Using cross-product matrices to compute the SVD

An improved ridge estimation (IRE) method for troposphere water vapor tomography

In this paper, a unified algebraic transformation approach is presented for designing parallel recursive and adaptive digital filters and singular value decomposition (SVD) algorithms. The approach is based on the explorations of some algebraic properties of the target algorithms' representations. Several typical modern digital signal processing examples are presented to illustrate the applications of the technique. They include the cascaded orthogonal recursive digital filter, the Givens rotation-based adaptive inverse QR algorithm for channel equalization, and the QR decomposition-based SVD algorithms. All three examples exhibit similar throughput constraints. There exist long feedback loops in the algorithms' signal flow graph representation, and the critical path is proportional to the size of the problem. Applying the proposed algebraic transformation techniques, parallel architectures are obtained for all three examples. For cascade orthogonal recursive filter, retiming transformation and orthogonal matrix decompositions (or pseudo-commutativity) are applied to obtain parallel filter architectures with critical path of five Givens rotations. For adaptive inverse QR algorithm, the commutativity and associativity of the matrix multiplications are applied to obtain parallel architectures with critical path of either four Givens rotations or three Givens rotations plus two multiply-add operations, whichever turns out to be larger. For SVD algorithms, retiming and associativity of the matrix multiplications are applied to derive parallel architectures with critical path of eight Givens rotations. The critical paths of all parallel architectures are independent of the problem size as compared with being proportional to the problem size in the original sequential algorithms. Parallelism is achieved at the expense of slight increase (or the same for the SVD case) in the algorithms' computational complexity.

Singular Value Decomposition (SVD) algorithm is a good method in Collaborative Filtering (CF) algorithm. But the traditional SVD method faces the matrix sparse problem, people solve this problem by Latent Factor Model (LMF). The data imbalances problem also troubles us. The algorithm of this paper combines Slope One and SVD algorithm to solve the data imbalances problem. First, we use Slope One algorithm to deal with the users who have few rating records. Then this kind of user will have more rating records. And we can use new data set to train the SVD model to predict the final results. The experiment results show that our algorithm is better than the traditional Slope One and SVD algorithm.

This paper proposes a new iterative integration regularization method for robust solution of ill-posed inverse problems. The proposed method is motivated from the fact that inversion of a positive definite matrix can be expressed in an integral form. Then, the development of the proposed method is mainly twofold. Firstly, two ways—including the linear iteration and the exponential ($$2^j$$) iteration—are invoked to compute the integral, of which the exponential iteration is often preferred due to its fast convergence. Secondly, after stability analysis, the proposed method is shown able to filter out the undesired effect of relatively small singular values, while preserving the desired terms of relatively large singular values, i.e., the proposed method has the guaranteed regularization effect. Numerical examples on three typical ill-posed problems are conducted with detailed comparison to some usual direct and iterative regularization methods. Final results have highlighted the proposed method: (a) due to the iterative nature, the proposed method often turns out to be more efficient than the conventional direct regularization methods including the Tikhonov regularization and the truncated singular value decomposition (TSVD), (b) the proposed method converges much faster than the Landweber method and (c) the regularization effect is guaranteed in the proposed method, while may not be in the conjugate gradient method for least squares problem (CGLS).

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.

Change point detection is essential to understand the time-evolving structure of dynamic networks. Recent research shows that a latent semantic indexing (LSI)-based algorithm effectively detects the change points of a dynamic network. The LSI-based method involves a singular value decomposition (SVD) on the data matrix. In a dynamic scenario, recomputing the SVD of a large matrix each time new data arrives is prohibitively expensive and impractical. A more efficient approach is to incrementally update the decomposition. However, in the classical incremental SVD (incSVD) algorithm, the information of the newly added columns is not fully considered in updating the right singular space, resulting in an approximation error which cannot be ignored. This paper proposes an enhanced incSVD (EincSVD) algorithm, in which the right singular matrix is calculated in an alternative way. An adaptive EincSVD (AEincSVD) algorithm is also proposed to further reduce the computational complexity. Theoretical analysis proves that the approximation error of the EincSVD is smaller than that of the incSVD. Simulation results demonstrate that the EincSVD and the AEincSVD perform much better than the incSVD on change point detection, and the performance of the EincSVD is comparable to the batch SVD algorithm.

Estimates of the past thermal state of the land surface are crucial to assess the magnitude of current anthropogenic climate change, as well as to assess the ability of Earth System Models to forecast the evolution of the climate near the ground, not included in standard meteorological records. Subsurface temperature data are able to retrieve long-term changes in surface energy balance –from decadal to millennial time scales, thus constituting an important record of the dynamics of the climate system that contributes low-frequency information to proxy-based paleoclimatic reconstructions. A broadly used technique to retrieve past temperature and heat flux histories from subsurface temperature profiles based on a Singular Value Decomposition (SVD) algorithm was able to take into account a limited number of sources of uncertainty, with recent works attempting to increase the number of factors considered in uncertainty estimates. Nevertheless, the SVD methodology did not define a statistical framework for aggregating inversions of individual profiles to derive global results, which lead to estimates of global and regional uncertainties that are difficult to interpret. To alleviate the lack of a conceptual framework for estimating uncertainties in past temperature and heat flux histories at regional and global scales, we combine a new bootstrapping sampling strategy with the broadly used SVD algorithm, and assess its performance against the original SVD technique and another technique based on generating perturbed parameter ensembles of inversions. The new bootstrap approach is able to reproduce the prescribed surface temperature series used to derive an artificial profile. Bootstrap results are also in agreement with the global mean surface temperature history and the global mean heat flux history retrieved in previous studies. Furthermore, the new bootstrap technique provides with a meaningful uncertainty range for the inversion of large sets of subsurface temperature profiles. We suggest the use of this new approach particularly for aggregating results from a number of individual profiles, and to this end, we release the programs used to derive all inversions in this study as a suite of codes labelled CIBOR 1.0: Codes for Inverting BORholes, version 1.0.

