Thresholding Approach for Low-Rank Correlation Matrix based on MM algorithm

Background: Low-rank approximation is a very useful approach for interpreting the features of a correlation matrix; however, a low-rank approximation may result in estimation far from zero even if the corresponding original value was far from zero. In this case, the results lead to misinterpretation. Methods: To overcome these problems, we propose a new approach to estimate a sparse low-rank correlation matrix based on threshold values combined with cross-validation. In the proposed approach, the MM algorithm was used to estimate the sparse low-rank correlation matrix, and a grid search was performed to select the threshold values related to sparse estimation. Results: Through numerical simulation, we found that the FPR and average relative error of the proposed method were superior to those of the tandem approach. For the application of microarray gene expression, the FPRs of the proposed approach with d=2,3, and 5 were 0.128, 0.139, and 0.197, respectively, while FPR of the tandem approach was 0.285. Conclusions: We propose a novel approach to estimate sparse low-rank correlation matrix. The advantage of the proposed method is that it provides results that are easy to interpret and avoid misunderstandings. We demonstrated the superiority of the proposed method through both numerical simulations and real examples.

Subspace Reduction for Stochastic Planar Elasticity

Stochastic eigenvalue problems are nonlinear and multiparametric. They require their own solution methods and remain one of the challenge problems in computational mechanics. For the simplest possible reference problems, the key is to have a cluster of at the low end of the spectrum. If the inputs, domain or material, are perturbed, the cluster breaks and tracing of the eigenpairs become difficult due to possible crossing of the modes. In this paper we have shown that the eigenvalue crossing can occur within clusters not only by perturbations of the domain, but also of material parameters. What is new is that in this setting, the crossing can be controlled; that is, the effect of the perturbations can actually be predicted. Moreover, the basis of the subspace is shown to be a well-defined concept and can be used for instance in low-rank approximation of solutions of problems with static loading. In our industrial model problem, the reduction in solution times is significant.

Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations

We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a “parabolic” type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. Furthermore, we show that these schemes can be interpreted as projector-splitting integrators and are strongly related to the scheme proposed in [29, 30], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions.

A Novel Compression Method of Spectral Data Matrix Based on the Low-Rank Approximation and the Fast Fourier Transform of the Singular Vectors

Storage, processing, and transfer of huge matrices are becoming challenging tasks in the process analytical technology and scientific research. Matrix compression can solve these problems successfully. We developed a novel compression method of spectral data matrix based on its low-rank approximation and the fast Fourier transform of the singular vectors. This method differs from the known ones in that it does not require restoring the low-rank approximated matrix for further Fourier processing. Therefore, the compression ratio increases. A compromise between the losses of the accuracy of the data matrix restoring and the compression ratio was achieved by selecting the processing parameters. The method was applied to multivariate chemometrics analysis of the cow milk for determining fat and protein content using two data matrices (the file sizes were 5.7 and 12.0 MB) restored from their compressed form. The corresponding compression ratios were about 52 and 114, while the loss of accuracy of the analysis was less than 1% compared with processing of the non-compressed matrix. A huge, simulated matrix, compressed from 400 MB to 1.9 MB, was successfully used for multivariate calibration and segment cross-validation. The data set simulated a large matrix of 10 000 low-noise infrared spectra, measured in the range 4000–400 cm−1 with a resolution of 0.5 cm−1. The corresponding file was compressed from 262.8 MB to 19.8 MB. The discrepancies between original and restored spectra were less than the standard deviation of the noise. The method developed in the article clearly demonstrated its potential for future applications to chemometrics-enhanced spectrometric analysis with limited options of memory size and data transfer rate. The algorithm used the standard routines of Matlab software.