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Van Vu

Publications and source records attributed to Van Vu.

At least 37 records · Page 2Linked to original sources
arXiv2021

Roots of random functions: A framework for local universality

We investigate the local distribution of roots of random functions of the form $F_n(z)= \sum_{i=1}^n ξ_i ϕ_i(z) $, where $ξ_i$ are independent random variables and $ϕ_i (z) $ are arbitrary analytic functions. Starting with the fundamental works of Kac and Littlewood-Offord in the 1940s, random functions of this type have been studied extensively in many fields of mathematics. We develop a robust framework to solve the problem by reducing, via universality theorems, the calculation of the distribution of the roots and the interaction between them to the case where $ξ_i$ are gaussian. In this special case, one can use the Kac-Rice formula and various other tools to obtain precise answers. Our framework has a wide range of applications, which include the most popular models of random functions, such as random trigonometric polynomials and all basic classes of random algebraic polynomials (Kac, Weyl, and elliptic). Each of these ensembles has been studied heavily by deep and diverse methods. Our method, for the first time, provides a unified treatment of all of them. Among the applications, we derive the first local universality result for random trigonometric polynomials with arbitrary coefficients. When restricted to the study of real roots, this result extends several recent results, proved for less general ensembles. For random algebraic polynomials, we strengthen several recent results of Tao and the second author, with significantly simpler proofs. As a corollary, we sharpen a classical result of Erd{ö}s and Offord on real roots of Kac polynomials, providing an optimal error estimate. Another application is a refinement of a recent result of Flasche and Kabluchko on the roots of random Taylor series.

math.PR↗
arXiv2021

VinDr-SpineXR: A deep learning framework for spinal lesions detection and classification from radiographs

Radiographs are used as the most important imaging tool for identifying spine anomalies in clinical practice. The evaluation of spinal bone lesions, however, is a challenging task for radiologists. This work aims at developing and evaluating a deep learning-based framework, named VinDr-SpineXR, for the classification and localization of abnormalities from spine X-rays. First, we build a large dataset, comprising 10,468 spine X-ray images from 5,000 studies, each of which is manually annotated by an experienced radiologist with bounding boxes around abnormal findings in 13 categories. Using this dataset, we then train a deep learning classifier to determine whether a spine scan is abnormal and a detector to localize 7 crucial findings amongst the total 13. The VinDr-SpineXR is evaluated on a test set of 2,078 images from 1,000 studies, which is kept separate from the training set. It demonstrates an area under the receiver operating characteristic curve (AUROC) of 88.61% (95% CI 87.19%, 90.02%) for the image-level classification task and a mean average precision (mAP@0.5) of 33.56% for the lesion-level localization task. These results serve as a proof of concept and set a baseline for future research in this direction. To encourage advances, the dataset, codes, and trained deep learning models are made publicly available.

eess.IV↗
arXiv2020

Random polynomials: central limit theorems for the real roots

The number of real roots has been a central subject in the theory of random polynomials and random functions since the fundamental papers of Littlewood-Offord and Kac in the 1940s. The main task here is to determine the limiting distribution of this random variable. In 1974, Maslova famously proved a central limit theorem (CLT) for the number of real roots of Kac polynomials. It has remained the only limiting theorem available for the number of real roots for more than four decades. In this paper, using a new approach, we derive a general CLT for the number of real roots of a large class of random polynomials with coefficients growing polynomially. Our result both generalizes and strengthens Maslova's theorem.

math.PR↗
arXiv2020

Reaching a Consensus on Random Networks: The Power of Few

A community of $n$ individuals splits into two camps, Red and Blue. The individuals are connected by a social network, which influences their colors. Everyday, each person changes his/her color according to the majority among his/her neighbors. Red (Blue) wins if everyone in the community becomes Red (Blue) at some point. We study this process when the underlying network is the random Erdos-Renyi graph $G(n, p)$. With a balanced initial state ($n/2$ person in each camp), it is clear that each color wins with the same probability. Our study reveals that for any constants $p$ and $\varepsilon$, there is a constant $C$ such that if one camp has $n/2 +C$ individuals, then it wins with probability at least $1 - \varepsilon$. The surprising key fact here is that $C$ does not depend on $n$, the population of the community. When $p=1/2$ and $\varepsilon =.1$, one can set $C$ as small as 6. If the aim of the process is to choose a candidate, then this means it takes only $6$ "defectors" to win an election unanimously with overwhelming odd.

math.CO↗
arXiv2019

Random matrix products: Universality and least singular values

We establish, under a moment matching hypothesis, the local universality of the correlation functions associated with products of $M$ independent iid random matrices, as $M$ is fixed, and the sizes of the matrices tend to infinity. This generalizes an earlier result of Tao and the third author for the case $M=1$. We also prove Gaussian limits for the centered linear spectral statistics of products of $M$ independent iid random matrices. This is done in two steps. First, we establish the result for product random matrices with Gaussian entries, and then extend to the general case of non-Gaussian entries by another moment matching argument. Prior to our result, Gaussian limits were known only for the case $M=1$. In a similar fashion, we establish Gaussian limits for the centered linear spectral statistics of products of independent truncated random unitary matrices. In both cases, we are able to obtain explicit expressions for the limiting variances. The main difficulty in our study is that the entries of the product matrix are no longer independent. Our key technical lemma is a lower bound on the least singular value of the translated linearization matrix associated with the product of $M$ normalized independent random matrices with independent and identically distributed subgaussian entries. This lemma is of independent interest.

