Computationally efficient methods for sparse tensor signal processing

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dc.contributor.authorWickramasingha, Ishan
dc.contributor.examiningcommitteeYahampath, Pradeepa (Electrical and Computer Engineering)en_US
dc.contributor.examiningcommitteeGoertzen, Andrew (Physics and Astronomy)en_US
dc.contributor.examiningcommitteeFieguth, Paul (Systems Design Engineering, University of Waterloo)en_US
dc.contributor.supervisorSherif, Sherif (Electrical and Computer Engineering)en_US
dc.date.accessioned2022-01-05T18:30:34Z
dc.date.available2022-01-05T18:30:34Z
dc.date.copyright2021-12-31
dc.date.issued2021-12-31en_US
dc.date.submitted2021-12-31T23:00:00Zen_US
dc.degree.disciplineElectrical and Computer Engineeringen_US
dc.degree.levelDoctor of Philosophy (Ph.D.)en_US
dc.description.abstractMany state-of-the-art algorithms typically solve Tensor (multi-dimensional) problems using linear algebra by vectorizing tensor signals. However, the size of the tensors increases in polynomial order with the tensor order (dimensions); Therefore, higher-order tensor problems could be challenging to store or solve practically. This research aimed to develop novel tensor-based methods using multilinear algebra to solve tensor problems efficiently with significantly lower computational resources. Sparse signal representations, typically obtained by solving sparse least-squares(LS) problems, result in simpler and faster processing and lower memory storage requirements. However, solving a sparse LS problem for a large tensor signal is computationally infeasible. Therefore, we develop the Tensor Least Angle Regression (T-LARS) algorithm, a generalization of Least Angle Regression (LARS) to solve large L0 or L1 constrained sparse multilinear least-squares(MLS) problems efficiently for all critical values of the regularization parameter. Sparse weighted MLS, generalize the sparse MLS problem, where a weights matrix incorporates prior information about parameters and data. We generalized the T-LARS to develop the Weighted Tensor Least Angle Regression (WT-LARS), an efficient algorithm to solve L0 or L1 constrained weighted MLS problems. The T-LARS could not initialize with solutions outside of the Pareto curve because it will violate the optimality conditions of T-LARS. Therefore, we developed the Tensor Dynamic Least Angle Regression (TD-LARS) algorithm, a multilinear generalization of the L1-Homotopy algorithm to efficiently solve L1 constrained MLS problems using nonzero initial solutions located on or off of the Pareto curve. We introduced the Multilinear Elastic Net by generalizing the one-dimensional Elastic Net to solve strictly convex L1 and L2 constrained MLS problems, and we developed the Tensor Elastic Net (T-NET) algorithm to solve the Multilinear Elastic Net problems efficiently. We also developed the tensor task-driven dictionary learning (T-TDDL) framework by generalizing the one-dimensional task-driven dictionary learning (TDDL) that could work as an efficient online data-driven or task-driven dictionary learning framework for solving large multi-dimensional supervised, semi-supervised and unsupervised machine learning problems. Experimental results show the validity and performance of T-LARS, WT-LARS, TD-LARS, and T-NET in obtaining sparse multilinear representations of tensor signals and the performance of T-TDDL in multi-dimensional regression and classification tasks.en_US
dc.description.noteFebruary 2022en_US
dc.identifier.citationI. Wickramasingha, M. Sobhy, A. Elrewainy, and S. S. Sherif, “Tensor least angle regression for sparse representations of multi-dimensional signals,” Neural Comput., vol. 32, no. 9, pp. 1697–1732, Sep. 2020,https://doi.org/10.1162/neco_a_01304.en_US
dc.identifier.urihttp://hdl.handle.net/1993/36150
dc.language.isoengen_US
dc.rightsopen accessen_US
dc.subjectTensor Signal Processingen_US
dc.subjectSparse Signal Representationsen_US
dc.subjectMultilinear Least Squaresen_US
dc.subjectTensor Least Angle Regressionen_US
dc.subjectWeighted Multilinear Least Squaresen_US
dc.subjectMultilinear Elastic Neten_US
dc.subjectOnline Tensor Task Driven Dictionary Learningen_US
dc.titleComputationally efficient methods for sparse tensor signal processingen_US
dc.typedoctoral thesisen_US
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