Considering Dependencies Amongst Genes Helps to Adjust the Significance Rank of DEGs

Lists of differentially expressed genes (DEGs) detected often show low reproducibility even in technique replicate experiments. The reproducibility is even lower for those real cancer data with large biological variations and limited number of samples. Since existing methods for identifying differentially expressed genes treat each gene separately, they cannot circumvent the problem of low reproducibility. Considering correlation structures of genes may help to mitigate the effect of errors on individual gene estimates and thus get more reliable lists of DEGs. We borrowed information from large amount of existing microarray data to define the expression dependencies amongst genes. We use this prior knowledge of dependencies amongst genes to adjust the significance rank of DEGs. We applied our method and four popular ranking algorithms including mean fold change (FC), SAM, t‐statistic and Wilcoxon rank sum‐test on two cancer microarray datasets. Our method achieved higher reproducibility than other methods across a range of sample sizes. Furthermore, our method obtained higher accuracy than other methods, especially when the sample size is small. The results demonstrate that considering the dependencies amongst genes helps to adjust the significance rank of genes and find those truly differentially expressed genes.     

A classification of subgroups of odd unitary groups

In this paper, under the Λ-stable rank condition, we discuss the classification of subgroups of odd unitary groups and get an analogue of Sandwich Theorem.     

The inertia of the symmetric approximation for low-rank matrices

In many areas of applied linear algebra, it is necessary to work with matrix approximations. A usual situation occurs when a matrix obtained from experimental or simulated data is needed to be approximated by a matrix that lies in a corresponding statistical model and satisfies some specific properties. In this short note, we focus on symmetric and positive-semidefinite approximations and we show that the positive and negative indices of inertia of the symmetric approximation and the rank of the positive-semidefinite approximation are always bounded from above by the rank of the original matrix.     

Some new non-additive maximum rank distance codes

In this paper, a construction of maximum rank distance (MRD) codes as a generalization of generalized Gabidulin codes is given. The family of the resulting codes is not covered properly by additive generalized twisted Gabidulin codes, and does not cover all twisted Gabidulin codes. When the basis field has more than two elements, this family includes also non-affine MRD codes, and such codes exist for all parameters. Therefore, these codes are the first non-additive MRD codes for most of the parameters.     

Computational Krylov‐based methods for large‐scale differential Sylvester matrix problems

In the present paper, we propose Krylov‐based methods for solving large‐scale differential Sylvester matrix equations having a low‐rank constant term. We present two new approaches for solving such differential matrix equations. The first approach is based on the integral expression of the exact solution and a Krylov method for the computation of the exponential of a matrix times a block of vectors. In the second approach, we first project the initial problem onto a block (or extended block) Krylov subspace and get a low‐dimensional differential Sylvester matrix equation. The latter problem is then solved by some integration numerical methods such as the backward differentiation formula or Rosenbrock method, and the obtained solution is used to build the low‐rank approximate solution of the original problem. We give some new theoretical results such as a simple expression of the residual norm and upper bounds for the norm of the error. Some numerical experiments are given in order to compare the two approaches.     

Convergence analysis of projected fixed‐point iteration on a low‐rank matrix manifold

In this paper, we analyze the convergence of a projected fixed‐point iteration on a Riemannian manifold of matrices with fixed rank. As a retraction method, we use the projector splitting scheme. We prove that the convergence rate of the projector splitting scheme is bounded by the convergence rate of standard fixed‐point iteration without rank constraints multiplied by the function of initial approximation. We also provide counterexample to the case when conditions of the theorem do not hold. Finally, we support our theoretical results with numerical experiments.