Some results on random circulant matrices

具有独立元素的反向循环矩阵线性特征值统计量的涨落

Fluctuation of linear eigenvalue statistics of reverse circulant matrices with independent entries

该文研究满足矩条件的独立元素反向循环矩阵的线性特征值统计量涨落，证明其经标准化后服从中心极限定理型结果，丰富了随机矩阵理论相关结论。

关于随机循环伯努利矩阵的奇异性概率

On the singularity probability of random circulant Bernoulli matrices

该研究针对所有概率参数取值，完整刻画了随机循环伯努利矩阵的渐近奇异性概率，填补了相关理论空白。

Bulk behaviour of skew-symmetric patterned random matrices

Limiting Spectral Distributions (LSD) of real symmetric patterned matrices have been well-studied. In this article, we consider skew-symmetric/anti-symmetric patterned random matrices and establish the LSDs of several common matrices. For the skew-symmetric Wigner, skew-symmetric Toeplitz and the skew-symmetric Circulant, the LSDs (on the imaginary axis) are the same as those in the symmetric cases. For the skew-symmetric Hankel and the skew-symmetric Reverse Circulant however, we obtain new LSDs. We also show the existence of the LSDs for the triangular versions of these matrices. We then int…

随机循环矩阵的范数

Norms of Randomized Circulant Matrices

该研究针对非齐次独立元素随机矩阵的算子范数双侧界问题，给出Rademacher矩阵下界并提出相关猜想，验证了随机n×n循环矩阵下猜想在相差log log n因子时成立，且特定假设下可消除该双重对数项。

大型随机矩形Toeplitz与循环矩阵的最小奇异值

The smallest singular value of large random rectangular Toeplitz and circulant matrices

本文研究大型随机矩形Toeplitz与循环矩阵的最小奇异值，证明其最小特征值收敛于零，并给出收敛速率的下界（快于多对数但慢于多项式）及矩形循环矩阵的多项式上界。

One-bit compressed sensing with partial Gaussian circulant matrices

In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy $δ$, $m\sim δ^{-4} s\log(N/sδ)$ measurements suffice to reconstruct the direction of any $s$-sparse vector up to accuracy $δ$ via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an $\ell_1/\ell_2$ RIP-property. Under a slightly worse dependence on $δ$, we establish stability with respect to approximate sparsity, as well as full vector recovery results.

The spectral laws of Hermitian block-matrices with large random blocks

We are going to study the limiting spectral measure of fixed dimensional Hermitian block-matrices with large dimensional Wigner blocks. We are going also to identify the limiting spectral measure when the Hermitian block-structure is Circulant. Using the limiting spectral measure of a Hermitian Circulant block-matrix we will show that the spectral measure of a Wigner matrix with $k-$weakly dependent entries need not to be the semicircle law in the limit.

局部次高斯随机变量、随机三角多项式与随机循环矩阵的Salem-Zygmund不等式

Salem-Zygmund Inequality for locally sub-Gaussian random variables, random trigonometric polynomials, and random circulant matrices

该研究将经典Salem-Zygmund不等式扩展至局部次高斯随机变量，以此证明Kac多项式根以高概率集中在单位圆外特定环域内，且随机循环矩阵最小奇异值满足相应概率下界。

On limiting spectral distribution and joint convergence of some patterned random matrices

This article deals with the limiting spectral distribution and joint convergence of reverse circulant and symmetric circulant matrices with independent entries. These results are already proved in articles Bose and Sen (2008) \cite{bose_sen_LSD_EJP}, and Bose, Hazra and Saha (2011) \cite{bose_saha_patter_JC_annals}, respectively. But this article provides alternate proofs of these results which are shorter than the existing proofs. Our method is mainly based on moment method.

随机Toeplitz和循环矩阵的条件数

Condition Numbers of Random Toeplitz and Circulant Matrices

研究高斯随机Toeplitz和循环矩阵的条件数估计，证明高斯随机Toeplitz矩阵的条件数不会随n指数增长，与经典结论相反。

$G$-循环分解的随机矩阵部分转置的自由性

Freeness for the $G$-circulant Decomposition of the Partial Transpose of Random Matrices

本文提出任意有限群上的左$G$-循环分解，证明随机矩阵部分转置的分量自由性，并刻画其分布，揭示同阶群结构对自由分解的影响。

大型随机矩形Toeplitz矩阵范数的精确渐近性

Precise asymptotics for the norm of large random rectangular Toeplitz matrices

该研究分析了独立元素的大型矩形随机Toeplitz和circulant矩阵的谱范数渐近行为，推导了对应极限结果，通过蒙特卡洛实验验证有限样本性能，并在高斯情形下开展高阶渐近分析提出极限分布猜想。

Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices

Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one…

随机循环矩阵条件数的渐近分布

The asymptotic distribution of the condition number for random circulant matrices

本文研究满足Lyapunov条件的独立同分布生成序列的随机循环矩阵，证明其归一化极值奇异值联合分布渐近收敛于Rayleigh与Gumbel分布的独立乘积，进而得到归一化条件数渐近服从Fréchet分布。

Robust one-bit compressed sensing with partial circulant matrices

We present optimal sample complexity estimates for one-bit compressed sensing problems in a realistic scenario: the procedure uses a structured matrix (a randomly sub-sampled circulant matrix) and is robust to analog pre-quantization noise as well as to adversarial bit corruptions in the quantization process. Our results imply that quantization is not a statistically expensive procedure in the presence of nontrivial analog noise: recovery requires the same sample size one would have needed had the measurement matrix been Gaussian and the noisy analog measurements been given as data.

The asymptotic trace norm of random circulants and the graph energy

We compute the expected normalized trace norm (matrix/graph energy) of random symmetric band circulant matrices and graphs in the limit of large sizes, and obtain explicit bounds on the rate of convergence to the limit, and on the probabilities of large deviations. We also show that random symmetric band Toeplitz matrices have the same limit norm assuming that their band widths remain small relative to their sizes. We compare the limit norms across a range of related random matrix and graph ensembles.

高斯部分循环矩阵的改进RIP界

Improved RIP Bounds for Gaussian Partial Circulant Matrices

该研究针对高斯部分循环矩阵，改进了其受限等距（RIP）界的理论结果，通过优化证明步骤将相关界中的一个对数因子替换，提升了RIP界的紧度。

Joint convergence of several copies of different patterned random matrices

We study the joint convergence of independent copies of several patterned matrices in the noncommutative probability setup. In particular, joint convergence holds for the well known Wigner, Toeplitz, Hankel, reverse circulant and symmetric circulant matrices. We also study some properties of the limits. In particular, we show that copies of Wigner becomes asymptotically free with copies of any of the above other matrices.

The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices

Given an ensemble of NxN random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N --> oo. While this has been proved for many thin patterned ensembles sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period…

Restricted Isometries for Partial Random Circulant Matrices

In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that…

Random non-Abelian G-circulant matrices. Spectrum of random convolution operators on large finite groups

We analyse the limiting behavior of the eigenvalue and singular value distribution for random convolution operators on large (not necessarily Abelian) groups, extending the results by M. Meckes for the Abelian case. We show that for regular sequences of groups the limiting distribution of eigenvalues (resp. singular values) is a mixture of eigenvalue (resp. singular value) distributions of Ginibre matrices with the directing measure being related to the limiting behavior of the Plancherel measure of the sequence of groups. In particular for the sequence of symmetric groups, the limiting distri…