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We are living in the era of "Big Data", an era characterized by a voluminous amount of available data. Such amount is mainly due to the continuing advances in the computational capabilities for capturing, storing, transmitting and processing data. However, it is not always the volume of data that matters, but rather the "relevant" information that resides in it. Exactly 70 years ago, Claude Shannon, the father of information theory, was able to quantify the amount of information in a communication scenario based on a probabilistic model of the data. It turns out that Shannon's theory can be adapted to various probability-based information processing fields, ranging from coding theory to machine learning. The computation of some information theoretic quantities, such as the mutual information, can help in setting fundamental limits and devising more efficient algorithms for many inference problems. This thesis deals with two different, yet intimately related, inference problems in the fields of coding theory and machine learning. We use Bayesian probabilistic formulations for both problems, and we analyse them in the asymptotic high-dimensional regime. The goal of our analysis is to assess the algorithmic performance on the first hand and to predict the Bayes-optimal performance on the second hand, using an information theoretic approach. To this end, we employ powerful analytical tools from statistical physics. The first problem is a recent forward-error-correction code called sparse superposition code. We consider the extension of such code to a large class of noisy channels by exploiting the similarity with the compressed sensing paradigm. Moreover, we show the amenability of sparse superposition codes to perform joint distribution matching and channel coding. In the second problem, we study symmetric rank-one matrix factorization, a prominent model in machine learning and statistics with many applications ranging from community detection to sparse principal component analysis. We provide an explicit expression for the normalized mutual information and the minimum mean-square error of this model in the asymptotic limit. This allows us to prove the optimality of a certain iterative algorithm on a large set of parameters. A common feature of the two problems stems from the fact that both of them are represented on dense graphical models. Hence, similar message-passing algorithms and analysis tools can be adopted. Furthermore, spatial coupling, a new technique introduced in the context of low-density parity-check (LDPC) codes, can be applied to both problems. Spatial coupling is used in this thesis as a "construction technique" to boost the algorithmic performance and as a "proof technique" to compute some information theoretic quantities. Moreover, both of our problems retain close connections with spin glass models studied in statistical mechanics of disordered systems. This allows us to use sophisticated techniques developed in statistical physics. In this thesis, we use the potential function predicted by the replica method in order to prove the threshold saturation phenomenon associated with spatially coupled models. Moreover, one of the main contributions of this thesis is proving that the predictions given by the "heuristic" replica method are exact. Hence, our results could be of great interest for the statistical physics community as well, as they help to set a rigorous mathematical foundation of the replica predictions.
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seeding'' perfect information on the replicas at the boundaries of the coupling chain. This extra information makes decoding easier near the boundaries, and this effect is then propagated into the coupling chain upon iterations of the decoding algorithm. Spatial coupling was also applied to various other problems that are governed by low-complexity message-passing algorithms, such as random constraint satisfaction problems, compressive sensing, and statistical physics. Each system has an associated algorithmic threshold and an optimal threshold. As with coding, once the underlying graphs are spatially coupled, the algorithms for these systems exhibit optimal performance. In this thesis, we analyze the performance of iterative low-complexity message-passing algorithms on general spatially coupled systems, and we specialize our results in coding theory applications. To do this, we express the evolution of the state of the system (along iterations of the algorithm) in a variational form, in terms of the so-called potential functional, in the continuum limit approximation. This thesis consists of two parts. In the first part, we consider the dynamic phase of the message-passing algorithm, in which iterations of the algorithm modify the state of the spatially coupled system. Assuming that the boundaries of the coupled chain are appropriately
seeded'', we find a closed-form analytical formula for the velocity with which the extra information propagates into the chain. We apply this result to coupled irregular LDPC code-ensembles with transmission over general BMS channels and to coupled general scalar systems. We perform numerical simulations for several applications and show that our formula gives values that match the empirical, observed velocity. This confirms that the continuum limit is an approximation well-suited to the derivation of the formula. In the second part of this thesis, we consider the static phase of the message-passing algorithm, when it can no longer modify the state of the system. We introduce a novel proof technique that employs displacement convexity, a mathematical tool from optimal transport, to prove that the potential functional is strictly displacement convex under an alternative structure in the space of probability measures. We hence establish the uniqueness of the state to which the spatially coupled system converges, and we characterize it. We apply this result to the (l,r)-regular Gallager ensemble with transmission over the BEC and to coupled general scalar systems.