The current availability of powerful computers and huge data sets is creating new opportunities in computational mathematics to bring together concepts and tools from graph theory, machine learning and signal processing, creating Data Analytics on Graphs. In discrete mathematics, a graph is merely a collection of points (nodes) and lines connecting some or all of them. The power of such graphs lies in the fact that the nodes can represent entities as diverse as the users of social networks or financial market data, and that these can be transformed into signals which can be analyzed using data analytics tools. Data Analytics on Graphs is a comprehensive introduction to generating advanced data analytics on graphs that allows us to move beyond the standard regular sampling in time and space to facilitate modelling in many important areas, including communication networks, computer science, linguistics, social sciences, biology, physics, chemistry, transport, town planning, financial systems, personal health and many others. The authors revisit graph topologies from a modern data analytics point of view, and proceed to establish a taxonomy of graph networks. With this as a basis, the authors show how the spectral analysis of graphs leads to even the most challenging machine learning tasks, such as clustering, being performed in an intuitive and physically meaningful way. The authors detail unique aspects of graph data analytics, such as their benefits for processing data acquired on irregular domains, their ability to finely-tune statistical learning procedures through local information processing, the concepts of random signals on graphs and graph shifts, learning of graph topology from data observed on graphs, and confluence with deep neural networks, multi-way tensor networks and Big Data. Extensive examples are included to render the concepts more concrete and to facilitate a greater understanding of the underlying principles. Aimed at readers with a good grasp of the fundamentals of data analytics, this book sets out the fundamentals of graph theory and the emerging mathematical techniques for the analysis of a wide range of data acquired on graph environments. Data Analytics on Graphs will be a useful friend and a helpful companion to all involved in data gathering and analysis irrespective of area of application.

Modern applications in engineering and data science are increasingly based on multidimensional data of exceedingly high volume, variety, and structural richness. However, standard machine learning and data mining algorithms typically scale exponentially with data volume and complexity of cross-modal couplings - the so called curse of dimensionality - which is prohibitive to the analysis of such large-scale, multi-modal and multi-relational datasets. Given that such data are often conveniently represented as multiway arrays or tensors, it is therefore timely and valuable for the multidisciplinary machine learning and data analytic communities to review tensor decompositions and tensor networks as emerging tools for dimensionality reduction and large scale optimization. This monograph provides a systematic and example-rich guide to the basic properties and applications of tensor network methodologies, and demonstrates their promise as a tool for the analysis of extreme-scale multidimensional data. It demonstrates the ability of tensor networks to provide linearly or even super-linearly, scalable solutions. The low-rank tensor network framework of analysis presented in this monograph is intended to both help demystify tensor decompositions for educational purposes and further empower practitioners with enhanced intuition and freedom in algorithmic design for the manifold applications. In addition, the material may be useful in lecture courses on large-scale machine learning and big data analytics, or indeed, as interesting reading for the intellectually curious and generally knowledgeable reader.