Showing papers by "Van Vu published in 2016"

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.

On the number of real roots of random polynomials

Abstract: Roots of random polynomials have been studied intensively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdős–Offord, showed that the expectation of the number of real roots is 2 πlog n + o(log n). In this paper, we determine the true nature of the error term by showing that the expectation equals 2 πlog n + O(1). Prior to this paper, the error term O(1) has been known only for polynomials with Gaussian coefficients.

Eigenvectors of random matrices: A survey

Abstract: Eigenvectors of large matrices (and graphs) play an essential role in combinatorics and theoretical computer science. The goal of this survey is to provide an up-to-date account on properties of eigenvectors when the matrix (or graph) is random.

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 et al. 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.

Sum-avoiding sets in groups

Abstract: Sum-avoiding sets in groups, Discrete Analysis 2016:15, 27 pp. Let $A$ be a subset of an Abelian group $G$. A subset $B\subset A$ is called _sum-avoiding in $A$_ if no two elements of $B$ add up to an element of $A$. Write $\phi(A)$ for the size of the largest sum-avoiding subset of $A$. If $G=\mathbb Z$ and $|A|=n$, then it is known that $\phi(A)$ must be at least $\log n(\log\log n)^{1/2-o(1)}$, and examples are known of sets for which $\phi(A)$ is at most $\exp(O(\sqrt{\log n}))$. These results are due to Xuancheng Shao and Imre Ruzsa, respectively. Reducing this gap to a reasonable size appears to be a very hard problem. If on the other hand, $G$ has torsion, then it is possible for $A$ to be large and finite while $\phi(A)$ is bounded. Indeed, if $A$ is any subgroup, then $\phi(A)=1$. However, this is not the end of the story, as the authors show. Suppose we know that $A$ is a subset of an Abelian group and that $\phi(A)\leq k$ for some fixed $k$. What can we say about $A$? Note that this condition expresses a kind of weak closure property: instead of saying that any two elements of $A$ add up to an element of $A$, it says that from any $k+1$ elements of $A$, two must add up to an element of $A$. A simple example of such a set that isn't itself a subgroup is a union of at most $k$ subgroups. Then given more than $k$ elements of $A$, two must belong to the same subgroup and hence add up to an element of $A$. In this paper, the authors show a converse to this easy observation: if $\phi(A)\leq k$, then there exist subgroups $H_1,\dots,H_m$ with $m\leq k$ such that $|A\cap H_i|\geq c|H_i|$ for each $i$, and all but at most $C$ elements of $A$ belong to $H_1\cup\dots\cup H_m$. Here, $c>0$ and $C$ are constants that depend on $k$ only. If you are familiar with the Balog-Szemeredi theorem and Freiman's theorem, then you might expect the proof of this result to be a fairly straightforward use of those tools. However, when one attempts to turn this thought into a proof, a significant difficulty arises, which the authors explain in their introduction. They resolve this difficulty by means of a complicated iterative argument -- in fact, it is sufficiently complicated that instead of desperately trying to keep control of all the parameters that would arise, they resort to the language of non-standard analysis. This tidies up the argument considerably, but at the price of yielding no bound at all for how the constants $c$ and $C$ depend on $k$. However, this is not a huge price, as they also say that if they had avoided non-standard analysis, then the bounds they would have obtained would have been extremely weak. The paper also contains a construction of arbitrarily large sets $A$ with $\phi(A)\leq k$ that contain no inverse pairs. This gives a negative answer to a question of Erdős. The construction, which is surprisingly simple, makes heavy use of the fact that their set lives in a group with order divisible by a small prime. They go on to show that this is necessary: if the order of $G$ has no small prime divisors, then Erdős's question has a positive answer.

