arXiv subjects

Van Vu

Publications and source records attributed to Van Vu.

At least 55 records · Page 3Linked to original sources

arXiv2016

Sumfree sets in groups: a survey

We discuss several questions concerning sum-free sets in groups, raised by Erdős in his survey "Extremal problems in number theory" (Proceedings of the Symp. Pure Math. VIII AMS) published in 1965. Among other things, we give a characterization for large sets $A$ in an abelian group $G$ which do not contain a subset $B$ of fixed size $k$ such that the sum of any two different elements of $B$ do not belong to $A$ (in other words, $B$ is sum-free with respect to $A$). Erdős, in the above mentioned survey, conjectured that if $|A|$ is sufficiently large compared to $k$, then $A$ contains two elements that add up to zero. This is known to be true for $k \leq 3$. We give counterexamples for all $k \ge 4$. On the other hand, using the new characterization result, we are able to prove a positive result in the case when $|G|$ is not divisible by small primes.

math.CO↗

arXiv2015

Anti-concentration for polynomials of independent random variables

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.

math.PR↗

arXiv2015

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

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.

cs.DS↗

arXiv2015

Random matrices: tail bounds for gaps between eigenvalues

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.

math.PR↗

arXiv2015

Dictionary Learning with Few Samples and Matrix Concentration

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.

math.PR↗

arXiv2015

Products of independent elliptic random matrices

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 Götze-Tikhomirov and O'Rourke-Soshnikov concerning the product of independent iid random matrices.

math.PR↗

arXiv2014

Random matrices have simple spectrum

Let $M_n = (ξ_{ij})_{1 \leq i,j \leq n}$ be a real symmetric random matrix in which the upper-triangular entries $ξ_{ij}, i<j$ and diagonal entries $ξ_{ii}$ are independent. We show that with probability tending to 1, $M_n$ has no repeated eigenvalues. As a corollary, we deduce that the Erd{\H o}s-Renyi random graph has simple spectrum asymptotically almost surely, answering a question of Babai.

math.PR↗

arXiv2014

Random walks with different directions: Drunkards beware !

As an extension of Polya's classical result on random walks on the square grids ($\Z^d$), we consider a random walk where the steps, while still have unit length, point to different directions. We show that in dimensions at least 4, the returning probability after $n$ steps is at most $n^{-d/2 - d/(d-2) +o(1)}$, which is sharp. The real surprise is in dimensions 2 and 3. In dimension 2, where the traditional grid walk is recurrent, our upper bound is $n^{-ω(1)}$, which is much worse than higher dimensions. In dimension 3, we prove an upper bound of order $n^{-4 +o(1)}$. We discover a new conjecture concerning incidences between spheres and points in $\R^3$, which, if holds, would improve the bound to $n^{-9/2 +o(1)}$, which is consistent % with the $d \ge 4$ case. to the $d \ge 4$ case. This conjecture resembles Szemerédi-Trotter type results and is of independent interest.

math.CO↗

arXiv2014

Real roots of random polynomials: expectation and repulsion

Let $P_{n}(x)= \sum_{i=0}^n ξ_i x^i$ be a Kac random polynomial where the coefficients $ξ_i$ are iid copies of a given random variable $ξ$. 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 $ξ$, that the expected number of real roots of $P_n(x)$ is exactly $\frac{2}π \log n +C +o(1)$, where $C$ is an absolute constant depending on the atom variable $ξ$. Prior to this paper, such a result was known only for the case when $ξ$ is Gaussian.

math.PR↗

arXiv2014

Random weighted projections, random quadratic forms and random eigenvectors

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 $\pm 1$ matrix is of order $O( \sqrt { \log n/n})$; (3) An estimate on the threshold for the local semi-circle law which is tight up to a $\sqrt {\log n}$ factor.

math.PR↗

arXiv2014

Local universality of zeroes of random polynomials

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 $f =\sum_{i=1}^n c_i ξ_i z^i$ and $\tilde f =\sum_{i=1}^n c_i \tilde ξ_i z^i$, where the $ξ_i$ and $\tilde ξ_i$ 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 $ξ_i, \tilde ξ_i$, 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.

