The thesis consists of three articles concerning reproducing kernels ofweighted spaces of polyanalytic functions on the complex plane. In the first paper, we study spaces of polyanalytic polynomials equipped with a Gaussianweight. In the remaining two papers, more general weight functions are considered. More precisely, we provide two methods to compute asymptotic expansions for the kernels near the diagonal and then apply the techniques to get estimates for reproducing kernels of polyanalytic polynomial spaces equipped with rather general weight functions. / <p>QC 20130513</p>

The mathematical modeling and performance analysis of cellular networks have seen a major paradigm shift with the application of stochastic geometry. The main purpose of stochastic geometry is to endow probability distributions on the locations of the base stations (BSs) and users in a network, which, in turn, provides an analytical handle on the performance evaluation of cellular networks. To preserve the tractability of analysis, the common practice is to assume complete spatial randomness} of the network topology. In other words, the locations of users and BSs are modeled as independent homogeneous Poisson point processes (PPPs). Despite its usefulness, the PPP-based network models fail to capture any spatial coupling between the users and BSs which is dominant in a multi-tier cellular network (also known as the heterogeneous cellular networks (HetNets)) consisting of macro and small cells. For instance, the users tend to form hotspots or clusters at certain locations and the small cell BSs (SBSs) are deployed at higher densities at these locations of the hotspots in order to cater to the high data demand. Such user-centric deployments naturally couple the locations of the users and SBSs. On the other hand, these spatial couplings are at the heart of the spatial models used in industry for the system-level simulations and standardization purposes. This dissertation proposes fundamentally new spatial models based on stochastic geometry which closely emulate these spatial couplings and are conductive for a more realistic and fine-tuned performance analysis, optimization, and design of cellular networks. First, this dissertation proposes a new class of spatial models for HetNets where the locations of the BSs and users are assumed to be distributed as Poisson cluster process (PCP). From the modeling perspective, the proposed models can capture different spatial couplings in a network topology such as the user hotspots and user BS coupling occurring due to the user-centric deployment of the SBSs. The PCP-based model is a generalization of the state-of-the-art PPP-based HetNet model. This is because the model reduces to the PPP-based model once all spatial couplings in the network are ignored. From the stochastic geometry perspective, we have made contributions in deriving the fundamental distribution properties of PCP, such as the distance distributions and sum-product functionals, which are instrumental for the performance characterization of the HetNets, such as coverage and rate. The focus on more refined spatial models for small cells and users brings to the second direction of the dissertation, which is modeling and analysis of HetNets with millimeter wave (mm-wave) integrated access and backhaul (IAB), an emerging design concept of the fifth generation (5G) cellular networks. While the concepts of network densification with small cells have emerged in the fourth generation (4G) era, the small cells can be realistically deployed with IAB since it solves the problem of high capacity wired backhaul of SBSs by replacing the last-mile fibers with mm-wave links. We have proposed new stochastic geometry-based models for the performance analysis of IAB-enabled HetNets. Our analysis reveals some interesting system-design insights: (1) the IAB HetNets can support a maximum number of users beyond which the data rate drops below the rate of a single-tier macro-only network, and (2) there exists a saturation point of SBS density beyond which no rate gain is observed with the addition of more SBSs. The third and final direction of this dissertation is the combination of machine learning and stochastic geometry to construct a new class of data driven network models which can be used in the performance optimization and design of a network. As a concrete example, we investigate the classical problem of wireless link scheduling where the objective is to choose an optimal subset of simultaneously active transmitters (Tx-s) from a ground set of Tx-s which will maximize the network-wide sum-rate. Since the optimization problem is NP-hard, we replace the computationally expensive heuristic by inferring the point patterns of the active Tx-s in the optimal subset after training a determinantal point process (DPP). Our investigations demonstrate that the DPP is able to learn the spatial interactions of the Tx-s in the optimal subset and gives a reasonably accurate estimate of the optimal subset for any new ground set of Tx-s. / Doctor of Philosophy / The high speed global cellular communication network is one of the most important technologies, and it continues to evolve rapidly with every new generation. This evolution greatly depends on observing performance-trends of the emerging technologies on the network models through extensive system-level simulations. Since these simulation models are extremely time-consuming and error prone, the complementary analytical models of cellular networks have been an area of active research for a long time. These analytical models are intended to provide crisp insights on the network behavior such as the dependence of network performance metrics (such as coverage or rate) on key system-level parameters (such as transmission powers, base