This chapter studies the problem of (unsupervised) community detection on large random graphs, with a focus on the dense graph setting for both stochastic block model and its degree-corrected variant. Discussion on sparse graphs are made, however, via a stats-physics-inspired heuristic approach.

This chapter covers large neural networks with random weights, in both feed-forward and recurrent settings. While being rather different from modern deep neural networks, these preliminary results shed new light on the interplay between data, network structure, and nonlinear neurons, leading to the somewhat surprising double descent phenomenon. The impact of gradient-based optimization method on the resulting network and more advanced consideration of deep networks are also discussed.

In this chapter, examples of concrete machine learning and statistical inference/estimation problems involving covariance matrix-based estimators are discussed. This includes the generalized likelihood ratio test, the popular linear and quadratic discriminant analysis, subspace methods such as MUSIC for direction of arrival estimation, covariance distance estimation, as well as robust covariance estimation.

This chapter discusses the generalized linear classifier that results from convex optimization problem and takes in general nonexplicit form. Random matrix theory is combined with leave-one-out arguments to handle the technical difficulty due to implicity. Again, counterintuitive phenomena arise in popular machine learning methods such as logistic regression or SMV in the large-dimensional setting, a well-defined solution may not even exist, and if it does, it behaves dramatically from its small-dimensional counterpart.

This chapter covers the basics of random matrix theory, within the unified framework of resolvent- and deterministic-equivalent approach. Historical and foundational random matrix results are presented in the proposed framework, together with heuristic derivations as well as detailed proofs. Topics such as statistical inference and spiked models are covered. The concentration-of-measure framework, as a newly born yet very flexible and powerful technical approach, is discussed at the end of the chapter.

This chapter discusses the fundamental kernel methods, with applications to supervised (kernel ridge regression or LS-SVM), semi-supervised (graph Laplacian-based learning), or unsupervised learning (such as kernel spectral clustering) schemes. By focusing on the typical examples of distance and inner-product-type kernels, we show how large-dimensional kernel approach differs from our small-dimensional intuition, and perhaps more importantly, how random matrix theory plays a central role in understanding and tuning various kernel-based methods.

This chapter discusses the fundamentally different mental images of large-dimensional machine learning (versus its small-dimensional counterpart), through the examples of sample covariance matrices and kernel matrices, on both synthetic and real data. Random matrix theory is presented as a flexible and powerful tool to assess, understand, and improve classical machine learning methods in this modern large-dimensional setting.

This chapter exploits concentration-of-measure approach for real data modeling, via the recent advance of deep generative adversarial networks (GANs). This assessment theoretically supports the surprisingly good match between theory and practice observed on real-world data in previous chapters. Conclusion on the universality of large-dimensional machine learning is drawn at the end of the chapter.

In this section, we study the second most investigated application of random matrix theory to wireless communications, namely multiple antenna systems, first introduced and motivated by the pioneering works of Telatar [Telatar, 1995] and Foschini [Foschini and Gans, 1998]. While large dimensional system analysis is easily defensible in CDMA networks, which typically allow for a large number of users with large orthogonal or random codes, it is not so for multiple antenna communications. Indeed, when it comes to applying approximated results provided by random matrix theory analysis, we expect that the typical system dimensions are of order ten to a thousand. However, for multiple input multiple output (MIMO) setups, the system dimensions can be of order 4, or even 2. Asymptotic results for such systems are then of minor interest. However, it will turn out in some specific scenarios that the difference between the ergodic capacity for multiple antenna systems and their respective deterministic equivalents is sometimes of order O(1/N), N being the typical system dimension. The per-receive antenna rate, which is of interest for studying the cost and gain of bringing additional antennas on finite size devices, can therefore be approximated within O(1/N2). This is a rather convenient rate, even for small N. In fact, as will be observed through simulations, the accuracy of the deterministic equivalents is often even better.

Quasi-static MIMO fading channels

We hereafter recall the foundations of multiple antenna communications.We first assume a simple point-to-point communication between a transmitter equipped with nt antennas and a receiver equipped with nr antennas.

Channel modeling is a fundamental field of research, as all communication models that were used so far in this book as well as in the whole mobile communication literature are based on such models. These models are by definition meant to provide an adequate representation of real practical communication environments. As such, i.i.d. Gaussian channel models are meant to represent the most scattered environment possible where multiple waveforms reflecting on a large number of scattering objects add up non-coherently and independently on the receive antennas. From this point of view, the Gaussian assumption is due to a loose application of the law of large numbers. Due to the mobility of the communication devices in the propagation environment, the statistical disposition of scatterers is rather uniform, hence the i.i.d. property. This is basically the physical arguments for using Gaussian i.i.d. models, along with confirmation by field measurements. An alternative explanation for Gaussian channel models will be provided in this chapter, which accounts for a priori information available at the system modeler, rather than for physical interpretations.

However, the complexity of propagation environments call for more involved models. This is why the Kronecker model comes into play to adequately model correlations arising at either communication end, supposedly independent from one another. This is also why the Rician model is of common use, as it can take into account possible line-of-sight components in the channel, which can often be assumed of constant fading value for a rather long time period, compared to the fast varying scattering part of the channel.