Showing papers by "Van Vu published in 2015"

Stochastic Block Model and Community Detection in Sparse Graphs: A spectral algorithm with optimal rate of recovery

Abstract: In this paper, we present and analyze a simple and robust spectral algorithm for the stochastic block model with k blocks, for any k fixed. Our algorithm works with graphs having constant edge density, under an optimal condition on the gap between the density inside a block and the density between the blocks. As a co-product, we settle an open question posed by Abbe et. al. concerning censor block models.

Local Universality of Zeroes of Random Polynomials

Abstract: In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials $f =\sum_{i=1}^n c_i \xi_i z^i$ and $\tilde f =\sum_{i=1}^n c_i \tilde \xi_i z^i$, where the $\xi_i$ and $\tilde \xi_i$ are iid random variables that match moments to second order, the coefficients $c_i$ are deterministic, and the degree parameter $n$ is large. Our results show, under some light conditions on the coefficients $c_i$ and the tails of $\xi_i, \tilde \xi_i$, that the correlation functions of the zeroes of $f$ and $\tilde f$ are approximately the same. As an application, we give some answers to the classical question `"How many zeroes of a random polynomials are real?" for several classes of random polynomial models. Our analysis relies on a general replacement principle, motivated by some recent work in random matrix theory. This principle enables one to compare the correlation functions of two random functions $f$ and $\tilde f$ if their log magnitudes $\log |f|, \log|\tilde f|$ are close in distribution, and if some non-concentration bounds are obeyed.

Random matrices: Universality of local spectral statistics of non-Hermitian matrices

Abstract: It is a classical result of Ginibre that the normalized bulk $k$-point correlation functions of a complex $n\times n$ Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on $\mathbb{C}$ with kernel $K_{\infty}(z,w):=\frac{1}{\pi}e^{-|z|^{2}/2-|w|^{2}/2+z\bar{w}}$ in the limit $n\to\infty$. In this paper, we show that this asymptotic law is universal among all random $n\times n$ matrices $M_{n}$ whose entries are jointly independent, exponentially decaying, have independent real and imaginary parts and whose moments match that of the complex Gaussian ensemble to fourth order. Analogous results at the edge of the spectrum are also obtained. As an application, we extend a central limit theorem for the number of eigenvalues of complex Gaussian matrices in a small disk to these more general ensembles. These results are non-Hermitian analogues of some recent universality results for Hermitian Wigner matrices. However, a key new difficulty arises in the non-Hermitian case, due to the instability of the spectrum for such matrices. To resolve this issue, we the need to work with the log-determinants $\log|\det(M_{n}-z_{0})|$ rather than with the Stieltjes transform $\frac{1}{n}\operatorname{tr}(M_{n}-z_{0})^{-1}$, in order to exploit Girko’s Hermitization method. Our main tools are a four moment theorem for these log-determinants, together with a strong concentration result for the log-determinants in the Gaussian case. The latter is established by studying the solutions of a certain nonlinear stochastic difference equation. With some extra consideration, we can extend our arguments to the real case, proving universality for correlation functions of real matrices which match the real Gaussian ensemble to the fourth order. As an application, we show that a real $n\times n$ matrix whose entries are jointly independent, exponentially decaying and whose moments match the real Gaussian ensemble to fourth order has $\sqrt{\frac{2n}{\pi}}+o(\sqrt{n})$ real eigenvalues asymptotically almost surely.

Random weighted projections, random quadratic forms and random eigenvectors

Abstract: We present a concentration result concerning random weighted projections in high dimensional spaces. As applications, we prove 1 New concentration inequalities for random quadratic forms. 2 The infinity norm of most unit eigenvectors of a random ±1 matrix is of order Ologn/n. 3 An estimate on the threshold for the local semi-circle law which is tight up to a logn factor. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 792-821, 2015

Products of Independent Elliptic Random Matrices

Abstract: For fixed $$m > 1$$ , we study the product of $$m$$ independent $$N \times N$$ elliptic random matrices as $$N$$ tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability $$1$$ , to the $$m$$ -th power of the circular law, regardless of the joint distribution of the mirror entries in each matrix. This leads to a new kind of universality phenomenon: the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves. Our result also generalizes earlier results of Gotze–Tikhomirov (On the asymptotic spectrum of products of independent random matrices, available at http://arxiv.org/abs/1012.2710 ) and O’Rourke–Soshnikov (J Probab 16(81):2219–2245, 2011) concerning the product of independent iid random matrices.

Stochastic Block Model and Community Detection in the Sparse Graphs: A spectral algorithm with optimal rate of recovery

Abstract: In this paper, we present and analyze a simple and robust spectral algorithm for the stochastic block model with $k$ blocks, for any $k$ fixed. Our algorithm works with graphs having constant edge density, under an optimal condition on the gap between the density inside a block and the density between the blocks. As a co-product, we settle an open question posed by Abbe et. al. concerning censor block models.

