Normal Matrices Research Articles

Depth-based sparse bundle adjustment.

It is demonstrated in this paper that due to error model inconsistency, a certain degree of accuracy loss would be incurred to the estimated parameters when the traditional bundle adjustment method is directly applied to the scenario where a fraction of observations is implicitly error free (e.g., the reference image points in commonly used least squares matching refinement). To this end, a depth-based object point model and corresponding depth-based sparse bundle adjustment method are proposed in this paper, in which the position of an object point is represented by its 1D depth relative to its reference image. A corresponding projection model is derived, the sparse block structures of normal equations are studied depending on whether there are shared image parameters to be optimized or not, and corresponding sparse solutions of the normal equations and parameter covariance matrices are derived. Compared with the traditional sparse bundle adjustment method, simulated experiments demonstrate that our method matches the error model of the target scenario, and thus can avoid further accuracy loss. Moreover, both simulated and real data experiments demonstrate that our method can effectively improve computational efficiency.

Alevin-fry unlocks rapid, accurate and memory-frugal quantification of single-cell RNA-seq data.

The rapid growth of high-throughput single-cell and single-nucleus RNA-sequencing (sc/snRNA-seq) technologies has produced a wealth of data over the past few years. The size, volume, and distinctive characteristics of these data necessitate the development of new computational methods to accurately and efficiently quantify sc/snRNA-seq data into count matrices that constitute the input to downstream analyses. We introduce the alevin-fry framework for quantifying sc/snRNA-seq data. In addition to being faster and more memory frugal than other accurate quantification approaches, alevin-fry ameliorates the memory scalability and false-positive expression issues that are exhibited by other lightweight tools. We demonstrate how alevin-fry can be effectively used to quantify sc/snRNA-seq data, and also how the spliced and unspliced molecule quantification required as input for RNA velocity analyses can be seamlessly extracted from the same preprocessed data used to generate regular gene expression count matrices.

Tail Bounds for ℓ1 Norm of Gaussian Random Matrices with Applications

As major components of the random matrix theory, Gaussian random matrices have been playing an important role in many fields, because they are both unitary invariant and have independent entries and can be used as models for multivariate data or multivariate phenomena. Tail bounds for eigenvalues of Gaussian random matrices are one of the hot study problems. In this paper, we present tail and expectation bounds for the ℓ1 norm of Gaussian random matrices, respectively. Moreover, the tail and expectation bounds for the ℓ1 norm of the Gaussian Wigner matrix are calculated based on the resulting bounds. Compared with existing results, our results are more suitable for the high‐dimensional matrix case. Finally, we study the tail bounds for the parameter vector of some existing regularization algorithms.

Pareto Optimal Design of the Tuned Mass Damper

Strong vibrations due to wind induced loads and earthquakes have undesirable effects on tall buildings and long-span bridges, and can cause discomfort, dizziness and anxiety for their occupants and users. The Single Tuned Mass Damper (STMD) is vibration control device used to mitigate strong motion. This study investigated the STMD from the perspective of multi-objective optimization. The main goal was to identify and characterize the Pareto set of optimal STMD parameters in terms of damping and stiffness. The optimization was performed using a multi-objective optimization algorithm based on gradient descent and successive bisecting of the search domain. <img src=image/14825382_01.gif> and <img src=image/14825382_02.gif> norms of the transfer matrices for structural displacement and acceleration were considered as design objectives. We found that for peak response reduction, irrespective of the excitation type, structural damping and STMD mass ratio, the Pareto set was linear and extended mainly in the direction of frequency ratio, while the damping ratio varied marginally within the set. For RMS displacement and peak acceleration minimization, the Pareto set was composed of two segments. Additionally, in all instances, the Pareto set was found to be bounded by the single-objective optimal STMDs for the considered design objectives. The implications of the findings were discussed and approximations of the Pareto optimal STMD were suggested.