station (BS) density) which serve as the prior knowledge for more fine-tuned simulations. Over the last decade, the analytical modeling of the cellular networks has been driven by stochastic geometry. The main purpose of stochastic geometry is to endow the locations of the base stations (BSs) and users with probability distributions and then leverage the properties of these distributions to average out the spatial randomness. This process of spatial averaging allows us to derive the analytical expressions of the system-level performance metrics despite the presence of a large number of random variables (such as BS and user locations, channel gains) under some reasonable assumptions. The simplest stochastic geometry based model of cellular networks, which is also the most tractable, is the so-called Poisson point process (PPP) based network model. In this model, users and BSs are assumed to be distributed as independent homogeneous PPPs. This is equivalent to saying that the users and BSs independently and uniformly at random over a plane. The PPP-based model turned out to be a reasonably accurate representation of the yesteryear’s cellular networks which consisted of a single tier of macro BSs (MBSs) intended to provide a uniform coverage blanket over the region. However, as the data-hungry devices like smart-phones, tablets, and application like online gaming continue to flood the consumer market, the network configuration is rapidly deviating from this baseline setup with different spatial interactions between BSs and users (also termed spatial coupling) becoming dominant. For instance, the user locations are far from being homogeneous as they are concentrated in specific areas like residential and commercial zones (also known as hotspots). Further, the network, previously consisting of a single tier of macro BSs (MBSs), is becoming increasingly heterogeneous with the deployment of small cell BSs (SBSs) with small coverage footprints and targeted to serve the user hotspots. It is not difficult to see that the network topology with these spatial couplings is quite far from complete spatial randomness which is the basis of the PPP-based models. The key contribution of this dissertation is to enrich the stochastic geometry-based mathematical models so that they can capture the fine-grained spatial couplings between the BSs and users. More specifically, this dissertation contributes in the following three research directions. Direction-I: Modeling Spatial Clustering. We model the locations of users and SBSs forming hotspots as Poisson cluster processes (PCPs). A PCP is a collection of offspring points which are located around the parent points which belong to a PPP. The coupling between the locations of users and SBSs (due to their user-centric deployment) can be introduced by assuming that the user and SBS PCPs share the same parent PPP. The key contribution in this direction is the construction of a general HetNet model with a mixture of PPP and PCP-distributed BSs and user distributions. Note that the baseline PPP-based HetNet model appears as one of the many configurations supported by this general model. For this general model, we derive the analytical expressions of the performance metrics like coverage probability, BS load, and rate as functions of the coupling parameters (e.g. BS and user cluster size). Direction-II: Modeling Coupling in Wireless Backhaul Networks. While the deployment of SBSs clearly enhances the network performance in terms of coverage, one might wonder: how long network densification with tens of thousands of SBSs can meet the everincreasing data demand? It turns out that in the current network setting, where the backhaul links (i.e. the links between the BSs and core network) are still wired, it is not feasible to densify the network beyond some limit. This backhaul bottleneck can be overcome if the backhaul links also become wireless and the backhaul and access links (link between user and BS) are jointly managed by an integrated access and backhaul (IAB) network. In this direction, we develop the analytical models of IAB-enabled HetNets where the key challenge is to tackle new types of couplings which exist between the rates on the wireless access and backhaul links. Such couplings exist due to the spatial correlation of the signal qualities of the two links and the number of users served by different BSs. Two fundamental insights obtained from this work are as follows: (1) the IAB HetNets can support a maximum number of users beyond which the network performance drops below that of a single-tier macro-only network, and (2) there exists a saturation point of SBS density beyond which no performance gain is observed with the addition of more SBSs. Direction-III: Modeling Repulsion. In this direction, we focus on modeling another aspect of spatial coupling imposed by the intra-point repulsion. Consider a device-to-device (D2D) communication scenario, where some users are transmitting some on-demand content locally cached in their devices using a common channel. Any reasonable multiple access scheme will ensure that two nearly users are never simultaneously active as they will cause severe mutual interference and thereby reducing the network-wide sum rate. Thus the active users in the network will have some spatial repulsion. The locations of these users can be modeled as determinantal point processes (DPPs). The key property of DPP is that it forms a bridge between stochastic geometry and machine learning, two otherwise non-overlapping paradigms for wireless network modeling and design. The main focus in this direction is to explore the learning framework of DPP and bring together advantages of stochastic geometry and machine learning to construct a new class of data-driven analytical network models.

