We show that for an $n\times n$ random symmetric matrix $A_n$, whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $ξ$ with mean $0$ and variance $1$, \[\mathbb{P}[s_n(A_n) \le ε/\sqrt{n}] \le O_ξ(ε^{1/8} + \exp(-Ω_ξ(n^{1/2}))) \quad \text{for all } ε\ge 0.\] This improves a result of Vershynin, who obtained such a bound with $n^{1/2}$ replaced by $n^{c}$ for a small constant $c$, and $1/8$ replaced by $(1/8) + η$ (with implicit constants also depending on $η> 0$). Furthermore, when $ξ$ is a Rademacher random variable, we prove that \[\mathbb{P}[s_n(A_n) \le ε/\sqrt{n}] \le O(ε^{1/8} + \exp(-Ω((\log{n})^{1/4}n^{1/2}))) \quad \text{for all } ε\ge 0.\] The special case $ε= 0$ improves a recent result of Campos, Mattos, Morris, and Morrison, which showed that $\mathbb{P}[s_n(A_n) = 0] \le O(\exp(-Ω(n^{1/2}))).$
The main innovation in our work are new notions of arithmetic structure -- the Median Regularized Least Common Denominator and the Median Threshold, which we believe should be more generally useful in contexts where one needs to combine anticoncentration information of different parts of a vector.
We derive estimates for the largest and smallest singular values of sparse rectangular $N\times n$ random matrices, assuming $\lim_{N,n\to\infty}\frac nN=y\in(0,1)$. We consider a model with sparsity parameter $p_N$ such that $Np_N\sim \log^{α}N$ for some $α>1$, and assume that the moments of the matrix elements satisfy the condition $\mathbf E|X_{jk}|^{4+δ}\le C<\infty$. We assume also that the entries of matrices we consider are truncated at the level $(Np_N)^{\frac12-\varkappa}$ with $\varkappa:=\fracδ{2(4+δ)}$.
Let $R_n$ be a $n \times n$ random matrix with i.i.d. subgaussian entries. Let $M$ be a $n \times n$ deterministic matrix with norm $\lVert M \rVert \le n^γ$ where $1/2<γ<1$. The goal of this paper is to give a general estimate of the smallest singular value of the sum $R_n + M$, which improves an earlier result of Tao and Vu.
We obtain lower tail estimates for the smallest singular value of random matrices with independent but non-identically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form \[ M = A\circ X + B = (a_{ij}ξ_{ij} + b_{ij}) \] where $X=(ξ_{ij})$ has iid centered entries of unit variance and $A$ and $B$ are fixed matrices. In our main result we obtain polynomial bounds on the smallest singular value of $M$ for the case that $A$ has bounded (possibly zero) entries, and $B= Z\sqrt{n}$ where $Z$ is a diagonal matrix with entries bounded away from zero. As …
We prove an estimate on the smallest singular value of a multiplicatively and additively deformed random rectangular matrix. Suppose $n\le N \le M \le ΛN$ for some constant $Λ\ge 1$. Let $X$ be an $M\times n$ random matrix with independent and identically distributed entries, which have zero mean, unit variance and arbitrarily high moments. Let $T$ be an $N\times M$ deterministic matrix with comparable singular values $c\le s_{N}(T) \le s_{1}(T) \le c^{-1}$ for some constant $c>0$. Let $A$ be an $N\times n$ deterministic matrix with $\|A\|=O(\sqrt{N})$. Then we prove that for any $ε>0$, the sm…
Let $A = (a_{ij})$ be a square $n\times n$ matrix with i.i.d. zero mean and unit variance entries. Rudelson and Vershynin showed that the upper bound for a smallest singular value $s_n(A)$ is of order $n^{-\frac12}$ with probability close to one under additional assumption on entries of $A$ that $\mathbb{E}a^4_{11} < \infty$. We remove the assumption on the fourth moment and show the upper bound assuming only $\mathbb{E}a^2_{11} = 1.$
Let $n,k\geq 1$ and let $G$ be the $n\times n$ random matrix with i.i.d. standard real Gaussian entries. We show that there are constants $c_k,C_k>0$ depending only on $k$ such that the smallest singular value of $G^k$ satisfies $$ c_k\,t\leq {\mathbb P}\big\{s_{\min}(G^k)\leq t^k\,n^{-1/2}\big\}\leq C_k\,t,\quad t\in(0,1], $$ and, furthermore, $$ c_k/t\leq {\mathbb P}\big\{\|G^{-k}\|_{HS}\geq t^k\,n^{1/2}\big\}\leq C_k/t,\quad t\in[1,\infty), $$ where $\|\cdot\|_{HS}$ denotes the Hilbert-Schmidt norm.
