A Step in the Right Dimension: Tensor Algebra and Applications
Newman, Elizabeth.
2019
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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 ... read moreproperties 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.read less - ID:
- wd376856j
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