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Optimization on Riemannian manifolds-the result of smooth geometry and optimization merging into one elegant modern framework-spans many areas of science and engineering, including machine learning, computer vision, signal processing, dynamical systems and scientific computing. This text introduces the differential geometry and Riemannian geometry concepts that will help students and researchers in applied mathematics, computer science and engineering gain a firm mathematical grounding to use these tools confidently in their research. Its charts-last approach will prove more intuitive from an optimizer's viewpoint, and all definitions and theorems are motivated to build time-tested optimization algorithms. Starting from first principles, the text goes on to cover current research on topics including worst-case complexity and geodesic convexity. Readers will appreciate the tricks of the trade for conducting research and for numerical implementations sprinkled throughout the book.
This extraordinary three-volume work, written in an engaging and rigorous style by a world authority in the field, provides an accessible, comprehensive introduction to the full spectrum of mathematical and statistical techniques underpinning contemporary methods in data-driven learning and inference. This final volume, Learning, builds on the foundational topics established in volume I to provide a thorough introduction to learning methods, addressing techniques such as least-squares methods, regularization, online learning, kernel methods, feedforward and recurrent neural networks, meta-learning, and adversarial attacks. A consistent structure and pedagogy is employed throughout this volume to reinforce student understanding, with over 350 end-of-chapter problems (including complete solutions for instructors), 280 figures, 100 solved examples, datasets and downloadable Matlab code. Supported by sister volumes Foundations and Inference, and unique in its scale and depth, this textbook sequence is ideal for early-career researchers and graduate students across many courses in signal processing, machine learning, data and inference.
This extraordinary three-volume work, written in an engaging and rigorous style by a world authority in the field, provides an accessible, comprehensive introduction to the full spectrum of mathematical and statistical techniques underpinning contemporary methods in data-driven learning and inference. This first volume, Foundations, introduces core topics in inference and learning, such as matrix theory, linear algebra, random variables, convex optimization and stochastic optimization, and prepares students for studying their practical application in later volumes. A consistent structure and pedagogy is employed throughout this volume to reinforce student understanding, with over 600 end-of-chapter problems (including solutions for instructors), 100 figures, 180 solved examples, datasets and downloadable Matlab code. Supported by sister volumes Inference and Learning, and unique in its scale and depth, this textbook sequence is ideal for early-career researchers and graduate students across many courses in signal processing, machine learning, statistical analysis, data science and inference.
With the rapid development of unmanned aerial vehicle (UAV), extensive attentions have been paid to UAV-aided data collection in wireless sensor networks. However, it is very challenging to maintain the information freshness of the sensor nodes (SNs) subject to the UAV’s limited energy capacity and/or the large network scale. This chapter introduces two modes of data collection: single and continuous data collection with the aid of UAV, respectively. In the former case, the UAVs are dispatched to gather sensing data from each SN just once according to a preplanned data collection strategy. To keep information fresh, a multistage approach is proposed to find a set of data collection points at which the UAVs hover to collect and the age-optimal flight trajectory of each UAV. In the later case, the UAVs perform data collection continuously and make real-time decisions on the uploading SN and flight direction at each step. A deep reinforcement learning (DRL) framework incorporating the deep Q-network (DQN) algorithm is proposed to find the age-optimal data collection solution subject to the maximum flight velocity and energy capacity of each UAV. Numerical results are presented to show the effectiveness of the proposed methods in different scenarios.
The Age of Information metric, which is a measure of the freshness of a continually updating piece of information as observed at a remote monitor, has been studied for a variety of different update monitoring systems. In this chapter, we introduce three network control mechanisms for controlling the age, namely, buffer size, packet deadlines, and packet management. In the case of packet deadlines, we analyze update monitoring system for the cases of a fixed deadline and a random exponential deadline and derive closed-form expressions for the average age. We also derive a closed-form expression for the optimal average deadline for the random exponential case.
