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We present an alternative strategy for truncating the higher-order singular value decomposition (T-HOSVD). An error expression for an approximate Tucker decomposition with orthogonal factor matrices is presented, leading us to propose a novel truncation strategy for the HOSVD, which we refer to as the sequentially truncated higher-order singular value decomposition (ST-HOSVD). This decomposition retains several favorable properties of the T-HOSVD, while reducing the number of operations required to compute the decomposition and practically always improving the approximation error. Three applications are presented, demonstrating the effectiveness of ST-HOSVD. In the first application, ST-HOSVD, T-HOSVD, and higher-order orthogonal iteration (HOOI) are employed to compress a database of images of faces. On average, the ST-HOSVD approximation was only 0.1\\% worse than the optimum computed by HOOI, while cutting the execution time by a factor of 20. In the second application, classification of handwritten digits, ST-HOSVD achieved a speedup factor of 50 over T-HOSVD during the training phase, and reduced the classification time and storage costs, while not significantly affecting the classification error. The third application demonstrates the effectiveness of ST-HOSVD in compressing results from a numerical simulation of a partial differential equation. In such problems, ST-HOSVD inevitably can greatly improve the running time. We present an example wherein the 2 hour 45 minute calculation of T-HOSVD was reduced to just over one minute by ST-HOSVD, representing a speedup factor of 133, while even improving the memory consumption.

A randomized quaternion singular value decomposition algorithm based on block Krylov iteration (RQSVD-BKI) is presented to solve the low-rank quaternion matrix approximation problem. The upper bounds of deterministic approximation error and expected approximation error for the RQSVD-BKI algorithm are also given. It is shown by numerical experiments that the running time of the RQSVD-BKI algorithm is smaller than that of the quaternion singular value decomposition, and the relative errors of the RQSVD-BKI algorithm are smaller than those of the randomized quaternion singular value decomposition algorithm in Liu et al. (SIAM J. Sci. Comput., 44(2): A870-A900 ( 2022 )) in some cases. In order to further illustrate the feasibility and effectiveness of the RQSVD-BKI algorithm, we use it to deal with the problem of color image inpainting.

A new asymmetric encryption system for double random phase encoding based on random weighted singular value decomposition and fractional Hartley transform domain has been proposed. Random weighted singular value decomposition is purely based upon random weights, isometric matrix and orthogonal triangular decomposition and all these fragments enhances the security of double random phase encoding cryptosystem. Random weights and orthogonal triangular decomposition are considered as heart of this cryptosystem. This system is carried out in fractional Hartley domain, where fractional orders play a vital role. On the receiver side, it is only possible to decrypt the image if anyone knows all the three components, its multiplication order, fractional order of fractional Hartley transform. Proposed cryptosystem is efficiently compared with singular value decomposition and truncated singular value decomposition. Similar to singular value decomposition and truncated singular value decomposition, proposed cryptosystem also yields three components. Because of random weights, these three components are highly differing from traditional singular value decomposition and truncated singular value decomposition components. Some analysis is offered to authenticate the opportunity.

The generalized singular value decomposition (GSVD) of a pair of matrices expresses each matrix as a product of an orthogonal, a diagonal, and a nonsingular matrix. The nonsingular matrix, which we denote by X T , is the same in both products. Available software for computing the GSVD scales the diagonal matrices and X T so that the squares of corresponding diagonal entries sum to one. This paper proposes a scaling that seeks to minimize the condition number of X T . The rescaled GSVD gives rise to new truncated GSVD methods, one of which is well suited for the solution of linear discrete ill-posed problems. Numerical examples show this new truncated GSVD method to be competitive with the standard truncated GSVD method as well as with Tikhonov regularization with regard to the quality of the computed approximate solution.

Learning large-scale data sets with high dimensionality is a main concern in research areas including machine learning, visual recognition, information retrieval, to name a few. In many practical uses such as images, video, audio, and text processing, we have to face with high-dimension and large-sample data problems. The trace-ratio problem is a key problem for feature extraction and dimensionality reduction to circumvent the high dimensional space. However, it has been long believed that this problem has no closed-form solution, and one has to solve it by using some inner-outer iterative algorithms that are very time consuming. Therefore, efficient algorithms for high-dimension and large-sample trace-ratio problems are still lacking, especially for dense data problems. In this work, we present a closed-form solution for the trace-ratio problem, and propose two algorithms to solve it. Based on the formula and the randomized singular value decomposition, we first propose a randomized algorithm for solving high-dimension and large-sample dense trace-ratio problems. For high-dimension and large-sample sparse trace-ratio problems, we then propose an algorithm based on the closed-form solution and solving some consistent under-determined linear systems. Theoretical results are established to show the rationality and efficiency of the proposed methods. Numerical experiments are performed on some real-world data sets, which illustrate the superiority of the proposed algorithms over many state-of-the-art algorithms for high-dimension and large-sample dimensionality reduction problems.