Singular value decomposition is the key tool in the analysis and understanding of linear regularization methods in Hilbert spaces. Besides simplifying computations it allows to provide a good understanding of properties of the forward problem compared to the prior information introduced by the regularization methods. In the last decade nonlinear variational approaches such as ` or total variation regularizations became quite prominent regularization techniques with certain properties being superior to standard methods. In the analysis of those, singular values and vectors did not play any role so far, for the obvious reason that these problems are nonlinear, together with the issue of defining singular values and singular vectors in the first place. In this paper however we want to start a study of singular values and vectors for nonlinear variational regularization of linear inverse problems, with particular focus on singular onehomogeneous regularization functionals. A major role is played by the smallest singular value, which we define as the ground state of an appropriate functional combining the (semi)norm introduced by the forward operator and the regularization functional. The optimality condition for the ground state further yields a natural generalization to higher singular values and vectors involving the subdifferential of the regularization functional, although we shall see that the Rayleigh principle may fail for higher singular values. Using those definitions of singular values and vectors, we shall carry over two main properties from the world of linear regularization. The first one is gaining information about scale, respectively the behavior of regularization techniques at different scales. This also leads to novel estimates at different scales, generalizing the estimates for the coefficients in the linear singular value expansion. The second one is to provide classes of exact solutions for variational regularization methods. We will show that all singular vectors can be reconstructed up to a scalar factor by the standard Tikhonov-type regularization approach even in the presence of (small) noise. Moreover, we will show that they can even be reconstructed without any bias by the recently popularized inverse scale space method.

We propose a new algorithm for the computation of a singular value decomposition (SVD) low-rank approximation of a matrix in the matrix product operator (MPO) format, also called the tensor train matrix format. Our tensor network randomized SVD (TNrSVD) algorithm is an MPO implementation of the randomized SVD algorithm that is able to compute dominant singular values and their corresponding singular vectors. In contrast to the state-of-the-art tensor-based alternating least squares SVD (ALS-SVD) and modified alternating least squares SVD (MALS-SVD) matrix approximation methods, TNrSVD can be up to 13 times faster while achieving better accuracy. In addition, our TNrSVD algorithm also produces accurate approximations in particular cases where both ALS-SVD and MALS-SVD fail to converge. We also propose a new algorithm for the fast conversion of a sparse matrix into its corresponding MPO form, which is up to 509 times faster than the standard tensor train SVD method while achieving machine precision accuracy. The efficiency and accuracy of both algorithms are demonstrated in numerical experiments.

Transfer-function realizations (TFR's) of multi-input, multi-output (MIMO) linear dynamic systems can be converted into minimum modal state variable realizations (MSR's) by the use of singular-value decompositions (SVD's) of the residue matrices for each pole. Each residue matrix is decomposed into the sum of outer products of vectors that form columns of the output matrix and rows of the input matrix of the MSR. If the rank of the residue matrix is r, there are r independent modes with the same eigenvalue. A MSR using only real numbers can be obtained by using a block diagonal dynamics matrix. When coupled repeated poles occur, the MSR is of the Jordan form. Starting from the residue matrix associated with the highest power of the repeated eigenvalue, the Jordan form is constructed by a series of singular value decompositions. The number of states in the MSR depends on the ranks of the residue matrices as determined by the SVD. The SVD quantifies the importance of each state by describing the error that results if states associated with small singular values are deleted. This method of converting TFR's into MSR's can be used on approximate TFR descriptions such as those determined from experimental data.

A combination of a multiresolution (MR) preconditioner and matrix decomposition algorithm – singular value decomposition (MDA-SVD) algorithm is proposed for the analysis of densely discretised electromagnetic (EM) scatterers. In realistic EM scattering problems, the discretisation may be very dense to accurately describe the geometric shape of targets. However, dense discretisations will generate two major drawbacks: (i) the badly conditioned method of moments (MoM) matrix which makes iterative solvers difficult to converge; (ii) the dense near-field MoM matrix for which the computational cost is large, even multilevel fast multipole algorithm (MLFMA) is applied. To remedy the first shortcoming of dense discretisations, an MR preconditioner can be applied to improve the condition of MoM matrices. To reduce the computational cost in solving the MoM matrix equation, an MDA-SVD algorithm can be applied to accelerate the matrix−vector multiplication. Compared with the traditional MLFMA, the MDA-SVD algorithm has no limit on the size of octree boxes which makes the MDA-SVD algorithm suitable for the analysis of dense discretisations. Therefore an MR preconditioner is combined with the MDA-SVD algorithm for fast analysis of dense discretisations. Numerical results indicate that the combination of the MR preconditioner and the MDA-SVD algorithm is efficient.

