Vishesh Jain, Ashwin Sah, Mehtaab Sawhney
Let $M_n$ be a random $n\times n$ matrix with i.i.d. $\text{Bernoulli}(1/2)$ entries. We show that for fixed $k\ge 1$, \[\lim_{n\to \infty}\frac{1}{n}\log_2\mathbb{P}[\text{corank }M_n\ge k] = -k.\]
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Rank deficiency of Bernoulli random matrices for growing corank
研究伯努利随机矩阵余秩至少为k的概率，其中k为O(√log n)阶，得到概率为(1-p+o_n(1))^(kn)。
We investigate the rank of random (symmetric) sparse matrices. Our main finding is that with high probability, any dependency that occurs in such a matrix is formed by a set of few rows that contains an overwhelming number of zeros. This allows us to obtain an exact estimate for the co-rank.
Let $M$ be a random $m \times n$ matrix with binary entries and i.i.d. rows. The weight (i.e., number of ones) of a row has a specified probability distribution, with the row chosen uniformly at random given its weight. Let $N(n,m)$ denote the number of left null vectors in ${0,1}^m$ for $M$ (including the zero vector), where addition is mod 2. We take $n, m \to \infty$, with $m/n \to α> 0$, while the weight distribution may vary with $n$ but converges weakly to a limiting distribution on ${3, 4, 5, ...}$; let $W$ denote a variable with this limiting distribution. Identifying $M$ with a hyperg…
Motivated by the work of Fulman and Goldstein, comparing the distribution of the corank of random matrices in $\mathbb F_q[n]$ with the limit distribution as $n \to \infty$, we define a countdown process, driven by independent geometric random variables related to random integer partitions. Analysis of this process leads to sharper bounds on the total variation distance.
We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and $\{0,1\}$-matrices over the rationals. The sufficient condition is generally necessary as well.
We extend the result of Markus, Spielman, and Srivastava about the sum of rank-one symmetric random matrices to the case when the isotropy assumption on the random matrices is relaxed.
We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. To be more precise, we consider random Hermitian matrices with independent Gaussian entries $M_{ij}, i\leq j$ with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large $N$ limit, various limiting distributions depending on both the eigenvalues of the matrix $(\mathbb{E}M_{ij})_{i,j=1}^N$ and its rank.
We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrices. We assume that the matrix entries have finite fourth moment and extend the results by Capitaine, Donati-Martin, and Féral for perturbations whose eigenvectors are delocalized.
Fluctuations of eigenvalues of a polynomial on Haar unitary and finite rank matrices
本文研究由有限秩确定性矩阵截断的大Haar酉矩阵上多项式特征值的涨落，在特征值均为单重时给出完整计算算法，并通过例子说明多重特征值情形下一般算法更为复杂。
We show that finite rank perturbations of certain random matrices fit in the framework of infinitesimal (type B) asymptotic freeness. This can be used to explain the appearance of free harmonic analysis (such as subordination functions appearing in additive free convolution) in computations of outlier eigenvalues in spectra of such matrices.
We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrice under the assumption that the off-diagonal matrix entries have uniformly bounded fifth moment and the diagonal entries have uniformly bounded third moment. Using our recent results on the fluctuation of resolvent entries [31],[28], and ideas from [9], we extend results by M.Capitaine, C.Donati-Martin, and D.Féral [12], [13].
In this paper we study how perturbing a matrix changes its non-negative rank. We prove that the non-negative rank is upper-semicontinuos and we describe some special families of perturbations. We show how our results relate to Statistics in terms of the study of Maximum Likelihood Estimation for mixture models.
We present a simple, yet useful result about the expected value of the determinant of random sum of rank-one matrices. Computing such expectations in general may involve a sum over exponentially many terms. Nevertheless, we show that an interesting and useful class of such expectations that arise in, e.g., D-optimal estimation and random graphs can be computed efficiently via computing a single determinant.
We prove a sharp $\sqrt{N}$ transition for the infinitesimal distribution of a periodically banded GUE matrix. For band widths $b_N = Ω(\sqrt{N})$, we further prove that our model is infinitesimally free from the matrix units and the normalized all-ones matrix. Our results allow us to extend previous work of Shlyakhtenko on finite-rank perturbations of Wigner matrices in the infinitesimal framework. For finite-rank perturbations of our model, we find outliers at the classical positions from the deformed Wigner ensemble.
We study the asymptotic behavior of outliers in the spectrum of bounded rank perturbations of large random matrices. In particular, we consider perturbations of elliptic random matrices which generalize both Wigner random matrices and non-Hermitian random matrices with iid entries. As a consequence, we recover the results of Capitaine, Donati-Martin, and Féral for perturbed Wigner matrices as well as the results of Tao for perturbed random matrices with iid entries. Along the way, we prove a number of interesting results concerning elliptic random matrices whose entries have finite fourth mome…
Robust recovery of low-rank matrices and low-tubal-rank tensors from noisy sketches
针对带噪声线性草图下的低秩矩阵恢复，本文分析双重草图算法，并基于非渐近随机矩阵理论首次给出误差保证，且推广至低秩矩阵逼近与低管秩张量恢复。
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