Analysis of the mean squared derivative cost function

In this paper, we investigate the mean squared derivative cost functions that arise in various applications such as in motor control, biometrics and optimal transport theory. We provide qualitative properties, explicit analytical formulas and computational algorithms for the cost functions. We also perform numerical simulations to illustrate the analytical results. In addition, as a by‐product of our analysis, we obtain an explicit formula for the inverse of a Wronskian matrix that is of independent interest in linear algebra and differential equations theory. Copyright © 2017 John Wiley & Sons, Ltd.     

One‐dimensional q‐Dirac equation

In this paper, we introduce a q‐analog of 1‐dimensional Dirac equation. We investigate the existence and uniqueness of the solution of this equation. Later, we discuss some spectral properties of the problem, such as formally self‐adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green function, existence of a countable sequence of eigenvalues, and eigenfunctions forming an orthonormal basis of . Finally, we give some examples.     

高维约束矩阵回归问题

高维约束矩阵回归是指高维情况下带非凸约束的多响应多预测统计回归问题,其数学模型是一个NP-难的矩阵优化,它在机器学习与人工智能、医学影像疾病诊疗、基因表达分析、脑神经网络、风险管理等领域有广泛应用.从高维约束矩阵回归的优化理论和算法两方面总结和评述这些新成果,同时,列出了相应的重要文献.     

On a difference equation of the second order with an exponential coefficient

This paper is devoted to a generalization of some previous results, as we completely solve a linear homogeneous difference equation of the second order with an exponential coefficient.     

Non-synchronous oscillations in four-dimensional nonlinear semelparous Leslie matrix models

In a certain class of semelparous Leslie matrix models, either a positive equilibrium is stable and an invariant set on the boundary of the nonnegative cone is unstable or vice versa generically if the model dimension is two or three. This dynamic dichotomy is expected to be failed in the four-dimensional case. Our study focuses on a semelparous Leslie matrix model with specific nonlinearities and rigorously proves that the dynamic dichotomy does not hold in the four-dimensional case. This result is derived by showing that the four-dimensional semelparous Leslie matrix model can be uniformly persistent with respect to the boundary of the nonnegative cone even if there exists an unstable positive equilibrium. In such a situation, there are no missing age-classes but population oscillation occurs.     

矩阵优化扰动性分析的若干进展

由于近年来实际问题特别是大数据应用的发展,矩阵优化问题越来越得到优化研究者,甚至是其他领域的研究者的高度关注,成为热点问题.优化问题的扰动性分析是优化理论研究的基础与核心,为包括算法设计在内的优化研究提供重要的理论基础.由于矩阵优化问题的非多面体性,使得相应扰动分析理论的研究本质上与经典的多面体优化问题(非线性规划)不同.结合文献[1,2],简要介绍矩阵优化扰动性分析方面取得的若干最新进展.     

Low rank matrix recovery from rank one measurements

We study the recovery of Hermitian low rank matrices $X\in {\mathbb{C}}^{n\times n}$ from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form $a_j{a}_j^{?}$ for some measurement vectors $a_1,\dots ,a_m$, i.e., the measurements are given by $b_j=\mathrm{tr}(X{a}_j{a}_j^{?})$. The case where the matrix $X=x{x}^{?}$ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements $b_j=|〈x,a_j〉{|}^2$) via the PhaseLift approach, which has been introduced recently. We derive bounds for the number m of measurements that guarantee successful uniform recovery of Hermitian rank r matrices, either for the vectors $a_j$, $j=1,\dots ,m$, being chosen independently at random according to a standard Gaussian distribution, or $a_j$ being sampled independently from an (approximate) complex projective t-design with $t=4$. In the Gaussian case, we require $m\ge Crn$ measurements, while in the case of 4-designs we need $m\ge \mathit{Cr}n\log ?(n)$. Our results are uniform in the sense that one random choice of the measurement vectors $a_j$ guarantees recovery of all rank r-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to Gross, Krahmer and Kueng. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.     

An analysis on the efficiency of Euler's method for computing the matrix pth root

It is shown that the matrix sequence generated by Euler's method starting from the identity matrix converges to the principal pth root of a square matrix, if all the eigenvalues of the matrix are in a region including the one for Newton's method given by Guo in 2010. The convergence is cubic if the matrix is invertible. A modification version of Euler's method using the Schur decomposition is developed. Numerical experiments show that the modified algorithm has the overall good numerical behavior.     

Local reconstruction of low‐rank matrices and subspaces

We study the problem of reconstructing a low‐rank matrix, where the input is an n × m matrix M over a field and the goal is to reconstruct a (near‐optimal) matrix that is low‐rank and close to M under some distance function Δ. Furthermore, the reconstruction must be local, i.e., provides access to any desired entry of by reading only a few entries of the input M (ideally, independent of the matrix dimensions n and m). Our formulation of this problem is inspired by the local reconstruction framework of Saks and Seshadhri (SICOMP, 2010). Our main result is a local reconstruction algorithm for the case where Δ is the normalized Hamming distance (between matrices). Given M that is ‐close to a matrix of rank (together with d and ), this algorithm computes with high probability a rank‐d matrix that is ‐close to M. This is a local algorithm that proceeds in two phases. The preprocessing phase reads only random entries of M, and stores a small data structure. The query phase deterministically outputs a desired entry by reading only the data structure and 2d additional entries of M. We also consider local reconstruction in an easier setting, where the algorithm can read an entire matrix column in a single operation. When Δ is the normalized Hamming distance between vectors, we derive an algorithm that runs in polynomial time by applying our main result for matrix reconstruction. For comparison, when Δ is the truncated Euclidean distance and , we analyze sampling algorithms by using statistical learning tools. A preliminary version of this paper appears appears in ECCC, see: http://eccc.hpi-web.de/report/2015/128/ © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 607–630, 2017