Wireless networks are increasingly interference-limited, which motivates the development of sophisticated interference management techniques. One recently discovered approach is interference alignment, which attains the maximum sum rate scaling (with signal-to-noise ratio) in many network configurations. Interference alignment is not yet well understood from an engineering perspective. Such design considerations include (i) partial rather than complete knowledge of channel state information, (ii) correlated channels, (iii) bursty packet-based network traffic that requires the frequent setup and tear down of sessions, and (iv) the spatial distribution and interaction of transmit/receive pairs. This dissertation aims to establish the benefits and limitations of interference alignment under these four considerations. The first contribution of this dissertation considers an isolated group of transmit/receiver pairs (a cluster) cooperating through interference alignment and derives the signal-to-interference-plus-noise ratio distribution at each receiver for each stream. This distribution is used to compare interference alignment to beamforming and spatial multiplexing (as examples of common transmission techniques) in terms of sum rate to identify potential switching points between them. This dissertation identifies such switching points and provides design recommendations based on severity of the correlation or the channel state information uncertainty. The second contribution considers transmitters that are not associated with any interference alignment cooperating group but want to use the channel. The goal is to retain the benefits of interference alignment amid interference from the out-of-cluster transmitters. This dissertation shows that when the out-of-cluster transmitters have enough antennas, they can access the channel without changing the performance of the interference alignment receivers. Furthermore, optimum transmit filters maximizing the sum rate of the out-of-cluster transmit/receive pairs are derived. When insufficient antennas exist at the out-of-cluster transmitters, several transmit filters that trade off complexity and sum rate performance are presented. The last contribution, in contrast to the first two, takes into account the impact of large scale fading and the spatial distribution of the transmit/receive pairs on interference alignment by deriving the transmission capacity in a decentralized clustered interference alignment network. Channel state information uncertainty and feedback overhead are considered and the optimum training period is derived. Transmission capacity of interference alignment is compared to spatial multiplexing to highlight the tradeoff between channel estimation accuracy and the inter-cluster interference; the closer the nodes to each other, the higher the channel estimation accuracy and the inter-cluster interference. / text

The coding and feedback inaccuracies of the channel state information (CSI) in limited feedback multiple-input multiple-output (MIMO) wireless systems can severely impact the achievable data rate and reliability. The CSI is mathematically represented as a Grassmann manifold or manifold of unitary matrices. These are non-Euclidean spaces with special constraints that makes efficient and high fidelity coding especially challenging. In addition, the CSI inaccuracies may occur due to digital representation, time variation, and delayed feedback of the CSI. To overcome these inaccuracies, the manifold structure of the CSI can be exploited. The objective of this dissertation is to develop a new signal processing techniques on the manifolds to harvest the benefits of MIMO wireless systems. First, this dissertation presents the Kerdock codebook design to represent the CSI on the Grassmann manifold. The CSI inaccuracy due to digital representation is addressed by the finite alphabet structure of the Kerdock codebook. In addition, systematic codebook construction is identified which reduces the resource requirement in MIMO wireless systems. Distance properties on the Grassmann manifold are derived showing the applicability of the Kerdock codebook to beam-forming and spatial multiplexing systems. Next, manifold-constrained algorithms to predict and encode the CSI with high fidelity are presented. Two prominent manifolds are considered; the Grassmann manifold and the manifold of unitary matrices. The Grassmann manifold is a class of manifold used to represent the CSI in MIMO wireless systems using specific transmission strategies. The manifold of unitary matrices appears as a collection of all spatial information available in the MIMO wireless systems independent of specific transmission strategies. On these manifolds, signal processing building blocks such as differencing and prediction are derived. Using the proposed signal processing tools on the manifold, this dissertation addresses the CSI coding accuracy, tracking of the CSI under time variation, and compensation techniques for delayed CSI feedback. Applications of the proposed algorithms in single-user and multiuser systems show that most of the spatial benefits of MIMO wireless systems can be harvested. / text