Norms of randomized circulant matrices

We investigate two-sided bounds for operator norms of random matrices with non-homogenous independent entries. We formulate a lower bound for Rademacher matrices and conjecture that it may be reversed up to a universal constant. We show that our conjecture holds up to loglogn factor for randomized n×n circulant matrices and that the double logarithm may be eliminated under some mild additional assumptions on the coefficients.

SwinFuse: A Residual Swin Transformer Fusion Network for Infrared and Visible Images

The existing deep learning fusion methods mainly concentrate on the convolutional neural networks, and few attempts are made with transformer. Meanwhile, the convolutional operation is a content-independent interaction between the image and convolution kernel, which may lose some important contexts and further limit fusion performance. Towards this end, we present a simple and strong fusion baseline for infrared and visible images, namely <i>Residual Swin Transformer Fusion Network</i>, termed SwinFuse. Our SwinFuse includes three parts: the global feature extraction, fusion layer and feature reconstruction. In particular, we build a fully attentional feature encoding backbone to model the long-range dependency, which is a pure transformer network and has a stronger representation ability compared with the convolutional neural networks. Moreover, we design a novel feature fusion strategy based on <i>L</i>₁-norm for sequence matrices, and measure the corresponding activity levels from row and column vector dimensions, which can well retain competitive infrared brightness and distinct visible details. Finally, we testify our SwinFuse with nine state-of-the-art traditional and deep learning methods on three different datasets through subjective observations and objective comparisons, and the experimental results manifest that the proposed SwinFuse obtains surprising fusion performance with strong generalization ability and competitive computational efficiency. The code will be available at https://github.com/Zhishe-Wang/SwinFuse.

Row-Average-Max-Norm of Fuzzy Matrix

Fuzzy matrices play a vital role in handling different models in an uncertain environment. In this paper, we have defined the row-average-max norm of fuzzy matrices. We also investigated some properties and lemma of the row-average-max norm of the fuzzy matrix.

On some algebraic properties of block Toeplitz matrices with commuting entries

Toeplitz matrices are ubiquitous and play important roles across many areas of mathematics. In this paper, we present some algebraic results concerning block Toeplitz matrices with block entries belonging to a commutative algebra $\AA$. The characterization of normal block Toeplitz matrices with entries from $\AA$ is also obtained.

Tensor completion via nonconvex tensor ring rank minimization with guaranteed convergence

In recent studies, the tensor ring (TR) rank has shown high effectiveness in tensor completion due to its ability of capturing the intrinsic structure within high-order tensors. A recently proposed TR rank minimization method is based on the convex relaxation by penalizing the weighted sum of nuclear norm of TR unfolding matrices. However, this method treats each singular value equally and neglects their physical meanings, which usually leads to suboptimal solutions in practice. To alleviate this weakness, this paper proposes an enhanced low-rank sparsity measure, which can more accurately approximate the TR rank and better promote the low-rankness of the solution. In specific, we apply the logdet function onto TR unfolding matrices to shrink less the larger singular values while shrink more the smaller ones. To solve the proposed nonconvex model efficiently, we develop an alternating direction method of multipliers algorithm and theoretically prove that, under some mild assumptions, our algorithm converges to a stationary point. Extensive experiments on color images, multispectral images, and color videos demonstrate that the proposed method outperforms several state-of-the-art competitors in both visual and quantitative comparison.

Optimal Rank‐1 Hankel Approximation in the Spectral Norm for Matrices with Multiple Largest Eigenvalue

AbstractWe extend a result from [2, 3] on the optimal rank‐1 Hankel approximation of real symmetric matrices with isolated largest eigenvalue to the case where the largest eigenvalue is not isolated. To illustrate our findings, we give an example where the optimal rank‐1 Hankel approximation is easily obtained and one where it does not exist.