In this chapter, we study the economic issues of fresh data trading markets, where the data freshness is captured by Age-of-Information (AoI). In our model, a destination user requests, and pays for, fresh data updates from a source provider. In this work, the destination incurs an age-related cost, modeled as a general increasing function of the AoI. To understand economic viability and profitability of fresh data markets, we consider a pricing mechanism to maximize the source’s profit, while the destination chooses a data update schedule to trade off its payments to the source and its age-related cost. The problem is exacerbated when the source has incomplete information regarding the destination’s age-related cost, which requires one to exploit (economic) mechanism design to induce the truthful information. This chapter attempts to build such a fresh data trading framework that centers around the following two key questions: (a) How should a source choose the pricing scheme to maximize its profit in a fresh data market under complete market information? (b) Under incomplete information, how should a source design an optimal mechanism to maximize its profit while ensuring the destination’s truthful report of its age-related cost information?
Optimization of information freshness in wireless networks has usually been performed based on queueing analysis that captures only the temporal traffic dynamics associated with the transmitters and receivers. However, the effect of interference, which is mainly dominated by the interferers’ geographic locations, is not well understood. This chapter presents a theoretical framework for the analysis of the Age of Information (AoI) from a joint queueing-geometry perspective. We also provide the design of a decentralized scheduling policy that exploits local observation to make transmission decisions that minimize the AoI. To quantify the performance, we derive analytical expressions for the average AoI. Numerical results validate the accuracy of the analyses as well as the efficacy of the proposed scheme in reducing the AoI.
In this chapter, we study the Age of Information (AoI) when the status updates of the underlying process of interest can be sampled at any time by the source node and are transmitted over an error-prone wireless channel. We assume the availability of perfect feedback that informs the transmitter about the success or failure of transmitted status updates and consider various retransmission strategies. More specifically, we study the scheduling of sampling and transmission of status updates in order to minimize the long-term average AoI at the destination under resource constraints. We assume that the underlying statistics of the system are not known, and hence, propose average-cost reinforcement learning algorithms for practical applications. Extensions of the results to a multiuser setting with multiple receivers and to an energy-harvesting source node are also presented, different reinforcement learning methods including deep Q Network (DQN) are exploited and their performances are demonstrated.
In this chapter, we study the value of information, a more comprehensive instrument than the age of information, for shaping the information flow in a networked control system subject to random processing delay. In addition, we establish a connection between these two instruments by presenting a condition under which the value of information is expressible in terms of the age of information. Nonetheless, we show that this condition is not achievable without a degradation in the performance of the system.
This chapter characterizes the average Age of Information (AoI) for the case of having multiple sources sharing a service facility with a single server. In particular, a simplified explanation of the SHS for AoI approach is provided to calculate the average age of updates of any source at the monitor. This approach is applied to various queueing systems including FCFS, M/M/1*, and M/M/1/2*, and the latter two with and without source priorities.
This chapter considers an application of age of information called AoCSI in which the channel states in a wireless network represent the information of interest and the goal is to maintain fresh estimates of these channel states at each node in the network. Rather than sampling some underlying time-varying process and propagating updates through a queue or graph, the AoCSI setting obtains direct updates of the channels as a by-product of wireless communication through standard physical layer channel estimation techniques. These CSI estimates are then disseminated through the network to provide global snapshots of the CSI to all of the nodes in the network. What makes the AoCSI setting unique is that disseminating some CSI updates and directly sampling/estimating other CSI occur simultaneously. Moreover, as illustrated in this chapter, there are inherent trade-offs on how much CSI should be disseminated in each transmission to minimize the average or maximum age.
Age-of-information (AoI) is used to characterize the freshness of information, and is critical for information monitoring, tracking, and control, which is typically required in many network applications, such as autonomous vehicles, virtual/augmented reality, and Internet-of-Things (IoT). Both inter-arrival times and delays of packets affect AoI performance, and thus traditional delay-efficient algorithms do not necessarily exhibit low AoI performance. This calls for “age-efficient” algorithm design in communication networks, which forms the focus of this chapter. In particular, we first discuss the recent advances in the age-efficient algorithm design for three different types of common network traffic: (i) elastic traffic (cf. Section 1.1): packets are allowed to be delivered without any specific deadline constraints; (ii) inelastic traffic (cf. Section 1.2): packets will be dropped if they are not delivered within a specific deadline; (iii) heterogeneous traffic (cf. Section 1.3): different packets may have different size. To facilitate our discussions, we explicitly consider the discrete-time model and emphasize the difference between age-efficient and delay-efficient algorithm design paradigms. Then, we examine “fresh” scheduling design for remote estimation with the goal of optimally balancing the trade-of between the estimation accuracy and the communication cost (cf. Section 1.4).