2DTPCA: A New Framework for Multilinear Principal Component Analysis

Ozdemir, Cagri; Hoover, Randy C.; Caudle, Kyle

doi:10.1109/icip42928.2021.9506729

Cited by 8 publications

(3 citation statements)

References 17 publications

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“…[31] further extends this idea to T-SVDM. [29] points out the importance of utilizing the orientation-dependence of T-SVDs to obtain information from different directions. Inspired by them, we derive a sparse model for T-SVDM and utilize the orient-dependence property in UFS.…”

Section: T-svds and Tensor Pcamentioning

confidence: 99%

Fast Sparse PCA via Positive Semidefinite Projection for Unsupervised Feature Selection

Zheng¹,

Zhang²,

Huo³

et al. 2022

Preprint

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<p>In the field of unsupervised feature selection, sparse principal component analysis (SPCA) methods have attracted more and more attention recently. Compared to spectral-based methods, SPCA methods don’t rely on the construction of similarity matrix and show better feature selection ability on real-world data. The original SPCA formulates a nonconvex optimization problem. And existing convex SPCA methods reformulate SPCA as a convex model by regarding reconstruction matrix as optimization variable. However, they are lack of constraints equivalent to the orthogonality restriction in SPCA, leading to larger solution space. In this paper, it’s proved that the optimal solution to a convex SPCA model falls onto the PSD cone. A new convex SPCA model with PSD constraint is then proposed. Further, a two-step fast optimization algorithm is presented to solve the proposed model. Firstly, adopt the derivative of objective function to obtain the unconstrained solution. Secondly, project the solution onto the PSD cone.The convergence of the proposed optimization algorithm is studied. With PSD projection, the proposed method takes less number of iterations and running time than other convex SPCA methods and has stronger feature selection ability. Experiments on both synthetic and real-world datasets demonstrate the effectiveness and efficiency of the proposed method. </p>

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Section: T-svds and Tensor Pcamentioning

confidence: 99%

Fast Sparse PCA via Positive Semidefinite Projection for Unsupervised Feature Selection

Zheng¹,

Zhang²,

Huo³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In [15] the authors develop a method to forecast higher-order tensors based on the Tucker decomposition and n-mode products (referred to as multilinear dynamical systems (MLDS)) [16,17]. The MLDS approach (based on dynamical systems theory and system identification methods) was extended in [18] by transitioning from the Tucker decomposition to a recently defined tensor product based on discrete transforms and mod-n convolution, referred to as the L-transform [19][20][21][22][23][24][25] (the details of which are outlined in Section 2). While both methods outlined in [15] and [18] (MLDS and L-MLDS respectively) show promise, they are both based on multilinear dynamical systems modeling as opposed to an autoregressive model as defined above.…”

Section: Introductionmentioning

confidence: 99%

Forecasting Multilinear Data via Transform-Based Tensor Autoregression

Cates¹,

Hoover²,

Caudle³

et al. 2022

Preprint

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In the era of big data, there is an increasing demand for new methods for analyzing and forecasting 2-dimensional data. The current research aims to accomplish these goals through the combination of time-series modeling and multilinear algebraic systems. We expand previous autoregressive techniques to forecast multilinear data, aptly named the L-Transform Tensor autoregressive (L-TAR for short). Tensor decompositions and multilinear tensor products have allowed for this approach to be a feasible method of forecasting. We achieve statistical independence between the columns of the observations through invertible discrete linear transforms, enabling a divide and conquer approach. We present an experimental validation of the proposed methods on datasets containing image collections, video sequences, sea surface temperature measurements, stock prices, and networks.

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“…Fundamental to t-SVD is the defined multiplication operator on third-order tensors (t-product) based upon Fourier theory and an algebra of circulants [17], [18]. Pose estimation and image classification applications were proposed in which principal component analysis (PCA) has been extended to third-order tensors to learn multilinear mappings from a third-order tensor input [19]- [22]. The extension of the t-SVD to order-n tensors was formulated and this extended structure was used in image deblurring and video facial recognition applications [23].…”

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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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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2DTPCA: A New Framework for Multilinear Principal Component Analysis

Cited by 8 publications

References 17 publications

Fast Sparse PCA via Positive Semidefinite Projection for Unsupervised Feature Selection

Fast Sparse PCA via Positive Semidefinite Projection for Unsupervised Feature Selection

Forecasting Multilinear Data via Transform-Based Tensor Autoregression

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

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