math.PR↗
arXiv2019

Random matrices: Probability of Normality

In this paper, we investigate the following question: How often is a random matrix normal? We consider a random $n\times n$ matrix, $M_n$, whose entries are i.i.d. Rademacher random variables (taking values $\{ \pm1 \}$ with probability $1/2$) and prove $$2^{-\left(0.5+o(1)\right)n^2} \le P\left(M_n \text{ is normal}\right) \le 2^{-(0.302+o(1))n^{2}}. $$ We conjecture that the lower bound is sharp.

math.PR↗
arXiv2018

Spectrum of complex networks

The study of complex networks has been one of the most active fields in science in recent decades. Spectral properties of networks (or graphs that represent them) are of fundamental importance. Researchers have been investigating these properties for many years, and, based on numerical data, have raised a number of questions about the distribution of the eigenvalues and eigenvectors. In this paper, we give the solution to some of these questions. In particular, we determine the limiting distribution of (the bulk of) the spectrum as the size of the network grows to infinity and show that the leading eigenvectors are strongly localized. We focus on the preferential attachment graph, which is the most popular mathematical model for growing complex networks. Our analysis is, on the other hand, general and can be applied to other models.

math.CO↗
arXiv2018

Sparse Random Matrices have Simple Spectrum

Let $M_n$ be a class of symmetric sparse random matrices, with independent entries $M_{ij} = δ_{ij} ξ_{ij}$ for $i \leq j$. $δ_{ij}$ are i.i.d. Bernoulli random variables taking the value $1$ with probability $p \geq n^{-1+δ}$ for any constant $δ> 0$ and $ξ_{ij}$ are i.i.d. centered, subgaussian random variables. We show that with high probability this class of random matrices has simple spectrum (i.e. the eigenvalues appear with multiplicity one). We can slightly modify our proof to show that the adjacency matrix of a sparse Erdős-Rényi graph has simple spectrum for $n^{-1+δ} \leq p \leq 1- n^{-1+δ}$. These results are optimal in the exponent. The result for graphs has connections to the notorious graph isomorphism problem.

math.PR↗
arXiv2017

Roots of random polynomials with coefficients having polynomial growth

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.

math.PR↗
arXiv2017

Law of Iterated Logarithm for random graphs

A milestone in Probability Theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables $\{t_i\}_{i=1}^{\infty}$ with mean $0$ and variance $1$ $$ \Pr \left[ \limsup_{n\rightarrow \infty} \frac{ \sum_{i=1}^n t_i }{σ_n \sqrt {2 \log \log n }} =1 \right] =1 . $$ In this paper we prove that LIL holds for various functionals of random graphs and hypergraphs models. We first prove LIL for the number of copies of a fixed subgraph $H$. Two harder results concern the number of global objects: perfect matchings and Hamiltonian cycles. The main new ingredient in these results is a large deviation bound, which may be of independent interest. For random $k$-uniform hypergraphs, we obtain the Central Limit Theorem (CLT) and LIL for the number of Hamilton cycles.

math.CO↗
arXiv2017

Central limit theorems for the real zeros of Weyl polynomials

We establish the central limit theorem for the number of real roots of the Weyl polynomial $P_n(x)=xi_0 + xi_1 x+ ... + xi_n (n!)^{(-1/2)} x^n$, where $xi_i$ are iid Gaussian random variables. The main ingredients in the proof are new estimates for the correlation functions of the real roots of $P_n$ and a comparison argument exploiting local laws and repulsion properties of these real roots.

math.PR↗
arXiv2017

Smooth analysis of the condition number and the least singular value

Let $\a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $\a$ and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M + N_{n}$, generalizing an earlier result of Spielman and Teng for the case when $\a$ is gaussian. Our investigation reveals an interesting fact that the "core" matrix $M$ does play a role on tail bounds for the least singular value of $M+N_{n} $. This does not occur in Spielman-Teng studies when $\a$ is gaussian. Consequently, our general estimate involves the norm $\|M\|$. In the special case when $\|M\|$ is relatively small, this estimate is nearly optimal and extends or refines existing results.

math.PR↗
arXiv2017

Sum-avoiding sets in groups

Let $A$ be a finite subset of an arbitrary additive group $G$, and let $ϕ(A)$ denote the cardinality of the largest subset $B$ in $A$ that is sum-avoiding in $A$ (that is to say, $b_1+b_2 \not \in A$ for all distinct $b_1,b_2 \in B$). The question of controlling the size of $A$ in terms of $ϕ(A)$ 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 $ϕ(A)$ 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 $ϕ(A)$ finite subgroups of $G$, or by fewer than $ϕ(A)$ 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 $ϕ(A)$ bounded, but give a positive result when $|G|$ is not divisible by small primes.

math.CO↗
arXiv2016

Packing perfect matchings in random hypergraphs

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.

math.CO↗
arXiv2016

Eigenvectors of random matrices: A survey

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.

math.PR↗
arXiv2016

Random matrices: Law of the iterated logarithm

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.

math.PR↗