大块矩阵的谱

Matrices Whose Inversions are Tridiagonal, Band or Block-Tridiagonal and Their Relationship with the Covariance Matrices of a Random Markov Processes (Fields)

The article discusses the matrices of the three forms whose inversions are: tridiagonal matrix, banded matrix or block-tridiagonal matrix and their relationships with the covariance matrices of measurements of ordinary (simple) Markov Random Processes (MRP), multiconnected MRP and vector MRP respectively. Such covariance matrices are frequently occurring in the problems of optimal filtering, extrapolation and interpolation of MRP and Markov Random Fields (MRF). It is shown, that the structure of these three forms of matrices has the same form, but the matrix elements in the first case are scal…

压缩感知中的Toeplitz块矩阵

Toeplitz Block Matrices in Compressed Sensing

压缩感知理论中的最新工作表明，$n\times N$独立同分布（IID）感知矩阵（其条目独立于某些概率分布绘制）保证了稀疏信号的高概率精确恢复，即使$n\ll N$。受信号处理应用的启发，随机滤波与Toeplitz感测矩阵的元素是从相同的分布被认为是，并证明也足以恢复稀疏信号从减少的样本完全具有高概率。本文考虑Toeplitz分块矩阵作为感知矩阵。它们自然出现在多通道和多维滤波应用中，并包括作为特殊情况的Toeplitz矩阵。它表明，准确重建的概率也很高。它们的性能进行了验证，使用模拟。

The Restricted Isometry Property for Random Block Diagonal Matrices

In Compressive Sensing, the Restricted Isometry Property (RIP) ensures that robust recovery of sparse vectors is possible from noisy, undersampled measurements via computationally tractable algorithms. It is by now well-known that Gaussian (or, more generally, sub-Gaussian) random matrices satisfy the RIP under certain conditions on the number of measurements. Their use can be limited in practice, however, due to storage limitations, computational considerations, or the mismatch of such matrices with certain measurement architectures. These issues have recently motivated considerable effort to…

Block-sparse Recovery of Semidefinite Systems and Generalized Null Space Conditions

This article considers the recovery of low-rank matrices via a convex nuclear-norm minimization problem and presents two null space properties (NSP) which characterize uniform recovery for the case of block-diagonal matrices and block-diagonal positive semidefinite matrices. These null-space conditions turn out to be special cases of a new general setup, which allows to derive the mentioned NSPs and well-known NSPs from the literature. We discuss the relative strength of these conditions and also present a deterministic class of matrices that satisfies the block-diagonal semidefinite NSP.

关于特征为2的交换环上一类特殊矩阵的猜想研究

Towards a conjecture on a special class of matrices over commutative rings of characteristic 2

本文证明了Keller和Rosemarin在Eurocrypt 2021提出的关于特征为2交换环上分块矩阵多项式零化度的猜想，确认了Starkad密码不变子空间维度的最优界，并揭示了Hadamard矩阵的代数结构及分块矩阵间的关系。

Sensing Matrix Design via Capacity Maximization for Block Compressive Sensing Applications

It is well established in the compressive sensing (CS) literature that sensing matrices whose elements are drawn from independent random distributions exhibit enhanced reconstruction capabilities. In many CS applications, such as electromagnetic imaging, practical limitations on the measurement system prevent one from generating sensing matrices in this fashion. Although one can usually randomized the measurements to some degree, these sensing matrices do not achieve the same reconstruction performance as the truly randomized sensing matrices. In this paper, we present a novel method, based up…