The smallest singular value and condition number play important roles in numerical linear algebra and the analysis of algorithms. In numerical analysis with randomness, many previous works make Gaussian assumptions, which are not general enough to reflect the arbitrariness of the input. To overcome this drawback, we prove the first quantitative universality for the smallest singular value and condition number of random matrices. Moreover, motivated by the study of smoothed analysis that random perturbation makes deterministic matrices well-conditioned, we consider an analog for random matrices…
Suppose $X$ is an $N \times n$ complex matrix whose entries are centered, independent, and identically distributed random variables with variance $1/n$ and whose fourth moment is of order ${\mathcal O}(n^{-2})$. In the first part of the paper, we consider the non-Hermitian matrix $X A X^* - z$, where $A$ is a deterministic matrix whose smallest and largest singular values are bounded below and above respectively, and $z\neq 0$ is a complex number. Asymptotic probability bounds for the smallest singular value of this model are obtained in the large dimensional regime where $N$ and $n$ diverge t…
We derive a lower bound on the smallest singular value of a random $d$-regular matrix, that is, the adjacency matrix of a random $d$-regular directed graph. More precisely, let $C_1<d< c_1 n/\log^2 n$ and let $\mathcal{M}_{n,d}$ be the set of all $0/1$-valued square $n\times n$ matrices such that each row and each column of a matrix $M\in \mathcal{M}_{n,d}$ has exactly $d$ ones. Let $M$ be uniformly distributed on $\mathcal{M}_{n,d}$. Then the smallest singular value $s_{n} (M)$ of $M$ is greater than $c_2 n^{-6}$ with probability at least $1-C_2\log^2 d/\sqrt{d}$, where $c_1$, $c_2$, $C_1$, a…
Let $A$ be the adjacency matrix of a uniformly random $d$-regular digraph on $n$ vertices, and suppose that $\min(d,n-d)\geqλn$. We show that for any $κ\geq 0$, \[\mathbb{P}[s_n(A)\leqκ]\leq C_λκ\sqrt{n}+2e^{-c_λn}.\] Up to the constants $C_λ, c_λ> 0$, our bound matches optimal bounds for $n\times n$ random matrices, each of whose entries is an i.i.d $\text{Ber}(d/n)$ random variable. The special case $κ= 0$ of our result confirms a conjecture of Cook regarding the probability of singularity of dense random regular digraphs.
We consider the least singular value of $M = R^* X T + U^* YV$, where $R,T,U,V$ are independent Haar-distributed unitary matrices and $X, Y$ are deterministic diagonal matrices. Under weak conditions on $X$ and $Y$, we show that the limiting distribution of the least singular value of $M$, suitably rescaled, is the same as the limiting distribution for the least singular value of a matrix of i.i.d. gaussian random variables. Our proof is based on the dynamical method used by Che and Landon to study the local spectral statistics of sums of Hermitian matrices.
The problem of estimating the smallest singular value of random square matrices is important in connection with matrix computations and analysis of the spectral distribution. In this survey, we consider recent developments in the study of quantitative invertibility in the non-Hermitian setting, and review some applications of this line of research.