With the increasing number of wireless devices and the phenomenal amount of data that is being generated by them, there is a growing interest in the wireless communications community to complement the traditional model-driven design approaches with data-driven machine learning (ML)-based solutions. However, managing the large-scale multi-dimensional data to maintain the efficiency and scalability of the ML algorithms has obviously been a challenge. Tensors provide a useful framework to represent multi-dimensional data in an integrated manner by preserving relationships in data across different dimensions. This thesis studies two new applications of tensors to ML for wireless communications where the tensor structure of the concerned data is exploited in novel ways. The first contribution of this thesis is a tensor learning-based low-complexity precoder codebook design technique for a full-dimension multiple-input multiple-output (FD-MIMO) system with a uniform planar antenna (UPA) array at the transmitter (Tx) whose channel distribution is available through a dataset. Represented as a tensor, the FD-MIMO channel is further decomposed using a tensor decomposition technique to obtain an optimal precoder which is a function of Kronecker-Product (KP) of two low-dimensional precoders, each corresponding to the horizontal and vertical dimensions of the FD-MIMO channel. From the design perspective, we have made contributions in deriving a criterion for optimal product precoder codebooks using the obtained low-dimensional precoders. We show that this product codebook design problem is an unsupervised clustering problem on a Cartesian Product Grassmann Manifold (CPM), where the optimal cluster centroids form the desired codebook. We further simplify this clustering problem to a $K$-means algorithm on the low-dimensional factor Grassmann manifolds (GMs) of the CPM which correspond to the horizontal and vertical dimensions of the UPA, thus significantly reducing the complexity of precoder codebook construction when compared to the existing codebook learning techniques. The second contribution of this thesis is a tensor-based bandwidth-efficient gradient communication technique for federated learning (FL) with convolutional neural networks (CNNs). Concisely, FL is a decentralized ML approach that allows to jointly train an ML model at the server using the data generated by the distributed users coordinated by a server, by sharing only the local gradients with the server and not the raw data. Here, we focus on efficient compression and reconstruction of convolutional gradients at the users and the server, respectively. To reduce the gradient communication overhead, we compress the sparse gradients at the users to obtain their low-dimensional estimates using compressive sensing (CS)-based technique and transmit to the server for joint training of the CNN. We exploit a natural tensor structure offered by the convolutional gradients to demonstrate the correlation of a gradient element with its neighbors. We propose a novel prior for the convolutional gradients that captures the described spatial consistency along with its sparse nature in an appropriate way. We further propose a novel Bayesian reconstruction algorithm based on the Generalized Approximate Message Passing (GAMP) framework that exploits this prior information about the gradients. Through the numerical simulations, we demonstrate that the developed gradient reconstruction method improves the convergence of the CNN model. / Master of Science / The increase in the number of wireless and mobile devices have led to the generation of massive amounts of multi-modal data at the users in various real-world applications including wireless communications. This has led to an increasing interest in machine learning (ML)-based data-driven techniques for communication system design. The native setting of ML is {em centralized} where all the data is available on a single device. However, the distributed nature of the users and their data has also motivated the development of distributed ML techniques. Since the success of ML techniques is grounded in their data-based nature, there is a need to maintain the efficiency and scalability of the algorithms to manage the large-scale data. Tensors are multi-dimensional arrays that provide an integrated way of representing multi-modal data. Tensor algebra and tensor decompositions have enabled the extension of several classical ML techniques to tensors-based ML techniques in various application domains such as computer vision, data-mining, image processing, and wireless communications. Tensors-based ML techniques have shown to improve the performance of the ML models because of their ability to leverage the underlying structural information in the data. In this thesis, we present two new applications of tensors to ML for wireless applications and show how the tensor structure of the concerned data can be exploited and incorporated in different ways. The first contribution is a tensor learning-based precoder codebook design technique for full-dimension multiple-input multiple-output (FD-MIMO) systems where we develop a scheme for designing low-complexity product precoder codebooks by identifying and leveraging a tensor representation of the FD-MIMO channel. The second contribution is a tensor-based gradient communication scheme for a decentralized ML technique known as federated learning (FL) with convolutional neural networks (CNNs), where we design a novel bandwidth-efficient gradient compression-reconstruction algorithm that leverages a tensor structure of the convolutional gradients. The numerical simulations in both applications demonstrate that exploiting the underlying tensor structure in the data provides significant gains in their respective performance criteria.

