Tensor Methods for Signal Reconstruction and Network Embedding

Abstract

Over the past few years, the avalanche of data along with advances in methodological and algorithmic design have triggered an increased interest in machine learning (ML) and signal processing (SP) research. How do we fuse and complete multi-dimensional signals? What is a concise and informative representation of entities in multi-dimensional networks? How do we develop efficient lightweight algorithms that handle very large data? These are important questions that have risen on the top of the scientific and engineering agenda of ML and SP communities. A plethora of methods has been proposed to answer such questions. While neural networks are the current trend and powerful non-linear data-driven tools, there exist principled alternatives, such as multi-linear tensor methods, that are also effective and oftentimes significantly outperform neural network approaches. In the era of data deluge, multi-dimensional data, also known as tensors, are ubiquitous in a number of engineering tasks and data analytics. Tensors can model various types of data in high-impact domains. Images, for example, are space-space-spectrum cubes that can be naturally represented as tensors. Different types of networks as knowledge graphs and networks with attributed nodes are also tailored to tensor modeling. On the other hand, tensor decompositions have proven essential tools in understanding, analyzing and processing multi-dimensional data. They offer a flexible analytical framework with solid foundations, as well as efficient algorithms that effectively handle multi-dimensional data. This thesis aims to answer the aforementioned questions by exploiting tensor modeling and decomposition tools. The objective is to propose elegant and effective solutions to a number of challenging machine learning and signal processing problems. In particular three main research thrusts are investigated: i) Hyperspectral super-resolution; ii) Tensor sampling and reconstruction; and iii) Network representation learning. For each of the thrusts, this thesis offers an efficient framework that is supported by theoretical analysis, algorithmic foundations and thorough experimental investigation.