Tensor decompositions have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-one outer products using either the CANDECOMP/PARAFAC, the Tucker model, or some variations thereof. The motivation of these decompositions is to find an approximate representation for a given tensor. The main propose of this paper is to develop two neural network models for finding an approximation based on t-product for a given third-order tensor. Theoretical analysis shows that each of the neural network models ensures the convergence performance. The computer simulation results further substantiate that the models can find effectively the left and right singular tensor subspace.

Digital watermarking has attracted increasing attentions as it has been the current solution to copyright protection and content authentication in today’s digital transformation, which has become an issue to be addressed in multimedia technology. In this paper, we propose an advanced image watermarking system based on the discrete wavelet transform (DWT) in combination with the singular value decomposition (SVD). Firstly, at the sender side, DWT is applied on a grayscale cover image and then eigendecomposition is performed on original HH (high–high) components. Similar operation is done on a grayscale watermark image. Then, two unitary and one diagonal matrices are combined to form a digital watermarked image applying inverse discrete wavelet transform (iDWT). The diagonal component of original image is transmitted through secured channel. At the receiver end, the watermark image is recovered using the watermarked image and diagonal component of the original image. Finally, we compare the original and recovered watermark image and obtained perfect normalized correlation. Simulation consequences indicate that the presented scheme can satisfy the needs of visual imperceptibility and also has high security and strong robustness against many common attacks and signal processing operations. The proposed digital image watermarking system is also compared to state-of-the-art methods to confirm the reliability and supremacy.

Two harmonic extraction based Jacobi–Davidson (JD) type algorithms are proposed to compute a partial generalized singular value decomposition (GSVD) of a large regular matrix pair. They are called cross product-free (CPF) and inverse-free (IF) harmonic JDGSVD algorithms, abbreviated as CPF-HJDGSVD and IF-HJDGSVD, respectively. Compared with the standard extraction based JDGSVD algorithm, the harmonic extraction based algorithms converge more regularly and suit better for computing GSVD components corresponding to interior generalized singular values. Thick-restart CPF-HJDGSVD and IF-HJDGSVD algorithms with some deflation and purgation techniques are developed to compute more than one GSVD components. Numerical experiments confirm the superiority of CPF-HJDGSVD and IF-HJDGSVD to the standard extraction based JDGSVD algorithm.

The rise of vision-threatening diabetic retinopathy (VTDR) underscores the imperative for advanced and efficient early detection mechanisms. With the integration of the Internet of Things (IoT) and 5G technologies, there is transformative potential for VTDR diagnosis, facilitating real-time processing of the burgeoning volume of fundus images (FIs). Combined with artificial intelligence (AI), this offers a robust platform for managing vast healthcare datasets and achieving unparalleled disease detection precision. Our study introduces a novel AI-driven VTDR detection framework that integrates multiple models through majority voting. This comprehensive approach encompasses pre-processing, data augmentation, feature extraction using a hybrid convolutional neural network-singular value decomposition (CNN-SVD) model, and classification through an enhanced SVM-RBF combined with a decision tree (DT) and K-nearest neighbor (KNN). Validated on the IDRiD dataset, our model boasts an accuracy of 99.89%, a sensitivity of 84.40%, and a specificity of 100%, marking a significant improvement over the traditional method. The convergence of the IoT, 5G, and AI technologies herald a transformative era in healthcare, ensuring timely and accurate VTDR diagnoses, especially in geographically underserved regions.

We propose an artificial intelligence approach based on deep neural networks to tackle a canonical 2D scalar inverse source problem. The learned singular value decomposition (L-SVD) based on hybrid autoencoding is considered. We compare the reconstruction performance of L-SVD to the Truncated SVD (TSVD) regularized inversion, which is a canonical regularization scheme, to solve an ill-posed linear inverse problem. Numerical tests referring to far-field acquisitions show that L-SVD provides, with proper training on a well-organized dataset, superior performance in terms of reconstruction errors as compared to TSVD, allowing for the retrieval of faster spatial variations of the source. Indeed, L-SVD accommodates a priori information on the set of relevant unknown current distributions. Different from TSVD, which performs linear processing on a linear problem, L-SVD operates non-linearly on the data. A numerical analysis also underlines how the performance of the L-SVD degrades when the unknown source does not match the training dataset.