The smallest singular value of random combinatorial matrices
Tuan Tran
备注：
27 pages + two appendices. Section 2.6 was rewritten. We also added an appendix by Jain, Sah and Sawhney. Update: We were recently informed that Proposition 3.4 has been proved earlier by Bero Roos [New inequalities for permanents and hafnians and some generalizations, arXiv:1906.06176v2]. As such we removed the proof of Proposition 3.4 and cited Bero Roos' paper instead
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 \[ \mathbb{P}\big\{Q_n \,\,\text{is singular}\big\} \le 2 e^{-cn}.\] To overcome the lack of independence between entries of $Q_n$, we introduce an arithmetic-combinatorial invariant of a pair of vectors, which we call a Combinatorial Least Common Denominator (CLCD). We prove a small ball probability inequality for the combinatorial statistic $\sum_{i=1}^{n}a_iv_{σ(i)}$ in terms of the CLCD of the pair $(a,v)$, where $σ$ is a uniformly random permutation of $\{1,2,\ldots,n\}$ and $a:=(a_1,\ldots,a_n), v:=(v_1,\ldots,v_n)$ are real vectors. This inequality allows us to derive strong anti-concentration properties for the distance between a fixed row of $Q_n$ and the linear space spanned by the remaining rows, and prove the main result.
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.
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 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 $\mathbb{E}a^2_{11} = 1.$
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 $n,k\geq 1$ 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.
Let $M_n$ denote a random symmetric $n \times n$ matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely conjectured that $M_n$ is singular with probability at most $(2+o(1))^{-n}$. On the other hand, the best known upper bound on the singularity probability of $M_n$, due to Vershynin (2011), is $2^{-n^c}$, for some unspecified small constant $c > 0$. This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showin…
An approximate Spielman-Teng theorem for the least singular value $s_n(M_n)$ of a random $n\times n$ square matrix $M_n$ is a statement of the following form: there exist constants $C,c >0$ such that for all $η\geq 0$, $\Pr(s_n(M_n) \leq η) \lesssim n^{C}η+ \exp(-n^{c})$. The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for $\{0,1\}$-valued matrices, each of whose rows is an independent vector with exactly $n/2$ zero components. This improves on previous work…
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 …
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…
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 $$\…
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…
Consider a random $n\times n$ zero-one matrix with "density" $p$, sampled according to one of the following two models: either every entry is independently taken to be one with probability $p$ (the "Bernoulli" model), or each row is independently uniformly sampled from the set of all length-$n$ zero-one vectors with exactly $pn$ ones (the "combinatorial" model). We give simple proofs of the (essentially best-possible) fact that in both models, if $\min(p,1-p)\geq (1+\varepsilon)\log n/n$ for any constant $\varepsilon>0$, then our random matrix is nonsingular with probability $1-o(1)$. In the B…
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…
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 …
Let $M_n$ be an $n\times n$ random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant $C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{-1}$, \begin{align*} {\mathbb P}\big\{\mbox{$M_n$ is singular}\big\}&=(1+o_n(1)){\mathbb P}\big\{\mbox{$M_n$ contains a zero row or column}\big\}\\ &=(2+o_n(1))n\,(1-p)^n, \end{align*} where $o_n(1)$ denotes a quantity which converges to zero as $n\to\infty$. We provide the corresponding upper and lower bounds on the smallest singular value of $M_n$ as well.