Polynomial eigenvalue estimation: numerical radii versus norms

Bounds on the moduli of polynomial eigenvalues are typically expressed in terms of norms of the coefficient matrices of the polynomials. It is shown that using the numerical radii of these matrices instead of their norms can lead to improvements of those bounds and also, in special cases, to information on the angular distribution of the eigenvalues.

Low rank approximation of positive semi-definite symmetric matrices using Gaussian elimination and volume sampling

Positive semi-definite matrices commonly occur as normal matrices of least squares problems in statistics or as kernel matrices in machine learning and approximation theory. They are typically large and dense. Thus algorithms to solve systems with such a matrix can be very costly. A core idea to reduce computational complexity is to approximate the matrix by one with a low rank. The optimal and well understood choice is based on the eigenvalue decomposition of the matrix. Unfortunately, this is computationally very expensive. Cheaper methods are based on Gaussian elimination but they require pivoting. We show how invariant matrix theory provides explicit error formulas for an averaged error based on volume sampling. The formula leads to ratios of elementary symmetric polynomials on the eigenvalues. We discuss several bounds for the expected norm of the approximation error and include examples where this expected error norm can be computed exactly.

A new generalized refinements of Young’s inequality

In this paper, we show a new generalized refinement of Young's inequality. As applications we give some new generalized refinements of Young type inequalities for the traces, determinants, and norms of positive definite matrices.

Variational quantum algorithms for trace norms and their applications

The trace norm of matrices plays an important role in quantum information and quantum computing. How to quantify it in today’s noisy intermediate scale quantum (NISQ) devices is a crucial task for information processing. In this paper, we present three variational quantum algorithms on NISQ devices to estimate the trace norms corresponding to different situations. Compared with the previous methods, our means greatly reduce the requirement for quantum resources. Numerical experiments are provided to illustrate the effectiveness of our algorithms.

Biomimetic matrix for the study of neuroblastoma cells: A promising combination of stiffness and retinoic acid

Neuroblastoma is the third most common pediatric cancer composed of malignant immature cells that are usually treated pharmacologically by all trans-retinoic acid (ATRA) but sometimes, they can spontaneously differentiate into benign forms. In that context, biomimetic cell culture models are warranted tools as they can recapitulate many of the biochemical and biophysical cues of normal or pathological microenvironments. Inspired by that challenge, we developed a neuroblastoma culture system based on biomimetic LbL films of physiological biochemical composition and mechanical properties. For that, we used chondroitin sulfate A (CSA) and poly-L-lysine (PLL) that were assembled and mechanically tuned by crosslinking with genipin (GnP), a natural biocompatible crosslinker, in a relevant range of stiffness (30–160 kPa). We then assessed the adhesion, survival, motility, and differentiation of LAN-1 neuroblastoma cells. Remarkably, increasing the stiffness of the LbL films induced neuritogenesis that was strengthened by the combination with ATRA. These results highlight the crucial role of the mechanical cues of the neuroblastoma microenvironment since it can dramatically modulate the effect of pharmacologic drugs. In conclusion, our biomimetic platform offers a promising tool to help fundamental understanding and pharmacological screening of neuroblastoma differentiation and may assist the design of translational biomaterials to support neuronal regeneration. Statement of significanceNeuroblastoma is one of the most common pediatric tumor commonly treated by the administration of all-trans-retinoic acid (ATRA). Unfortunately, advanced neuroblastoma often develop ATRA resistance. Accordingly, in the field of pharmacological investigations on neuroblastoma, there is a tremendous need of physiologically relevant cell culture systems that can mimic normal or pathological extracellular matrices. In that context, we developed a promising matrix-like cell culture model that provides new insights on the crucial role of mechanical properties of the microenvironment upon the success of ATRA treatment on the neuroblastoma maturation. We were able to control adhesion, survival, motility, and differentiation of neuroblastoma cells. More broadly, we believe that our system will help the design of in vitro pharmacological screening strategy.

