1 - Implicitly restarted arnoldi/lanczos methods and large scale svd applications

This paper concerns accurate computation of the singular value decomposition (SVD) of an $m\times n$ matrix $A$ . As is well known, cross-product matrix based SVD algorithms compute large singular values accurately but generally deliver poor small singular values. A new novel cross-product matrix based SVD method is proposed: (a) Use a backward stable algorithm to compute the eigenpairs of $A^{\rm T}A$ and take the square roots of the large eigenvalues of it as the large singular values of $A$ ; (b) form the Rayleigh quotient of $A^{\rm T}A$ with respect to the matrix consisting of the computed eigenvectors associated with the computed small eigenvalues of $A^{\rm T}A$ ; (c) compute the eigenvalues of the Rayleigh quotient and take the square roots of them as the small singular values of $A$ . A detailed quantitative error analysis is conducted on the method. It is proved that if small singular values are well separated from the large ones then the method can compute the small ones accurately up to the order of the unit roundoff $\epsilon$ . An algorithm is developed that is not only cheaper than the standard Golub–Reinsch and Chan SVD algorithms but also can update or downdate a new SVD by adding or deleting a row and compute certain refined Ritz vectors for large matrix eigenproblems at very low cost. Several variants of the algorithm are proposed that compute some or all parts of the SVD. Typical numerical examples confirm the high accuracy of our algorithm.

MATLAB is an interactive computer program that serves as a convenient laboratory for computations involving matrices. It provides easy access to matrix software developed by the LINPACK and EISPACK projects [1--3]. The capabilities range from standard tasks such as solving simultaneous linear equations and inverting matrices, through symmetric and nonsymmetric eigenvalue problems, to fairly sophisticated matrix tools such as the singular value decomposition.

Modern deep neural networks (DNNs) often require high memory consumption and large computational loads. In order to deploy DNN algorithms efficiently on edge or mobile devices, a series of DNN compression algorithms have been explored, including factorization methods. Factorization methods approximate the weight matrix of a DNN layer with the multiplication of two or multiple low-rank matrices. However, it is hard to measure the ranks of DNN layers during the training process. Previous works mainly induce low-rank through implicit approximations or via costly singular value decomposition (SVD) process on every training step. The former approach usually induces a high accuracy loss while the latter has a low efficiency. In this work, we propose SVD training, the first method to explicitly achieve low-rank DNNs during training without applying SVD on every step. SVD training first decomposes each layer into the form of its full-rank SVD, then performs training directly on the decomposed weights. We add orthogonality regularization to the singular vectors, which ensure the valid form of SVD and avoid gradient vanishing/exploding. Low-rank is encouraged by applying sparsity-inducing regularizers on the singular values of each layer. Singular value pruning is applied at the end to explicitly reach a low-rank model. We empirically show that SVD training can significantly reduce the rank of DNN layers and achieve higher reduction on computation load under the same accuracy, comparing to not only previous factorization methods but also state-of-the-art filter pruning methods.

Lightweight scheme of secure outsourcing SVD of a large matrix on cloud

Image enhancement and reconstruction is an important field of research in digital image analysis. To increase the quality of low-contrast images, a variety of image-enhancing technologies are available. Among these image enhancement techniques, singular value decomposition and discrete wavelet transform are the popular approaches for enhancing low-quality images. In this study, we have developed a modified singular value decomposition approach to enhance low-contrast and low-resolution close-circuit television images. Low-quality and low-resolution images with singular values near zero are used as input in the proposed approach. On the selected data, a threshold value was determined in the singular values of the diagonal matrix. Finally, the proposed modified singular value decomposition technique was applied to enhance the value of images with low quality and resolution. The datasets include 90 facial images of different individuals with low quality and resolution from a university database. The performance of the proposed approach is assessed using image entropy, peak signal to noise ratio, mean square error, contrast measurement and time computation. During the experiments performed on the obtained dataset, it was found that the proposed modified singular value decomposition approach outperformed the existing singular value decomposition, discrete wavelet transform-singular value decomposition and stationary wavelet transform methods.

Matrix factorization plays an important role in scientific computation. The widely used one is singular value decomposition (SVD) which approximates the original data matrix with three lower rank matrices with orthogonality constraints. Recently nonnegative matrix factorization (NMF) considering the nonnegativity of data makes the results more interpretable than those of SVD. However NMF finds only two factor matrices and there is no significant index as singular values of SVD which can be used for sorting learned basis vectors. In this paper we take into account the nonnegativity for SVD and propose nonnegative SVD (NNSVD). The preliminary results on the microarray data of spermatogenesis show that NNSVD has advantages of both SVD and NMF.