The cluster number can directly affect the clustering effect and its application in real-world scenarios. Its determination is one of the key issues in cluster analysis. According to singular value decomposition (SVD), the characteristic directions of larger singular values likely represent the primary data patterns, trends, or structures corresponding to the main information. In clustering analysis, the main information and structure are likely related to the cluster structure itself. The number of larger singular values may correspond to the number of clusters, and their main information may correspond to different clusters. Based on this, a singular-value-based cluster number detection method is proposed. First, the transferred K-nearest neighbors (TKNN) density formula is proposed to address the limitation of the DPC algorithm in failing to identify centroids in sparse clusters of unbalanced datasets. Second, core data are selected by the DPC algorithm with a modified density formula to better capture the data distribution. Third, based on the selected core data, a sparse similarity matrix is constructed to further highlight the relationships between data and enhance the distribution of data features. Finally, SVD is performed on the sparse similarity matrix to obtain singular values, the cumulative contribution rate is introduced to determine the number of relatively large singular values (i.e., the cluster number). Experimental results show that our method is superior in determining the cluster number for datasets with complex shapes.

In order to solve the problems of low prediction accuracy and long training time that are common in the existing analog circuit fault diagnosis models, this paper proposes a new combination of wavelet packet feature extraction, singular value decomposition(SVD) and dimensionality reduction and support vector machine(SVM) classification method. This method selects wavelet packet analysis with higher accuracy than traditional wavelet analysis, extracts features of analog circuit fault data, and normalizes the extracted feature data; then uses singular value decomposition method to perform fault data matrix decompose to achieve the purpose of dimensionality reduction. The size of the singular value obtained by decomposition reflects the characteristics of the fault information. Selecting the matrix with the largest singular value as a sample can express the fault characteristics more accurately and efficiently; finally, use the support vector machine to decompose the fault after the singular value. The matrix is trained and classified, so as to realize the fault diagnosis of the analog circuit. The simulation experiment results show that, compared with the current diagnosis models such as BAGRNN, the SVD model proposed in this paper improves the fault diagnosis rate of analog circuits, effectively reduces the amount of matrix calculation, and speeds up the diagnosis.

Purpose:The Bayesian estimation algorithm improves the precision of bolus tracking perfusion imaging. However, this algorithm cannot directly calculate Tmax, the time scale widely used to identify ischemic penumbra, because Tmax is a non-physiological, artificial index that reflects the tracer arrival delay (TD) and other parameters. We calculated Tmax from the TD and mean transit time (MTT) obtained by the Bayesian algorithm and determined its accuracy in comparison with Tmax obtained by singular value decomposition (SVD) algorithms.Methods:The TD and MTT maps were generated by the Bayesian algorithm applied to digital phantoms with time-concentration curves that reflected a range of values for various perfusion metrics using a global arterial input function. Tmax was calculated from the TD and MTT using constants obtained by a linear least-squares fit to Tmax obtained from the two SVD algorithms that showed the best benchmarks in a previous study. Correlations between the Tmax values obtained by the Bayesian and SVD methods were examined.Results:The Bayesian algorithm yielded accurate TD and MTT values relative to the true values of the digital phantom. Tmax calculated from the TD and MTT values with the least-squares fit constants showed excellent correlation (Pearson’s correlation coefficient = 0.99) and agreement (intraclass correlation coefficient = 0.99) with Tmax obtained from SVD algorithms.Conclusions:Quantitative analyses of Tmax values calculated from Bayesian-estimation algorithm-derived TD and MTT from a digital phantom correlated and agreed well with Tmax values determined using SVD algorithms.

Face recognition under the influence of complex illumination is a challenging problem to be solved. The common treatments for minimizing the affection of illumination variation are illumination preprocessing and illumination insensitive extraction techniques. However, the methods proposed previously present low performances. To realize high-accuracy recognition under varying illumination, this paper proposes a novel illumination processing algorithm called REC&SIG-SVD algorithm. Above all, singular value decomposition (SVD) is utilized to obtain preliminary high-frequency and low-frequency features of the face image in logarithm domain. This study proposes Sigmoid function which satisfies the principle of diminishing marginal utility to normalize singular values, aiming at calculating effective high-frequency features. Furthermore, this paper proposes a novel illumination normalization method to process low-frequency features, which is based on retina modeling cooperate with an advanced contrast limited adaptive histogram equalization (CLAHE). Meanwhile, enhancement on high-frequency features is realized by threshold-value filtering. Last but not least, the normalized high-frequency and enhanced low-frequency features are reassembled to form the normalized face image. The comparative trials based on Yale B and CMU PIE databases are conducted for our algorithm and other similar techniques as well as deep learning methods. The experimental results demonstrate that REC&SIG-SVD algorithm shows outstanding recognition performance.