A Multidimensional Principal Component Analysis via the C-Product Golub–Kahan–SVD for Classification and Face Recognition

Abstract

Face recognition and identification are very important applications in machine learning. Due to the increasing amount of available data, traditional approaches based on matricization and matrix PCA methods can be difficult to implement. Moreover, the tensorial approaches are a natural choice, due to the mere structure of the databases, for example in the case of color images. Nevertheless, even though various authors proposed factorization strategies for tensors, the size of the considered tensors can pose some serious issues. Indeed, the most demanding part of the computational effort in recognition or identification problems resides in the training process. When only a few features are needed to construct the projection space, there is no need to compute a SVD on the whole data. Two versions of the tensor Golub–Kahan algorithm are considered in this manuscript, as an alternative to the classical use of the tensor SVD which is based on truncated strategies. In this paper, we consider the Tensor Tubal Golub–Kahan Principal Component Analysis method which purpose it to extract the main features of images using the tensor singular value decomposition (SVD) based on the tensor cosine product that uses the discrete cosine transform. This approach is applied for classification and face recognition and numerical tests show its effectiveness.

A tensor bidiagonalization method for higher‐order singular value decomposition with applications

“…In the case of tensors, harmonic Ritz lateral slices furnish approximations of eigenvectors of  H ⋆  associated with harmonic Ritz tubes of  H ⋆ . The harmonic Ritz tubes θj of  H ⋆  associated with the partial tensor tridiagonalization defined in (14) are the eigentubes of the generalized eigenvalue problem ( (…”

“…Using this identification, PCA for third‐order tensors that represent color images is structurally very similar to PCA for matrices that represent grayscale images. The latter is described in Reference 14.…”

“…Table 1 displays the error in the four largest approximate singular tubes computed by augmentation by Ritz lateral slices (referred to as Ritz in the table) and by the partial Lanczos bidiagonalization/Golub‐Kahan algorithm (referred to as GK in the table) as described in Reference 14, but using the t‐product. These errors are given by for $$$ i=1,2,3,4 $$$ with $$$ m=20 $$$, where , and $$$ \boldsymbol{\Sigma} \left(i,i,:\right) $$$ denote, respectively, the $$$ {i}^{th} $$$ approximated singular tube, and the $$$ {i}^{th} $$$ exact singular tube given from the t‐SVD.…”

“…The need to compute the SVD or a partial SVD of a tensor arises in a variety of applications, including image restoration, tensor completion, 10 robust tensor principal component analysis, 11 tensor compression, 12 and recognition of color faces. [13][14][15] These applications require knowledge of the largest singular values and associated lateral tensor singular slices.…”

“…Generalizations of the singular value decomposition (SVD) to tensors are described in Reference 5 using the $$$ n $$$‐mode product (the so‐called HOSVD), and in References 6,8 using the tensor c‐product and t‐product. The need to compute the SVD or a partial SVD of a tensor arises in a variety of applications, including image restoration, tensor completion, 10 robust tensor principal component analysis, 11 tensor compression, 12 and recognition of color faces 13‐15 . These applications require knowledge of the largest singular values and associated lateral tensor singular slices.…”