Multilinear Discriminant Analysis Through Tensor-Tensor Eigendecomposition

Caudle, Kyle; Hoover, Randy C.; Braman, Karen

doi:10.1109/icmla.2018.00093

Cited by 7 publications

(3 citation statements)

References 38 publications

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“…The final tool necessary for a multilinear LDA is to define a tensor-tensor eigenvalue decomposition. In our previous work, tensor-tensor eigendecomposition has been introduced for third order tensor based upon on the discrete Fourier transform (DFT) [33]. In this subsection, we extend this work to order-n tensor by utilization of the new tensor operators defined in Section II-B.…”

Section: Order-n Tensor Eigendecompositionmentioning

confidence: 99%

“…In this paper, we introduce a new framework reffered to as High-Order Multilinear Discriminant Analysis (HOMLDA) for order-n tensors based on tensor-tensor decomposition as opposed to utilizing a Tucker representation. Our proposed approach builds upon our prior multilinear discriminant analysis (MLDA) approach defined for third-order tensors [33]. Moreover, the within-class scatter tensor computed for HOMLDA may be close to singular and cause poor classification performance.…”

Section: Introductionmentioning

confidence: 99%

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High-Order Multilinear Discriminant Analysis via Order-$\textit{n}$ Tensor Eigendecomposition

Ozdemir¹,

Hoover²,

Caudle³

et al. 2022

Preprint

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Higher-order data with high dimensionality is of immense importance in many areas of machine learning, computer vision, and video analytics. Multidimensional arrays (commonly referred to as tensors) are used for arranging higher-order data structures while keeping the natural representation of the data samples. In the past decade, great efforts have been made to extend the classic linear discriminant analysis for higherorder data classification generally referred to as multilinear discriminant analysis (MDA). Most of the existing approaches are based on the Tucker decomposition and n-mode tensormatrix products. The current paper presents a new approach to tensor-based multilinear discriminant analysis referred to as High-Order Multilinear Discriminant Analysis (HOMLDA). This approach is based upon the tensor decomposition where an ordern tensor can be written as a product of order-n tensors and has a natural extension to traditional linear discriminant analysis (LDA). Furthermore, the resulting framework, HOMLDA, might produce a within-class scatter tensor that is close to singular. Thus, computing the inverse inaccurately may distort the discriminant analysis. To address this problem, an improved method referred to as Robust High-Order Multilinear Discriminant Analysis (RHOMLDA) is introduced. Experimental results on multiple data sets illustrate that our proposed approach provides improved classification performance with respect to the current Tucker decomposition-based supervised learning methods.

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Section: Order-n Tensor Eigendecompositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High-Order Multilinear Discriminant Analysis via Order-$\textit{n}$ Tensor Eigendecomposition

Ozdemir¹,

Hoover²,

Caudle³

et al. 2022

Preprint

Self Cite

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“…This concept was generalized to functions of third-order tensors in [23], based on the tensor t-product formalism [5,19,20]; see also [25] for a further extension to so-called generalized tensor functions, which are functions of tensors with non-square faces. Functions (and generalized functions) of tensors have applications in deblurring of color images [28], tensor neural networks [24,27], multilinear dynamical systems [14], and the computation of the tensor nuclear norm [4].…”

Section: Introductionmentioning

confidence: 99%

The Fréchet derivative of the tensor t-function

Lund¹,

Schweitzer²

2023

Preprint

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The tensor t-function, a formalism that generalizes the well-known concept of matrix functions to third-order tensors, is introduced in [K. Lund, The tensor t-function: a definition for functions of third-order tensors, Numer. Linear Algebra Appl. 27 (3), e2288]. In this work, we investigate properties of the Fréchet derivative of the tensor t-function and derive algorithms for its efficient numerical computation. Applications in condition number estimation and nuclear norm minimization are explored. Numerical experiments implemented by the t-Frechet toolbox hosted at https://gitlab.com/katlund/t-frechet illustrate properties of the t-function Fréchet derivative as well as the efficiency and accuracy of the proposed algorithms.

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