Cyclooxygenase-2 regulates PTHrP transcription in human articular chondrocytes and is involved in the pathophysiology of osteoarthritis in rats.

BackgroundCyclooxygenase-2 (COX-2) inhibitors are prescribed for the management of osteoarthritis (OA)-associated pain and inflammation. However, the role of COX-2 in normal and osteoarthritic articular chondrocytes has not been well investigated. We hypothesize that COX-2 plays a role in articular chondrocytes under normal conditions and during OA progression.MethodsIn vivo COX-2 levels in articular cartilage of normal and papain-induced osteoarthritic rats were compared. The role of COX-2 in human articular chondrocytes (HACs) was tested in vitro by COX-2 overexpression or activity inhibition. The levels of COX-2 and marker gene for normal function or articular cartilage degeneration were evaluated: mRNA by qRT-PCR; proteins by western blotting or immunohistochemistry; and glycosaminoglycan (GAG) by Safranin O–fast green staining. Parathyroid hormone-related protein (PTHrP) promoter activity was detected with luciferase reporter assays.ResultsIn the OA rat study, COX-2 and PTHrP were simultaneously increased in osteoarthritic rat chondrocytes, while increased PTHrP levels were reduced by celecoxib, a COX-2 selective inhibitor. The levels of normal cartilage matrices, GAG and type II collagen decreased, while markers of degeneration, collagen type X and MMP13 were elevated in osteoarthritic articular chondrocytes. Celecoxib rescued the loss of GAG and the increased collagen type X and MMP13 levels. In vitro, COX-2 overexpression in HACs significantly increased Col2a1, Col10a1, PTHrP and MMP13 mRNA expression, which was decreased when COX-2 activity was suppressed. More importantly, COX-2 overexpression upregulated the PTHrP transcription, mRNA expression and protein levels.ConclusionCOX-2 plays a pathophysiological role by preventing terminal differentiation of articular chondrocytes by upregulating PTHrP expression at the early stage of OA progression.The Translational potential of this articleCOX2 up-regulates PTHrP expression in normal and OA articular chondrocytes.

Robust tensor train component analysis

SummaryRobust Principal Component Analysis plays a key role in various fields such as image and video processing, data mining, and hyperspectral data analysis. In this paper, we study the problem of robust tensor train (TT) principal component analysis from partial observations, which aims to decompose a given tensor into the low TT rank and sparse components. The decomposition of the proposed model is used to find the hidden factors and help alleviate the curse of dimensionality via a set of connected low‐rank tensors. A relaxation model is to minimize a weighted combination of the sum of nuclear norms of unfolding matrices of core tensors and the tensor norm. A proximal alternating direction method of multipliers is developed to solve the resulting model. Furthermore, we show that any cluster point of the convergent subsequence is a Karush‐Kuhn‐Tucker point of the proposed model under some conditions. Extensive numerical examples on both synthetic data and real‐world datasets are presented to demonstrate the effectiveness of the proposed approach.

Algebraic techniques for the least squares problems in elliptic complex matrix theory and their applications

In this study, we introduce concepts of norms of elliptic complex matrices and derive the least squares solution, the pure imaginary least squares solution, and the pure real least squares solution with the least norm for the elliptic complex matrix equation by using the real representation of elliptic complex matrices. To prove the authenticity of our results and to distinguish them from existing ones, some illustrative examples are also given. Elliptic numbers are generalized form of complex and so real numbers. Thus, the obtained results extend, generalize, and complement some known least squares solutions results from the literature.

Optimal properties of tensor product of B-bases

It is proved the optimal conditioning for the ∞-norm of collocation matrices of the tensor product of normalized B-bases among the tensor product of all normalized totally positive bases of the corresponding space of functions. Bounds for the minimal eigenvalue and singular value and illustrative numerical examples are also included.

Power boundedness in the maximum norm of stability matrices for ADI methods

Power boundedness in the maximum norm of stability matrices for ADI methods