Real roots of random polynomials: expectation and repulsion

Abstract: Let $P_{n}(x)= \sum_{i=0}^n \xi_i x^i$ be a Kac random polynomial where the coefficients $\xi_i$ are iid copies of a given random variable $\xi$. Our main result is an optimal quantitative bound concerning real roots repulsion. This leads to an optimal bound on the probability that there is a double root. As an application, we consider the problem of estimating the number of real roots of $P_n$, which has a long history and in particular was the main subject of a celebrated series of papers by Littlewood and Offord from the 1940s. We show, for a large and natural family of atom variables $\xi$, that the expected number of real roots of $P_n(x)$ is exactly $\frac{2}{\pi} \log n +C +o(1)$, where $C$ is an absolute constant depending on the atom variable $\xi$. Prior to this paper, such a result was known only for the case when $\xi$ is Gaussian.

Random matrices: tail bounds for gaps between eigenvalues

Abstract: Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for random matrices with discrete entries and the first super-polynomial bound on the probability that a random graph has simple spectrum, along with several applications.

Dictionary Learning with Few Samples and Matrix Concentration

Abstract: Let $A$ be an $n \times n$ matrix, $X$ be an $n \times p$ matrix and $Y = AX$. A challenging and important problem in data analysis, motivated by dictionary learning and other practical problems, is to recover both $A$ and $X$, given $Y$. Under normal circumstances, it is clear that this problem is underdetermined. However, in the case when $X$ is sparse and random, Spielman, Wang and Wright showed that one can recover both $A$ and $X$ efficiently from $Y$ with high probability, given that $p$ (the number of samples) is sufficiently large. Their method works for $p \ge C n^2 \log^ 2 n$ and they conjectured that $p \ge C n \log n$ suffices. The bound $n \log n$ is sharp for an obvious information theoretical reason. In this paper, we show that $p \ge C n \log^4 n$ suffices, matching the conjectural bound up to a polylogarithmic factor. The core of our proof is a theorem concerning $l_1$ concentration of random matrices, which is of independent interest. Our proof of the concentration result is based on two ideas. The first is an economical way to apply the union bound. The second is a refined version of Bernstein's concentration inequality for the sum of independent variables. Both have nothing to do with random matrices and are applicable in general settings.

Roots of random polynomials with arbitrary coefficients

Abstract: In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coefficients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even when the coefficients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.

Roots of random polynomials with coefficients having polynomial growth

Abstract: In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coeffcients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even when the coeffcients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.

Non-abelian Littlewood-Offord inequalities

Abstract: In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalized by generations of researchers, with applications in several areas of mathematics. In this paper, we present the first non-abelian analogue of Littlewood-Offord result, a sharp anti-concentration inequality for products of independent random variables.

Random Matrices: l1 Concentration and Dictionary Learning with Few Samples

Abstract: Let X be a sparse random matrix of size n by p (p >> n). We prove that if p > C n log4 n, then with probability 1-o(1), |XT v|1 is close to its expectation for all vectors v in Rn (simultaneously). The bound on p is sharp up to the polylogarithmic factor. The study of this problem is directly motivated by an application. Let A be an n by n matrix, X be an n by p matrix and Y = AX. A challenging and important problem in data analysis, motivated by dictionary learning and other practical problems, is to recover both A and X, given Y. Under normal circumstances, it is clear that this problem is underdetermined. However, in the case when X is sparse and random, Spiel man, Wang and Wright showed that one can recover both A and X efficiently from Y with high probability, given that p (the number of samples) is sufficiently large. Their method works for p > C n2 log2 n and they conjectured that p > C n log n suffices. The bound n log n is sharp for an obvious information theoretical reason. The matrix concentration result verifies the Spiel man et. Al. Conjecture up to a log3 n factor. Our proof of the concentration result is based on two ideas. The first is an economical way to apply the union bound. The second is a refined version of Bernstein's concentration inequality for a sum of independent variables. Both have nothing to do with random matrices and are applicable in general settings.

Anti-concentration for polynomials of independent random variables

Abstract: We prove anti-concentration results for polynomials of independent random variables with arbitrary degree. Our results extend the classical Littlewood-Offord result for linear polynomials, and improve several earlier estimates. We discuss applications in two different areas. In complexity theory, we prove near optimal lower bounds for computing the Parity, addressing a challenge in complexity theory posed by Razborov and Viola, and also address a problem concerning OR functions. In random graph theory, we derive a general anti-concentration result on the number of copies of a fixed graph in a random graph.

Anti-concentration for polynomials of Rademacher random variables and applications in complexity theory

Abstract: We prove anti-concentration results for polynomials of independent random variables with arbitrary degree. Our results extend the classical Littlewood-Offord result for linear polynomials, and improve several earlier estimates. We discuss applications in two different areas. In complexity theory, we prove near optimal lower bounds for computing the Parity, addressing a challenge in complexity theory posed by Razborov and Viola, and also address a problem concerning OR functions. In random graph theory, we derive a general anti-concentration result on the number of copies of a fixed graph in a random graph.

Anti-concentration for random polynomials.

Abstract: We prove anti-concentration results for polynomials of independent Rademacher random variables, with arbitrary degree. Our results extend the classical Littlewood-Offord result for linear polynomials, and improve several earlier estimates. As an application, we address a challenge in complexity theory posed by Razborov and Viola.