Tensorizatie met applicaties in blinde signaalscheiding

The value of data cannot be underestimated in our current digital age. Data mining techniques have allowed various priceless technological advances, influencing our daily lives to a significant extent. An important aspect of data mining is data representation. While vectors and matrices can be used to represent one-way and two-way data, respectively, so-called tensors are well suited to represent multiway data. The capabilities of recently developed tensor tools such as tensor decompositions surpass the power of their vector and matrix counterparts. It follows that these tools are already established in domains such as signal processing, statistics and machine learning. Tensor tools obviously require a tensor. In various applications such as source separation and data clustering, only one- or two-way data is available. While classic matrix tools sometimes fall short, tensor tools have the ability to, for example, uniquely identify underlying components. The main goal of this thesis consists of investigating how one can purposefully use tensor tools and exploit their powerful tensor properties in such applications given only a single vector or matrix. One approach encompasses a so-called tensorization step by first mapping the given data to a tensor. A number of tensorization techniques have appeared in the literature such as Hankelization and higher-order statistics. Not every mapping is meaningful though, and the effectiveness of a technique strongly depends on the problem at hand. In this thesis we present a comprehensive overview of both existing and novel tensorization techniques. We uncover relations between the properties of the given data and the properties of the tensor obtained after tensorization, and provide connections with tensor tools. We showcase the power of tensorization in the context of instantaneous and convolutive blind signal separation, including fetal heart rate extraction, direction-of-arrival estimation and blind separation of 16-QAM signals, and provide theoretical working conditions. Other applications are touched upon as well, such as data and graph clustering and the training of neural networks. Furthermore, we exploit our expertise in tensor-based optimization to propose a novel technique for nonnegative matrix factorization. Throughout the thesis, particular attention is paid to the use of tensorization in a large-scale context, leading to efficient representations of structured tensors and algorithms that are able to cope with large tensors after tensorization.status: publishe