Abstract: Identifying geometric patterns in data has been a perennial task in signal processing and machine learning. Learning from geometric data patterns is even more crucial nowadays, in our data-analytics and artificial-intelligence era. To meet the needs of the data analytics/processing community, the present tutorial plans to offer an overview of the wide landscape of methods for learning from data in manifolds. Two manifold-modeling cases with wide applicability for the learning tasks of clustering, classification and regression will be considered: (i) Riemannian manifolds with well-known structure, such as the sets of all tensors of fixed rank and all positive semidefinite matrices, and (ii) manifolds with structure which is unknown to the user but needs to be identified to serve as a modeling assumption for highly non-linear data structures. Without diving into involved mathematical concepts, this tutorial aspires to present in simple terms state-of-the-art methods and recent developments, including recent efforts via the popular nowadays neural networks. Although the relevant applications are numerous and diverse, ranging from computer-vision tasks to system identification in robotics, the present tutorial builds on the important applications of dynamic magnetic resonance imaging and time-series analysis in brain networks to clarify arguments and solidify concepts.

Abstract: The objective of this tutorial is to introduce attendees to recent results involving monotonic and (weakly) scalable mappings in applications in the wireless domain. These mappings, known as interference mappings in the wireless community and popularized by Yates in the 90’s, are increasingly being used to solve utility optimization problems in current and future wireless systems. For example, they have been used to estimate load in 4G networks and, more recently, to solve power control problems in massive MIMO and cell-less systems, key technologies to 5G and beyond-5G networks. From a mathematical perspective, common to these problems is that their solutions can be characterized as fixed points of possibly normalized interference mappings, which opens up the opportunity to address wireless network problems with a unified framework.

In this tutorial we start with a brief introduction to the now classic axiomatic framework of Yates, and then we proceed to show that this framework and many recent generalizations can be directly related to problems that have been independently studied in mathematics. This fact enables us to bring powerful mathematical machinery to the wireless domain. In particular, we will show that the concept of nonlinear spectral radius of interference mappings can be used to identify whether a given network configuration is able to support the required traffic, to provide bounds for the convergence speed of existing iterative algorithms for load estimation, and to indicate whether a wireless network is likely to be operating in an interference limited regime or a noise limited regime. We will also show that some weighted max-min utility optimization problems can be solved with simple fixed point algorithms if we use achievable rates as the utility, and not the signal-to-interference noise ratio as commonly done, even though the resulting optimization problems use seemingly difficult-to-handle nonlinear mappings in their formulation. Although the focus of this tutorial is on wireless systems, we note that the theory to be presented can address problems in different fields, including, for example, in the analysis of some deep neural networks.