The protection of sensitive and confidential data become a challenging task in the present scenario as more and more digital data is stored and transmitted between the end users. The privacy is vitally necessary in case of medical data, which contains the important information of the patients. In this article, a novel biometric inspired medical encryption technique is proposed based on newly introduced parameterized all phase orthogonal transformation (PR-APBST), singular value, and QR decomposition. The proposed technique utilizes the biometrics of the patient/owner to generate a key management system to obtain the parameters involved in the proposed technique. The medical image is then encrypted employing PR-APBST, QR and singular value decomposition and is ready for secure transmission or storage. Finally, a reliable decryption process is employed to reconstruct the original medical image from the encrypted image. The validity and feasibility of the proposed framework have been demonstrated using an extensive experiments on various medical images and security analysis.

The partitioning of the overall molecular charge distribution into atom centered monopole charges, while quantum mechanically ill-defined, is nevertheless a technique which finds applications in several broad classes of chemical problems. Charges derived from fits to electrostatic potentials have an intuitive appeal since, in principle, these could be derived from either theoretical or experimental data. It has been noted, however, that such potential derived charges can be conformationally dependent in ways that do not appear to reflect the changes in the molecular wavefunction. Both the algorithm used for selecting points at which the molecular electrostatic potential will be fit and the density of points used in the fit have been suggested to influence the resultant charges. Recently [Stouch TR, Williams DE (1992) J Comp Chem 13: 622-32; Stouch TR, Williams DE (1993) J Comp Chem 14: 858-66] it has been noted that numerical difficulties may make it impossible to fit all the atomic charges in a molecule. Singular value decomposition (SVD) of the linear least squares matrices used in fitting atom based monopoles to molecular electrostatic potentials provides a tool for evaluating the integrity of the calculated charges. Based on the SVD analysis for a selected group of molecules we have noted particularly that increasing the molecular size reduces the fraction of charges which can be validly assigned. Users of PD derived charges, especially those who are using those charges for tasks other than reproduction of the MEP, should be aware that there is a high probability that a significant portion of those charges are statistically unreliable. Therefore, charges in many biological molecules, such as sugars, prove to be difficult to obtain by potential derived (PD) methods such as CHELP or CHELPG. Results from the SVD can be used to both assess PD charges and to generate an improved, albeit incomplete, set. Improved PD fits are presented for a series of simple saccharides.

The singular value decomposition (SVD) of an imaging system is a computationally intensive calculation for tomographic imaging systems due to the large dimensionality of the system matrix. The computation often involves memory and storage requirements beyond those available to most end users. We have developed a method that reduces the dimension of the SVD problem towards the goal of making the calculation tractable for a standard desktop computer. In the presence of discrete rotational symmetry we show that the dimension of the SVD computation can be reduced by a factor equal to the number of collection angles for the tomographic system. In this paper we present the mathematical theory for our method, validate that our method produces the same results as standard SVD analysis, and finally apply our technique to the sensitivity matrix for a clinical CT system. The ability to compute the full singular value spectra and singular vectors will augment future work in system characterization, image-quality assessment and reconstruction techniques for tomographic imaging systems.

Availability of diagnostic codes in Electronic Health Records (EHRs) is crucial for patient care as well as reimbursement purposes. However, entering them in the EHR is tedious, and some clinical codes may be overlooked. Given an incomplete list of clinical codes, we investigate the performance of ML methods on predicting the complete ones, and assess the added predictive value of including other clinical patient data in this task. We used the MIMIC-III dataset and frame the task of completing the clinical codes as a recommendation problem. We consider various autoencoder approaches plus two strong baselines; item co-occurrence and Singular Value Decomposition (SVD). Inputs are 1) a record’s known clinical codes, 2) the codes plus variables. The co-occurrence-based approach performed slightly better (F1 score = 0.26, Mean Average Precision [MAP] = 0.19) than the SVD (F1 = 0.24, MAP = 0.18). However, the adversarial autoencoder achieved the best performance when using the codes plus variables (F1 = 0.32, MAP = 0.25). Adversarial autoencoders performed best in terms of F1 and were equal to vanilla and denoising autoencoders in term of MAP. Using clinical variables in addition to the incomplete codes list, improves the predictive performance of the models.

