul Turke

Analysis of Spatially-Coupled Counter Braids

A counter braid (CB) is a novel counter architecture introduced by Lu et al. in 2007 for per-flow measurements on high-speed links. CBs achieve an asymptotic compression rate (under optimal decoding) that matches the entropy lower bound of the flow size distribution. Spatially-coupled CBs (SC-CBs) have recently been proposed. In this work, we further analyze single-layer CBs and SC-CBs using an equivalent bipartite graph representation of CBs. On this equivalent representation, we show that the potential and area thresholds are equal. We also show that the area under the extended belief propagation (BP) extrinsic information transfer curve (defined for the equivalent graph), computed for the expected residual CB graph when a peeling decoder equivalent to the BP decoder stops, is equal to zero precisely at the area threshold. This, combined with simulations and an asymptotic analysis of the Maxwell decoder, leads to the conjecture that the area threshold is in fact equal to the Maxwell decoding threshold and hence a lower bound on the maximum a posteriori (MAP) decoding threshold. Finally, we present some numerical results and give some insight into the apparent gap of the BP decoding threshold of SC-CBs to the conjectured lower bound on the MAP decoding threshold.Comment: To appear in the IEEE Information Theory Workshop, Jeju Island, Korea, October 201