Sum-avoiding sets in groups

Abstract: Let $A$ be a finite subset of an arbitrary additive group $G$, and let $\phi(A)$ denote the cardinality of the largest subset $B$ in $A$ that is sum-avoiding in $A$ (that is to say, $b_1+b_2 ot \in A$ for all distinct $b_1,b_2 \in B$). The question of controlling the size of $A$ in terms of $\phi(A)$ in the case when $G$ was torsion-free was posed by Erdős and Moser. When $G$ has torsion, $A$ can be arbitrarily large for fixed $\phi(A)$ due to the presence of subgroups. Nevertheless, we provide a qualitative answer to an analogue of the Erdős-Moser problem in this setting, by establishing a structure theorem, which roughly speaking asserts that $A$ is either efficiently covered by $\phi(A)$ finite subgroups of $G$, or by fewer than $\phi(A)$ finite subgroups of $G$ together with a residual set of bounded cardinality. In order to avoid a large number of nested inductive arguments, our proof uses the language of nonstandard analysis. We also answer negatively a question of Erdős regarding large subsets $A$ of finite additive groups $G$ with $\phi(A)$ bounded, but give a positive result when $|G|$ is not divisible by small primes.

Normal vector of a random hyperplane

Abstract: Let v_1,...,v_{n-1} be n-1 independent vectors in R^n (or C^n). We study x, the unit normal vector of the hyperplane spanned by the v_i. Our main finding is that x resembles a random vector chosen uniformly from the unit sphere, under some randomness assumption on the v_i. Our result has applications in random matrix theory. Consider an n by n random matrix with iid entries. We first prove an exponential bound on the upper tail for the least singular value, improving the earlier linear bound by Rudelson and Vershynin. Next, we derive optimal delocalization for the eigenvectors corresponding to eigenvalues of small modulus.

Packing perfect matchings in random hypergraphs

Abstract: We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as \emph{online sprinkling}. As an illustrative application of this method, we show that for any fixed integer $k\geq 3$, the binomial $k$-uniform random hypergraph $H^{k}_{n,p}$ contains $N:=(1-o(1))\binom{n-1}{k-1}p$ edge-disjoint perfect matchings, provided $p\geq \frac{\log^{C}n}{n^{k-1}}$, where $C:=C(k)$ is an integer depending only on $k$. Our result for $N$ is asymptotically best optimal and for $p$ is optimal up to the $polylog(n)$ factor.

Sumfree sets in groups

Abstract: Let $A$ be a finite subset of an arbitrary additive group $G$, and let $\phi(A)$ denote the cardinality of the largest subset $B$ in $A$ that is sum-avoiding in $A$ (that is to say, $b_1+b_2 ot \in A$ for all distinct $b_1,b_2 \in B$). The question of controlling the size of $A$ in terms of $\phi(A)$ in the case when $G$ was torsion-free was posed by Erdős and Moser. When $G$ has torsion, $A$ can be arbitrarily large for fixed $\phi(A)$ due to the presence of subgroups. Nevertheless, we provide a qualitative answer to an analogue of the Erdős-Moser problem in this setting, by establishing a structure theorem, which roughly speaking asserts that $A$ is either efficiently covered by $\phi(A)$ finite subgroups of $G$, or by fewer than $\phi(A)$ finite subgroups of $G$ together with a residual set of bounded cardinality. In order to avoid a large number of nested inductive arguments, our proof uses the language of nonstandard analysis. We also answer negatively a question of Erdős regarding large subsets $A$ of finite additive groups $G$ with $\phi(A)$ bounded, but give a positive result when $|G|$ is not divisible by small primes.

Sumfree sets in groups: a survey

Abstract: 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.

Random matrices: Law of the iterated logarithm

Abstract: The theory of random matrices contains many central limit theorems. We have central limit theorems for eigenvalues statistics, for the log-determinant and log-permanent, for limiting distribution of individual eigenvalues in the bulk, and many others. In this notes, we discuss the following problem: Is it possible to prove the law of the iterated logarithm? We illustrate this possibility by showing that this is indeed the case for the log of the permanent of random Bernoulli matrices and pose open questions concerning several other matrix parameters.

Random walks with different directions

Abstract: As an extension of Polya’s classical result on random walks on the square grids (\({\mathbf {Z}}^d\)), 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 \(n^{-\omega (1) }\), which is much worse than in higher dimensions. In dimension 3, we prove an upper bound of order \(n^{-4 +o(1) }\). We find a new conjecture concerning incidences between spheres and points in \({\mathbf {R}}^3\), which, if holds, would improve the bound to \(n^{-9/2 +o(1) }\), which is consistent to the \(d \ge 4\) case. This conjecture resembles Szemeredi-Trotter type results and is of independent interest.