Object detection has been a fundamental part of many emerging technologies, such as autonomous vehicles, robotics, and security. As deep learning is the main reason behind the leap of performance in object detection, it has mostly been associated with a post-processing step of non-maximum suppression (NMS) to reduce the number of resulting bounding boxes output from the network to, ideally, one box per object. As non-maximum suppression blindly suppress the overlap with a pre-defined threshold, it introduces the problem of suppressing false negatives in crowded scenes by choosing a high threshold, or vice versa. This problem is critical, especially in the autonomous vehicle industry, as this concerns the safety of passengers. The problem of the machine understanding whether these bounding boxes belong to the same object or two near-by objects is still not directly solvable. Although a lot of previous research tried to invent a new box-reduction method, every method has its own drawbacks while solving the problem. That is why, until now, many researchers are still using non‐maximum suppression. In this research, a literature review was carried out to determine the best NMS alternatives. Then, an approach for box reduction based on determinantal point process (DPP) was implemented. Furthermore, an evaluation pipeline was introduced for experimental analysis for the differences between NMS and DPP. Although NMS shows a better performance in terms of precision and recall, DPP chooses better fitting bounding boxes. / Objektdetektering har varit en grundläggande del av många nya tekniker, t.ex. autonoma fordon, robotik och säkerhet. Eftersom djupinlärning är den främsta orsaken till den stora prestandaskillnaden vid objektsdetektering har den oftast varit förknippad med ett efterbehandlingssteg med icke-maximal undertryckning (NMS) för att minska antalet resulterande avgränsande rutor som produceras av nätverket till, idealt sett, en ruta per objekt. Eftersom icke-maximal undertryckning blint undertrycker överlappningen med ett fördefinierat tröskelvärde, uppstår problemet med att undertrycka falskt negativa resultat i överfulla scener genom att välja ett högt tröskelvärde, eller tvärtom. Detta problem är kritiskt, särskilt inom industrin för autonoma fordon, eftersom det gäller passagerarnas säkerhet. Problemet med att maskinen ska förstå om dessa avgränsande rutor tillhör samma objekt eller två närliggande objekt är fortfarande inte direkt lösbart. Även om man i tidigare forskning har försökt hitta en ny metod för att reducera boxar, har varje metod sina egna nackdelar när den löser problemet. Det är därför som många forskare fram till nu fortfarande använder sig av icke-maximalt undertryckande. I denna forskning gjordes en litteraturstudie för att fastställa de bästa NMS-alternativen. Därefter implementerades en metod för boxförminskning baserad på determinant punktprocess (DPP). Dessutom infördes en utvärderingsledning för experimentell analys av skillnaderna mellan NMS och DPP. Även om NMS visar en bättre prestanda när det gäller precision och återkallande, väljer DPP bättre passande avgränsande lådor.