We study the distribution of the least singular value associated to an ensemble of sparse random matrices. Our motivating example is the ensemble of $N\times N$ matrices whose entries are chosen independently from a Bernoulli distribution with parameter $p$. These matrices represent the adjacency matrices of random Erdős--Rényi digraphs and are sparse when $p\ll 1$. We prove that in the regime $pN\gg 1$, the distribution of the least singular value is universal in the sense that it is independent of $p$ and equal to the distribution of the least singular value of a Gaussian matrix ensemble. We…
In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from $n$ independent observations of a $p$-dimensional time series with iid components converge almost surely to $(1+\sqrtγ)^2$ and $(1-\sqrtγ)^2$, respectively, as $n \to \infty$, if $p/n\to γ\in (0,1]$ and the truncated variance of the entry distribution is 'almost slowly varying', a condition we describe via moment properties of self-normalized sums. Moreover, the empirical spectral distributions of these sample correlation matrices converge weakly, with probability 1, to the Marchenko-P…
We are concerned with the small ball behavior of the smallest singular value of random matrices. Often, establishing such results involves, in some capacity, a discretization of the unit sphere. This requires bounds on the norm of the matrix, and the latter bounds require strong assumptions on the distribution of the entries, such as bounded fourth moments (for a weak estimate), sub-gaussian tails (for a strong estimate), and structural assumptions such as mean zero and variance one. Recently, Rebrova and Tikhomirov developed a discretization procedure which does not rely on strong tail assump…
We study largest singular values of large random matrices, each with mean of a fixed rank $K$. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It provides a decomposition of the largest $K$ singular values into the deterministic rate of growth, random centered fluctuations given as explicit linear combinations of the entries of the matrix, and a term negligible in probability. We use this representation to establish asymptotic normality of the largest singular values for random matrices with means that ha…
Let $A$ be an $n\times n$ real matrix, and let $M$ be an $n\times n$ random matrix whose entries are i.i.d sub-Gaussian random variables with mean $0$ and variance $1$. We make two contributions to the study of $s_n(A+M)$, the smallest singular value of $A+M$. (1) We show that for all $ε\geq 0$, $$\mathbb{P}[s_n(A + M) \leq ε] = O(ε\sqrt{n}) + 2e^{-Ω(n)},$$ provided only that $A$ has $Ω(n)$ singular values which are $O(\sqrt{n})$. This extends a well-known result of Rudelson and Vershynin, which requires all singular values of $A$ to be $O(\sqrt{n})$. (2) We show that any bound of the form $$\…
In this paper, the exact values of the structured singular values of some generalized stochastic complex matrices is explicit in term of the constant row (column) sum.
We show that for an $n\times n$ random matrix $A$ with independent uniformly anti-concentrated entries, such that $\mathbb{E} ||A||^2_{HS}\leq K n^2$, the smallest singular value $σ_n(A)$ of $A$ satisfies $$ P\left( σ_n(A)\leq \frac{\varepsilon}{\sqrt{n}} \right) \leq C\varepsilon+2e^{-cn},\quad \varepsilon \ge 0. $$ This extends earlier results of Rudelson and Vershynin, and Rebrova and Tikhomirov by removing the assumption of mean zero and identical distribution of the entries across the matrix, as well as the recent result of Livshyts, where the matrix was required to have i.i.d. rows. Our …
In this paper we give a simple, short, and self-contained proof for a non-trivial upper bound on the probability that a random $\pm 1$ symmetric matrix is singular.
Let $Q_n$ be a random $n\times n$ matrix with entries in $\{0,1\}$ whose rows are independent vectors of exactly $n/2$ zero components. We show that the smallest singular value $s_n(Q_n)$ of $Q_n$ satisfies \[ \mathbb{P}\Big\{s_n(Q_n)\le \frac{\varepsilon}{\sqrt{n}}\Big\} \le C\varepsilon + 2 e^{-cn} \quad \forall \varepsilon \ge 0, \] which is optimal up to the constants $C,c>0$. This improves on earlier results of Ferber, Jain, Luh and Samotij, as well as Jain. In particular, for $\varepsilon=0$, we obtain the first exponential bound in dimension for the singularity probability \[ \mathbb{P}…
In this paper, we investigate extreme singular values of the analysis matrix of a Gabor frame $(g, Λ)$ with a random window $g$. Columns of such matrices are time and frequency shifts of $g$, and $Λ\subset \mathbb{Z}_M\times\mathbb{Z}_M$ is the set of time-frequency shift indices. Our aim is to obtain bounds on the singular values of such random time-frequency structured matrices for various choices of the frame set $Λ$, and to investigate their dependence on the structure of $Λ$, as well as on its cardinality. We also compare the results obtained for Gabor frame analysis matrices with the res…
For each $n$, let $M_n$ be an $n\times n$ random matrix with independent $\pm 1$ entries. We show that ${\mathbb P}\{\mbox{$M_n$ is singular}\}=(1/2+o_n(1))^n$, which settles an old problem. Some generalizations are considered.
Let $M_n$ be drawn uniformly from all $\pm 1$ symmetric $n \times n$ matrices. We show that the probability that $M_n$ is singular is at most $\exp(-c(n\log n)^{1/2})$, which represents a natural barrier in recent approaches to this problem. In addition to improving on the best-known previous bound of Campos, Mattos, Morris and Morrison of $\exp(-c n^{1/2})$ on the singularity probability, our method is different and considerably simpler.
For large random matrices $X$ with independent, centered entries but not necessarily identical variances, the eigenvalue density of $XX^*$ is well-approximated by a deterministic measure on $\mathbb{R}$. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [arXiv:1606.07353] to the vicinity of these singularities.
