A Step in the Right Dimension: Tensor Algebra and Applications - Newman, Elizabeth. (Tufts University. Tisch Library., 2019)
As data have become more complex to reflect multi-way relationships in the real world, tensors have become essential to reveal latent content in multidimensional data. In this thesis, we focus on a tensor framework based on the *_M-product, a general class of tensor-tensor products which imposes algebraic structure in a high-dimensional space. The induced *_M-algebra inherits matrix-mimetic properties and offers provably optimal, compressed representations. To emphasize the advantages of working in an algebraically-consistent multidimensional framework, we begin by discussing some traditional tensor-based approaches in Chapter 1. While these classic techniques offer compressed representations of data, they lack theoretical guarantees of optimality. In Chapter 2, we discuss the *_M-product and associated tensor algebra originally introduced in (Kernfeld, Kilmer, and Aeron, 2015; Kilmer and Martin 2011). Using our well-defined *_M-product, in Chapter 3 we discuss two provably optimal tensor decompositions: the tensor singular value decomposition (t-SVDM) and a variant called the t-SVDMII. Beyond optimality, we prove that the t-SVDM provides a superior representation over the optimal matrix decomposition (matrix SVD) and other traditional tensor decompositions. To take advantage of the superiority of our the t-SVDM, we present a novel classification algorithm in Chapter 4 which uses local t-SVDMs to optimally represent each class. In Chapter 5, we develop a tensor neural network (t-NN) based on the *_M-product which offers parametric advantages over traditional neural networks. We emphasize the efficacy of mimeticity by extending a stable neural network architecture introduced in (Haber and Ruthotto, 2019) to our t-NN framework. In Chapter 6, we discuss representing tensors sparsely through tensor dictionary learning, a non-negative tensor factorization technique (Soltani, Kilmer, and Hansen, 2016). We present a tensor-based Modified Residual Norm Steepest Descent (MRNSD) algorithm which promotes sparse representations, and explore the quality of the representation for various tensor dictionaries.; Thesis (Ph.D.)--Tufts University, 2019.; Submitted to the Dept. of Mathematics.; Advisor: Misha Kilmer.; Committee: Lior Horesh, Haim Avron, Christoph Borgers, and James Murphy.; Keywords: Mathematics, and Applied mathematics.