math.PR↗

arXiv2014

A simple SVD algorithm for finding hidden partitions

Finding a hidden partition in a random environment is a general and important problem, which contains as subproblems many famous questions, such as finding a hidden clique, finding a hidden coloring, finding a hidden bipartition etc. In this paper, we provide a simple SVD algorithm for this purpose, answering a question of McSherry. This algorithm is very easy to implement and works for sparse graphs with optimal density.

math.CO↗

arXiv2014

Universality of local eigenvalue statistics in random matrices with external source

Consider a random matrix of the form $W_n = M_n + D_n$, where $M_n$ is a Wigner matrix and $D_n$ is a real deterministic diagonal matrix ($D_n$ is commonly referred to as an external source in the mathematical physics literature). We study the universality of the local eigenvalue statistics of $W_n$ for a general class of Wigner matrices $M_n$ and diagonal matrices $D_n$. Unlike the setting of many recent results concerning universality, the global semicircle law fails for this model. However, we can still obtain the universal sine kernel formula for the correlation functions. This demonstrates the remarkable phenomenon that local laws are more resilient than global ones. The universality of the correlation functions follows from a four moment theorem, which we prove using a variant of the approach used earlier by Tao and Vu.

math.PR↗

arXiv2014

On the number of real roots of random polynomials

Roots of random polynomials have been studied exclusively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdos-Offord, showed that the expectation of the number of real roots is $\frac{2}π \log n + o(\log n)$. In this paper, we determine the true nature of the error term by showing that the expectation equals $\frac{2}π\log n + O(1)$. Prior to this paper, such estimate has been known only in the gaussian case, thanks to works of Edelman and Kostlan.

math.PR↗

arXiv2014

Random matrices: Law of the determinant

Let $A_n$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\det A_n|$ satisfies a central limit theorem. More precisely, \begin{eqnarray*}\sup_{x\in {\mathbf {R}}}\biggl|{\mathbf {P}}\biggl(\frac{\log(|\det A_n|)-({1}/{2})\log (n-1)!}{\sqrt{({1}/{2})\log n}}\le x\biggr)-{\mathbf {P}}\bigl(\mathbf {N}(0,1)\le x\bigr)\biggr|\\\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}

math.PR↗

arXiv2013

Random matrices: Sharp concentration of eigenvalues

Let $W_n= \frac{1}{\sqrt n} M_n$ be a Wigner matrix whose entries have vanishing third moment, normalized so that the spectrum is concentrated in the interval $[-2,2]$. We prove a concentration bound for $N_I = N_I(W_n)$, the number of eigenvalues of $W_n$ in an interval $I$. Our result shows that $N_I$ decays exponentially with standard deviation at most $O(\log^{O(1)} n)$. This is best possible up to the constant exponent in the logarithmic term. As a corollary, the bulk eigenvalues are localized to an interval of width $O(\log^{O(1)} n/n)$; again, this is optimal up to the exponent. These results strengthen recent results of Erdos, Yau and Yin (under the extra assumption of vanishing third

math.PR↗

arXiv2013

The spectrum of random kernel matrices: universality results for rough and varying kernels

We consider random matrices whose entries are f( ) or f(||Xi-Xj||^2) for iid vectors Xi in R^p with normalized distribution. Assuming that f is sufficiently smooth and the distribution of Xi's is sufficiently nice, El Karoui [17] showed that the spectral distributions of these matrices behave as if f is linear in the Marchenko--Pastur limit. When Xi's are Gaussian vectors, variants of this phenomenon were recently proved for varying kernels, i.e. when f may depend on p, by Cheng and Singer [13]. Two results are shown in this paper: first it is shown that for a large class of distributions the regularity assumptions on f in El Karoui's results can be reduced to minimal; and secondly it is shown that the Gaussian assumptions in Cheng--Singer's result can be removed, answering a question posed in [13] about the universality of the limiting spectral distribution.

math.PR↗

arXiv2012

A central limit theorem for the determinant of a Wigner matrix

We establish a central limit theorem for the log-determinant $\log|\det(M_n)|$ of a Wigner matrix $M_n$, under the assumption of four matching moments with either the GUE or GOE ensemble. More specifically, we show that this log-determinant is asymptotically distributed like $N(\log \sqrt{n!} - 1/2 \log n, 1/2 \log n)_\R$ when one matches moments with GUE, and $N(\log \sqrt{n!} - 1/4 \log n, 1/4 \log n)_\R$ when one matches moments with GOE.

math.PR↗