Properties of pattern matrices with applications to structured systems

Doubly structured mapping problems of the form $Δx=y$ and $Δ^*z=w$

For a given class of structured matrices $\mathbb S$, we find necessary and sufficient conditions on vectors $x,w\in \C^{n+m}$ and $y,z \in \C^{n}$ for which there exists $Δ=[Δ_1~Δ_2]$ with $Δ_1 \in \mathbb S$ and $Δ_2 \in \C^{n,m}$ such that $Δx=y$ and $Δ^*z=w$. We also characterize the set of all such mappings $Δ$ and provide sufficient conditions on vectors $x,y,z$, and $w$ to investigate a $Δ$ with minimal Frobenius norm. The structured classes $\mathbb S$ we consider include (skew)-Hermitian, (skew)-symmetric, pseudo(skew)-symmetric, $J$-(skew)-symmetric, pseudo(skew)-Hermitian, positive …

结构化低秩逼近的精确解

Exact Solutions in Structured Low-Rank Approximation

结构化低秩逼近是在线性空间中的所有固定秩矩阵中最小化到给定矩阵的加权弗罗贝纽斯距离的问题。我们研究精确解这个问题的计算代数几何的方式。一个特别的重点在于汉克尔矩阵，西尔维斯特矩阵和一般的线性空间。

Rank-$1$ matrix differential equations for structured eigenvalue optimization

A new approach to solving eigenvalue optimization problems for large structured matrices is proposed and studied. The class of optimization problems considered is related to computing structured pseudospectra and their extremal points, and to structured matrix nearness problems such as computing the structured distance to instability or to singularity. The structure can be a general linear structure and includes, for example, large matrices with a given sparsity pattern, matrices with given range and co-range, and Hamiltonian matrices. Remarkably, the eigenvalue optimization can be performed o…

有限域上特殊结构多项式矩阵的素性概率及其在线性系和卷积码中的应用

The Probability of Primeness for Specially Structured Polynomial Matrices over Finite Fields with Applications to Linear Systems and Convolutional Codes

计算了有限域上具有一定结构的随机多项式矩阵分别是右素数或左素数的概率。特别地，我们给出了n个非奇异多项式矩阵互左互质的概率的一个渐近公式。这些结果被用来估计可达和可观测的线性系的数量以及非灾难性的卷积码的数量。此外，我们能够获得一个渐近公式的概率，并联线性系是可达的。

Denoising and Completion of Structured Low-Rank Matrices via Iteratively Reweighted Least Squares

We propose a new Iteratively Reweighted Least Squares (IRLS) algorithm for the problem of completing or denoising low-rank matrices that are structured, e.g., that possess a Hankel, Toeplitz or block-Hankel/Toeplitz structure. The algorithm optimizes an objective based on a non-convex surrogate of the rank by solving a sequence of quadratic problems. Our strategy combines computational efficiency, as it operates on a lower dimensional generator space of the structured matrices, with high statistical accuracy which can be observed in experiments on hard estimation and completion tasks. Our expe…

Compact representations of structured BFGS matrices

For general large-scale optimization problems compact representations exist in which recursive quasi-Newton update formulas are represented as compact matrix factorizations. For problems in which the objective function contains additional structure, so-called structured quasi-Newton methods exploit available second-derivative information and approximate unavailable second derivatives. This article develops the compact representations of two structured Broyden-Fletcher-Goldfarb-Shanno update formulas. The compact representations enable efficient limited memory and initialization strategies. Two…

通过列生成方法优化正定矩阵的结构子集

Optimization over Structured Subsets of Positive Semidefinite Matrices via Column Generation

本文提出通过线性规划和二次锥规划近似正定矩阵锥的方法，利用列生成思想改进初始近似，并应用于非凸多项式优化和离散优化问题。

Efficient Structured Matrix Recovery and Nearly-Linear Time Algorithms for Solving Inverse Symmetric $M$-Matrices

In this paper we show how to recover a spectral approximations to broad classes of structured matrices using only a polylogarithmic number of adaptive linear measurements to either the matrix or its inverse. Leveraging this result we obtain faster algorithms for variety of linear algebraic problems. Key results include: $\bullet$ A nearly linear time algorithm for solving the inverse of symmetric $M$-matrices, a strict superset of Laplacians and SDD matrices. $\bullet$ An $\tilde{O}(n^2)$ time algorithm for solving $n \times n$ linear systems that are constant spectral approximations of Laplac…

同时结构模型及其在稀疏低秩矩阵中的应用

Simultaneously Structured Models with Application to Sparse and Low-rank Matrices

给定少量线性观测值的结构化模型的恢复问题近年来得到了很好的研究。示例包括恢复稀疏或组稀疏向量、低秩矩阵以及稀疏和低秩矩阵的和等。在信号处理和机器学习的各种应用中，已知感兴趣的模型同时以几种方式构造，例如，同时稀疏和低秩的矩阵。通常，促进每个个体结构的范数是已知的，并且允许使用按顺序的最优数量的测量（例如，稀疏性的$\ell_1$范数，矩阵秩的核范数）。因此，合理的做法是尽量减少这些规范的组合。我们表明，令人惊讶的是，如果我们使用这些规范的多目标优化，那么我们可以做得更好，顺序明智的，比一个算法，利用目前的结构之一。这个结果表明，为了充分利用多结构，我们需要一个全新的凸松弛，即不是一个用于每个结构的凸松弛的函数。然后，我们专门我们的结果的情况下，稀疏和低秩矩阵。我们发现，一个非凸制定的问题可以恢复模型从非常少的测量，这是对矩阵的自由度的顺序，而从组合的$\ell_1$和核规范的凸问题需要更多的测量。这证明了在这种情况下，凸和非凸恢复问题的性能之间存在顺序差距。我们的框架适用于任意的结构诱导规范，以及广泛的测量合奏。这使我们能够给出稀疏相位恢复和低秩张量完成等问题的性能界限。

通过矩形块Hankel矩阵推广的Hurwitz稳定性准则用于非monic矩阵多项式

A generalized Hurwitz stability criterion via rectangular block Hankel matrices for nonmonic matrix polynomials

本文提出了一种基于矩形块Hankel矩阵的非monic矩阵多项式Hurwitz稳定性准则，通过列降阶方法扩展了传统稳定性判据。

结构化协方差估计的几何方法

Geometric methods for estimation of structured covariances

本文采用基于Wasserstein/Bures/Hellinger距离的几何方法估计Toeplitz结构协方差矩阵，通过求解线性矩阵不等式实现高效计算，并与最大似然和Burg方法在谱线功率谱估计上进行了比较。

逆双非负矩阵的符号模式

Sign Patterns of Inverse Doubly-Nonnegative Matrices

本文研究了逆双非负矩阵的符号模式，提出了一个必要且充分的条件，用于判断符号矩阵是否对应于逆双非负矩阵，并证明了当双非负矩阵的图是树时，其逆矩阵具有唯一的符号模式，可通过图的双色染色来表达。