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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.

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.

A cross-product free (CPF) Jacobi–Davidson type method is proposed to compute a partial generalized singular value decomposition (GSVD) of a large regular matrix pair { A , B } , called CPF-JDGSVD. It implicitly solves the mathematically equivalent generalized eigenvalue problem of the cross-product matrix pair { A T A , B T B } using the Rayleigh–Ritz projection method but does not form the cross-product matrices explicitly, and thus avoids the possible accuracy loss of the computed generalized singular values and generalized singular vectors. The method is an inner-outer iteration method, where the expansion of the right searching subspace forms the inner iterations that approximately solve the correction equations involved and the outer iterations extract approximate GSVD components with respect to the subspaces. A convergence result is established for the outer iterations, compact bounds are derived for the condition numbers of the correction equations, and the least solution accuracy requirements on the inner iterations are found, which can maximize the overall efficiency of CPF-JDGSVD as much as possible. Based on them, practical stopping criteria are designed for the inner iterations. A thick-restart CPF-JDGSVD algorithm with deflation and purgation is developed to compute several GSVD components of { A , B } associated with the generalized singular values closest to a given target τ . Numerical experiments illustrate the efficiency of the algorithm.

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.

In view of the fact that vibration signals of rolling bearings are much contaminated by noise in the early failure period, this paper presents a new denoising SVD-VMD method by combining singular value decomposition (SVD) and variational mode decomposition (VMD). SVD is used to determine the structure of the underlying model, which is referred to as signal and noise subspaces, and VMD is used to decompose the original signal into several band-limited modes. Then the effective components are selected from these modes to reconstruct the denoised signal according to the difference spectrum (DS) of singular values and kurtosis values. Simulated signals and experimental signals of roller bearing faults have been analyzed using this proposed method and compared with SVD-DS. The results demonstrate that the proposed method can effectively retain the useful signals and denoise the bearing signals in extremely noisy backgrounds.

For the computation of the generalized singular value decomposition (GSVD) of a large matrix pair ( A , B ) of full column rank, the GSVD is commonly formulated as two mathematically equivalent generalized eigenvalue problems, so that a generalized eigensolver can be applied to one of them and the desired GSVD components are then recovered from the computed generalized eigenpairs. Our concern in this paper is, in finite precision arithmetic, which generalized eigenvalue formulation is numerically preferable to compute the desired GSVD components more accurately. We make a detailed perturbation analysis on the two formulations and show how to make a suitable choice between them. Numerical experiments illustrate the results obtained.

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.

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.