In this thesis we consider concentration inequalities and the concentration of measure phenomenon from a variety of angles. Sharp tail bounds on the deviation of Lipschitz functions of independent random variables about their mean are well known. We consider variations on this theme for dependent variables on the Boolean cube. In recent years negatively associated probability distributions have been studied as potential generalizations of independent random variables. Results on this class of distributions have been sparse at best, even when restricting to the Boolean cube. We consider the class of negatively associated distributions topologically, as a subset of the general class of probability measures. Both the weak (distributional) topology and the total variation topology are considered, and the simpler notion of negative correlation is investigated. The concentration of measure phenomenon began with Milman's proof of Dvoretzky's theorem, and is therefore intimately connected to the field of high-dimensional convex geometry. Recently this field has found application in the area of compressed sensing. We consider these applications and in particular analyze the use of Gordon's min-max inequality in various compressed sensing frameworks, including the Dantzig selector and the matrix uncertainty selector. Finally we consider the use of concentration inequalities in developing a theoretically sound anomaly detection algorithm. Our method uses a ranking procedure based on KNN graphs of given data. We develop a max-margin learning-to-rank framework to train limited complexity models to imitate these KNN scores. The resulting anomaly detector is shown to be asymptotically optimal in that for any false alarm rate α, its decision region converges to the α-percentile minimum volume level set of the unknown underlying density.

During the last half century there has been a resurgence of interest in Monge's 18th century mass transportation problem, with most of the activity limited to continuous spaces. This thesis, consequently, develops techniques based on mass transportation for the purpose of obtaining tight concentration inequalities in a discrete setting. Such inequalities on n-fold products of graphs, equipped with product measures, have been well investigated using combinatorial and probabilistic techniques, the most notable being martingale techniques. The emphasis here, is instead on the analytic viewpoint, with the precise contribution being as follows. We prove that the modified log-Sobolev inequality implies the transportation inequality in the first systematic comparison of the modified log-Sobolev inequality, the Poincar inequality, the transportation inequality, and a new variance transportation inequality. The duality shown by Bobkov and Gtze of the transportation inequality and a generating function inequality is then utilized in finding the asymptotically correct value of the subgaussian constant of a cycle, regardless of the parity of the length of the cycle. This result tensorizes to give a tight concentration inequality on the discrete torus. It is interesting in light of the fact that the corresponding vertex isoperimetric problem has remained open in the case of the odd torus for a number of years. We also show that the class of bounded degree expander graphs provides an answer, in the affirmative, to the question of whether there exists an infinite family of graphs for which the spread constant and the subgaussian constant differ by an order of magnitude. Finally, a candidate notion of a discrete Ricci curvature for finite Markov chains is given in terms of the time decay of the Wasserstein distance of the chain to its stationarity. It can be interpreted as a notion arising naturally from a standard coupling of Markov chains. Because of its natural definition, ease of calculation, and tensoring property, we conclude that it deserves further investigation and development. Overall, the thesis demonstrates the utility of using the mass transportation problem in discrete isoperimetric and functional inequalities.

The object of this thesis is the study of constrained measurement systems of signals having low-dimensional structure using analytic tools from Compressed Sensing (CS). Realistic measurement systems usually have architectural constraints that make them differ from their idealized, well-studied counterparts. Nonetheless, these measurement systems can exploit structure in the signals that they measure. Signals considered in this research have low-dimensional structure and can be broken down into two types: static or dynamic. Static signals are either sparse in a specified basis or lying on a low-dimensional manifold (called manifold-modeled signals). Dynamic signals, exemplified as states of a dynamical system, either lie on a low-dimensional manifold or have converged onto a low-dimensional attractor. In CS, the Restricted Isometry Property (RIP) of a measurement system ensures that distances between all signals of a certain sparsity are preserved. This stable embedding ensures that sparse signals can be distinguished one from another by their measurements and therefore be robustly recovered. Moreover, signal-processing and data-inference algorithms can be performed directly on the measurements instead of requiring a prior signal recovery step. Taking inspiration from the RIP, this research analyzes conditions on realistic, constrained measurement systems (of the signals described above) such that they are stable embeddings of the signals that they measure. Specifically, this thesis focuses on four different types of measurement systems. First, we study the concentration of measure and the RIP of random block diagonal matrices that represent measurement systems constrained to make local measurements. Second, we study the stable embedding of manifold-modeled signals by existing CS matrices. The third part of this thesis deals with measurement systems of dynamical systems that produce time series observations. While Takens' embedding result ensures that this time series output can be an embedding of the dynamical systems' states, our research establishes that a stronger stable embedding result is possible under certain conditions. The final part of this thesis is the application of CS ideas to the study of the short-term memory of neural networks. In particular, we show that the nodes of a recurrent neural network can be a stable embedding of sparse input sequences.