We monitored the occupancy of a functionally important non-coordinated water molecule in the distal heme pocket of sperm whale myoglobin over the pH range 4.3-9.4. Water occupancy was assessed by using time-resolved spectroscopy to detect the perturbation of the heme visible band absorption spectrum caused by water entry after CO photodissociation ( Goldbeck, R. A., Bhaskaran, S., Ortega, C., Mendoza, J. L., Olson, J. S., Soman, J., Kliger, D. S., and Esquerra, R. M. (2006) Proc. Natl. Acad. Sci. U. S. A. 103, 1254-1259 ). We found that the water occupancy observed during the time interval between ligand photolysis and diffusive recombination decreased by nearly 20% as the pH was lowered below 6. This decrease accounted for most of the concomitant increase in the observed CO bimolecular recombination rate constant, as the lower water occupancy presented a smaller kinetic barrier to CO entry into the pocket at lower pH. These results were consistent with a model in which the distal histidine, which stabilizes the water molecule within the distal pocket by accepting a hydrogen bond, tends to swing out of the pocket upon protonation and destabilize the water occupancy at low pH. Extrapolation of this model to lower pH suggests that the additional increase in ligand association rate constant observed previously in stopped-flow studies at pH 3 may also be due in part to reduced distal water occupancy concomitant with further His64 protonation and coupled protein conformational change.

Image compression is preferred for less storage space, sending image on the internet and often for medical imaging in health industry, technical drawings or comics. Because of wide popularity of sharing media in internet, compression plays a major role in the watermarking algorithm. This paper presents a compression technique through image watermarking algorithm based on Non-subsampled Contourlet Transform (NSCT) and Singular Value Decomposition (SVD). NSCT and SVD both are explored for image watermarking. For image compression, only Singular Value Decomposition is used. The NSCT can give more accurate coefficients of the edges and contours in image which is shift invariant. After getting the coefficients from NSCT decomposition SVD is applied to break the matrix in a convenient way so that image compression can be done without losing meaningful data. The compressed singular values of NSCT decomposed coefficients of cover image are added with the compressed singular values of NSCT decomposed coefficient of gray key image. This helps to improve the performance of the compression using watermarking method. In terms of PSNR and compression ratio, the performance of this scheme is analyzed. The proposed algorithm is robust and suitable for copyright protection and data transmission against compression applications.

This paper proposes a new solution integrating energy function into singular value decomposition (SVD) for image de-noising. The singular values on the diagonal matrix obtained through SVD represent different components in image. By selecting the proper singular values that represent signal and discarding the ones that represent noise, the additive noise of an image can be eliminated effectively. In order to obtain the optimal number of the singular values for image reconstruction and to eliminate the noise, the paper presents a minimum energy model. This model is used to obtain the optimum number for de-noising through calculating the minimum in the defined energy curve. The experiment results show that the established model is effective in the circumstance that the image has simple/regular structure/pattern.

When seismic waves encounter abrupt points of stratum or lithology in the process of propagation, such as fault edges, pinch-out points, or the protrusion of unconformity surfaces, the Snell law will break down and many diffractions will be generated. However, the diffraction information is typically masked by strong reflections; thus, separating diffractions is one key issue for diffraction imaging. The traditional singular value decomposition (SVD) method with the ability for wavefield decomposition and reconstruction is useful in removing strong reflections of large singular values. However, the continuously changing characteristics of the singular value of steep-slope reflections make it difficult to choose a suitable parameter for separating diffractions. To solve the problem of singular value selection in diffractions reconstruction by the SVD method, we have developed a Cook-distance SVD method by mathematically least-squares measuring the contribution of every singular value. The Cook-distance SVD can amplify the difference between continuously changing singular values and easily find the singular values representing diffractions. Synthetic examples and field data applications demonstrate that our Cook-distance SVD method can avoid parameter testing and has good performance in diffraction separation and can highlight reservoir-involved small-scale geologic edges and scatterers.

Attenuation effects can be significant in photoacoustic tomography since the generated pressure signals are broadband, and ignoring them may lead to image artifacts and blurring. La Rivière et al. [Opt. Lett. 31(6), pp. 781-783, (2006)] had previously derived a method for modeling the attenuation effect and correcting for it in the image reconstruction. This was done by relating the ideal, unattenuated pressure signals to the attenuated pressure signals via an integral operator. We derive an integral operator relating the attenuated pressure signals to the absorbed optical energy for a planar measurement geometry. The matrix operator relating the two quantities is a function of the temporal frequency, attenuation coefficient and the two-dimensional spatial frequency. We perform singular-value decomposition (SVD) of this integral operator to study the problem further. We find that the smallest singular values correspond to wavelet-like eigenvectors in which most of the energy is concentrated at times corresponding to greater depths in tissue. This allows us to characterize the ill-posedness of recovering the absorbed optical energy distribution at different depths in an attenuating medium. This integral equation can be inverted using standard SVD methods, and the initial pressure distribution can be recovered. We conduct simulations and derive an algorithm for image reconstruction using SVD for a planar measurement geometry. We also study the noise and resolution properties of this image-reconstruction method.