Dictionary learning and multidimensional processing for tensor data
Creators: name: Shakeri, Zahra
Publisher: name: No Publisher Supplied
Year: 2019

description: Modern machine learning and signal processing relies on finding meaningful and succinct representations of data. While most works in the literature have focused on finding representations of vector data, many of today's data are collected using various sensors and have a multidimensional structure. This dissertation addresses the problem of feature learning for tensor (i.e., multiway) data, which are defined as data having multiple modes. The work presented in this dissertation aims to study the theoretical and algorithmic aspects of dictionary learning from tensor data and further investigate the computational aspects of exploiting the structure of tensor data in wireless communication systems. The dissertation has been divided into three main parts.The first part of the dissertation is focused on the theoretical aspects of Kronecker-structured dictionary learning from tensor data. Here, the structure of tensor data is exploited by requiring that the dictionary underlying the vectorized versions of tensor data samples be Kronecker structured. That is, it is comprised of coordinate dictionaries that independently transform various modes of the tensor data. The presented results are primarily stated in terms of lower and upper bounds on the sample complexity of dictionary learning, defined as the number of samples needed to reconstruct the true structured dictionary underlying the tensor data from noisy samples. These results highlight the effects of different parameters on the sample complexity of the problem and also bring out the potential advantages of structured dictionary learning from tensor data. The second part of this dissertation focuses on extending the Kronecker-structured dictionary learning model to a less restrictive class of dictionaries referred to as low-separation-rank dictionary learning, while still exploiting the structure of tensor data in the underlying dictionary. Various computational algorithms are developed to learn such dictionaries in cases where tensor data are available in batch or are streaming in an online manner. Numerical experiments are provided to demonstrate the performance of the provided algorithms for synthetic tensor data representation and real-world image data denoising. These experiments highlight the advantages of the low-separation-rank dictionary learning model over Kronecker-structured dictionary learning for complex data classes such as images in the denoising problem.The final part of the dissertation focuses on another application of sparse representations of tensor data and studies the sparse channel estimation problem in massive multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems. By modeling the underlying wireless channel as a tensor, a sparse tensor recovery technique is used to estimate the channel using lower computational resources and storage at the receiver compared to vectorized representation methods. Numerical experiments are provided to compare the performance of the estimation algorithms corresponding to vectorized and tensor formulations. These results also highlight the effects of various training signal parameters on the channel estimation performance.