Ce mémoire de thèse concerne l'interpolation sur les variétés de Grassmann et ses applications à la réduction de modèles en mécanique et plus généralement aux systèmes d'équations aux dérivées partielles d'évolution. Après une description de la méthode POD, nous introduisons les fondements théoriques en géométrie des variétés de Grassmann, qui seront utilisés dans le reste de la thèse. Ce chapitre donne à ce mémoire à la fois une rigueur mathématique au niveau des algorithmes mis au point, leur domaine de validité ainsi qu'une estimation de l'erreur en distance grassmannienne, mais également un caractère auto-contenu "self-contained" du manuscrit. Ensuite, on présente la méthode d'interpolation sur les variétés de Grassmann introduite par David Amsallem et Charbel Farhat. Cette méthode sera le point de départ des méthodes d'interpolation que nous développerons dans les chapitres suivants. La méthode de Amsallem-Farhat consiste à choisir un point d'interpolation de référence, envoyer l'ensemble des points d'interpolation sur l'espace tangent en ce point de référence via l'application logarithme géodésique, effectuer une interpolation classique sur cet espace tangent, puis revenir à la variété de Grassmann via l'application exponentielle géodésique. On met en évidence par des essais numériques l'influence du point de référence sur la qualité des résultats. Dans notre premier travail, nous présentons une version grassmannienne d'un algorithme connu dans la littérature sous le nom de Pondération par Distance Inverse (IDW). Dans cette méthode, l'interpolé en un point donné est considéré comme le barycentre des points d'interpolation où les coefficients de pondération utilisés sont inversement "proportionnels" à la distance entre le point considéré et les points d'interpolation. Dans notre méthode, notée IDW-G, la distance géodésique sur la variété de Grassmann remplace la distance euclidienne dans le cadre standard des espaces euclidiens. L'avantage de notre algorithme, dont on a montré la convergence sous certaines conditions assez générales, est qu'il ne requiert pas de point de référence contrairement à la méthode de Amsallem-Farhat. Pour remédier au caractère itératif (point fixe) de notre première méthode, nous proposons une version directe via la notion de barycentre généralisé. Notons enfin que notre algorithme IDW-G dépend nécessairement du choix des coefficients de pondération utilisés. Dans notre second travail, nous proposons une méthode qui permet un choix optimal des coefficients de pondération, tenant compte de l'auto-corrélation spatiale de l'ensemble des points d'interpolation. Ainsi, chaque coefficient de pondération dépend de tous les points d'interpolation et non pas seulement de la distance entre le point considéré et un point d'interpolation. Il s'agit d'une version grassmannienne de la méthode de Krigeage, très utilisée en géostatique. La méthode de Krigeage grassmannienne utilise également le point de référence. Dans notre dernier travail, nous proposons une version grassmannienne de l'algorithme de Neville qui permet de calculer le polynôme d'interpolation de Lagrange de manière récursive via l'interpolation linéaire entre deux points. La généralisation de cet algorithme sur une variété grassmannienne est basée sur l'extension de l'interpolation entre deux points (géodésique/droite) que l'on sait faire de manière explicite. Cet algorithme ne requiert pas le choix d'un point de référence, il est facile d'implémentation et très rapide. De plus, les résultats numériques obtenus sont remarquables et nettement meilleurs que tous les algorithmes décrits dans ce mémoire. / This dissertation deals with interpolation on Grassmann manifolds and its applications to reduced order methods in mechanics and more generally for systems of evolution partial differential systems. After a description of the POD method, we introduce the theoretical tools of grassmannian geometry which will be used in the rest of the thesis. This chapter gives this dissertation a mathematical rigor in the performed algorithms, their validity domain, the error estimate with respect to the grassmannian distance on one hand and also a self-contained character to the manuscript. The interpolation on Grassmann manifolds method introduced by David Amsallem and Charbel Farhat is afterward presented. This method is the starting point of the interpolation methods that we will develop in this thesis. The method of Amsallem-Farhat consists in chosing a reference interpolation point, mapping forward all interpolation points on the tangent space of this reference point via the geodesic logarithm, performing a classical interpolation on this tangent space and mapping backward the interpolated point to the Grassmann manifold by the geodesic exponential function. We carry out the influence of the reference point on the quality of the results through numerical simulations. In our first work, we present a grassmannian version of the well-known Inverse Distance Weighting (IDW) algorithm. In this method, the interpolation on a point can be considered as the barycenter of the interpolation points where the used weights are inversely proportional to the distance between the considered point and the given interpolation points. In our method, denoted by IDW-G, the geodesic distance on the Grassmann manifold replaces the euclidean distance in the standard framework of euclidean spaces. The advantage of our algorithm that we show the convergence undersome general assumptions, does not require a reference point unlike the method of Amsallem-Farhat. Moreover, to carry out this, we finally proposed a direct method, thanks to the notion of generalized barycenter instead of an earlier iterative method. However, our IDW-G algorithm depends on the choice of the used weighting coefficients. The second work deals with an optimal choice of the weighting coefficients, which take into account of the spatial autocorrelation of all interpolation points. Thus, each weighting coefficient depends of all interpolation points an not only on the distance between the considered point and the interpolation point. It is a grassmannian version of the Kriging method, widely used in Geographic Information System (GIS). Our grassmannian Kriging method require also the choice of a reference point. In our last work, we develop a grassmannian version of Neville's method which allow the computation of the Lagrange interpolation polynomial in a recursive way via the linear interpolation of two points. The generalization of this algorithm to grassmannian manifolds is based on the extension of interpolation of two points (geodesic/straightline) that we can do explicitly. This algorithm does not require the choice of a reference point, it is easy to implement and very quick. Furthermore, the obtained numerical results are notable and better than all the algorithms described in this dissertation.