During the last two decades we have witnessed considerable activity in building bridges between the fields of information theory/communications, computer science, and statistical physics. This is due to the realization that many fundamental concepts and notions in these fields are in fact related and that each field can benefit from the insight and techniques developed in the others. For instance, the notion of channel capacity in information theory, threshold phenomena in computer science, and phase transitions in statistical physics are all expressions of the same concept. Therefore, it would be beneficial to develop a common framework that unifies these notions and that could help to leverage knowledge in one field to make progress in the others. A particularly striking example is the celebrated belief propagation algorithm. It was independently invented in each of these fields but for very different purposes. The realization of the commonality has benefited each of the areas. We investigate polarization and spatial coupling: two techniques that were originally invented in the context of channel coding (communications) thus resulting for the first time in efficient capacity-achieving codes for a wide range of channels. As we will discuss, both techniques play a fundamental role also in computer science and statistical physics and so these two techniques can be seen as further fundamental building blocks that unite all three areas. We demonstrate applications of these techniques, as well as the fundamental phenomena they provide. In more detail, this thesis consists of two parts. In the first part, we consider the technique of polarization and its resultant class of channel codes, called polar codes. Our main focus is the analysis and improvement of the behavior of polarization towards the most significant aspects of modern channel-coding theory: scaling laws, universality, and complexity (quantization). For each of these aspects, we derive fundamental laws that govern the behavior of polarization and polar codes. Even though we concentrate on applications in communications, the analysis that we provide is general and can be carried over to applications of polarization in computer science and statistical physics. As we will show, our investigations confirm some of the inherent strengths of polar codes such as their robustness with respect to quantization. But they also make clear in which aspects further improvement of polar codes is needed. For example, we will explain that the scaling behavior of polar codes is quite slow compared to the optimal one. Hence, further research is required in order to enhance the scaling behavior of polar codes towards optimality. In the second part of this thesis, we investigate spatial coupling. By now, there exists already a considerable literature on spatial coupling in the realm of information theory and communications. We therefore investigate mainly the impact of spatial coupling on the fields of statistical physics and computer science. We consider two well-known models. The first is the Curie-Weiss model that provides us with the simplest model for understanding the mechanism of spatial coupling in the perspective of statistical physics. Many fundamental features of spatial coupling can be simply explained here. In particular, we will show how the well-known Maxwell construction in statistical physics manifests itself through spatial coupling. We then focus on a much richer class of graphical models called constraint satisfaction problems (CSP) (e.g., K-SAT and Q-COL). These models are central to computer science. We follow a general framework: First, we introduce interpolation procedures for proving that the coupled and standard (un-coupled) models are fundamentally related, in that their static properties (such as their SAT/UNSAT threshold) are the same. We then use tools from spin glass theory (cavity method) to demonstrate the so-called phenomenon of threshold saturation in these coupled models. Finally, we present the algorithmic implications and argue that all these features provide a new avenue for obtaining better, provable, algorithmic lower bounds on static thresholds of the individual standard CSP models. We consider simple decimation algorithms (e.g., the unit clause propagation algorithm) for the coupled CSP models and provide a machinery to analyze these algorithms. These analyses enable us to observe that the algorithmic thresholds on the coupled model are significantly improved over the standard model. For some models (e.g., 3-SAT, 3-COL), these coupled algorithmic thresholds surpass the best lower bounds on the SAT/UNSAT threshold in the literature and provide us with a new lower bound. We conclude by pointing out that although we only considered some specific graphical models, our results are of general nature hence applicable to a broad set of models. In particular, a main contribution of this thesis is to firmly establish both polarization, as well as spatial coupling, in the common toolbox of information theory/communication, statistical physics, and computer science.
EPFL2013
