Related papers: Anti-concentration in most directions
Let $X_1,\ldots,X_n$ be independent identically distributed random vectors in $\mathbb{R}^d$. We consider upper bounds on $\max_x \mathbb{P}(a_1X_1+\cdots+a_nX_n=x)$ under various restrictions on $X_i$ and the weights $a_i$. When…
We prove a stability version of Harper's cube vertex isoperimetric inequality, showing that subsets of the cube with vertex boundary close to the minimum possible are close to (generalised) Hamming balls. Furthermore, we obtain a local…
We present some extensions of Bernstein's concentration inequality for random matrices. This inequality has become a useful and powerful tool for many problems in statistics, signal processing and theoretical computer science. The main…
Works of Hooley and Heath-Brown imply a near-optimal bound on the number $N$ of integral solutions to $x_1^3+\dots+x_6^3 = 0$ in expanding regions, conditional on automorphy and GRH for certain Hasse--Weil $L$-functions; for regions of…
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…
Multiple players are each given one independent sample, about which they can only provide limited information to a central referee. Each player is allowed to describe its observed sample to the referee using a channel from a family of…
Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise…
Concentration inequalities, a major tool in probability theory, quantify how much a random variable deviates from a certain quantity. This paper proposes a systematic convex optimization approach to studying and generating concentration…
The paper focuses on general properties of parametric minimum contrast estimators. The quality of estimation is measured in terms of the rate function related to the contrast, thus allowing to derive exponential risk bounds invariant with…
This is an expository survey on recent sum-product results in finite fields. We present a number of sum-product or "expander" results that say that if $|A| > p^{2/3}$ then some set determined by sums and product of elements of $A$ is nearly…
We study the approximability of Max Ones when the number of variable occurrences is bounded by a constant. For conservative constraint languages (i.e., when the unary relations are included) we give a complete classification when the number…
Let $\{X_{i,j}:(i,j)\in\mathbb N^2\}$ be a two-dimensional array of independent copies of a random variable $X$, and let $\{N_n\}_{n\in\mathbb N}$ be a sequence of natural numbers such that $\lim_{n\to\infty}e^{-cn}N_n=1$ for some $c>0$.…
The purpose of this note is to show how simple Optimal Transport arguments, on the real line, can be used in Superconcentration theory. This methodology is efficient to produce sharp non-asymptotic variance bounds for various functionals…
Concentration inequalities for the sample mean, like those due to Bernstein, Hoeffding, and Bentkus, are valid for any sample size but overly conservative, yielding confidence intervals that are unnecessarily wide. The central limit theorem…
High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and…
We prove new strong converse results in a variety of group testing settings, generalizing a result of Baldassini, Johnson and Aldridge. These results are proved by two distinct approaches, corresponding to the non-adaptive and adaptive…
We consider the following data perturbation model, where the covariates incur multiplicative errors. For two $n \times m$ random matrices $U, X$, we denote by $U \circ X$ the Hadamard or Schur product, which is defined as $(U \circ X)_{ij}…
Let $Q_n$ be the $n$-dimensional Hamming cube and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically \[2n2^{N/4},\] as was conjectured by Ilinca and the first author in connection with a question of…
We show tight lower bounds for the entire trade-off between space and query time for the Approximate Near Neighbor search problem. Our lower bounds hold in a restricted model of computation, which captures all hashing-based approaches. In…
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 $\sigma_n(A)$ of $A$ satisfies $$ P\left( \sigma_n(A)\leq…