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Bo'az Klartag

Publications and source records attributed to Bo'az Klartag.

At least 19 recordsLinked to original sources
arXiv2024

The strong data processing inequality under the heat flow

Let $ν$ and $μ$ be probability distributions on $\mathbb{R}^n$, and $ν_s,μ_s$ be their evolution under the heat flow, that is, the probability distributions resulting from convolving their density with the density of an isotropic Gaussian random vector with variance $s$ in each entry. This paper studies the rate of decay of $s\mapsto D(ν_s\|μ_s)$ for various divergences, including the $χ^2$ and Kullback-Leibler (KL) divergences. We prove upper and lower bounds on the strong data-processing inequality (SDPI) coefficients corresponding to the source $μ$ and the Gaussian channel. We also prove generalizations of de Brujin's identity, and Costa's result on the concavity in $s$ of the differential entropy of $ν_s$. As a byproduct of our analysis, we obtain new lower bounds on the mutual information between $X$ and $Y=X+\sqrt{s} Z$, where $Z$ is a standard Gaussian vector in $\mathbb{R}^n$, independent of $X$, and on the minimum mean-square error (MMSE) in estimating $X$ from $Y$, in terms of the Poincaré constant of $X$.

cs.IT↗
arXiv2023

Horocyclic Brunn-Minkowski inequality

Given two non-empty subsets $A$ and $B$ of the hyperbolic plane $\mathbb{H}^2$, we define their horocyclic Minkowski sum with parameter $λ=1/2$ as the set $[A:B]_{1/2} \subseteq \mathbb{H}^2$ of all midpoints of horocycle curves connecting a point in $A$ with a point in $B$. These horocycle curves are parameterized by hyperbolic arclength, and the horocyclic Minkowski sum with parameter $0 < λ<1$ is defined analogously. We prove that when $A$ and $B$ are Borel-measurable, $$ \sqrt{ Area( [A:B]_λ )} \geq (1-λ) \cdot \sqrt{ Area(A) } + λ\cdot \sqrt{ Area(B) }, $$ where $Area$ stands for hyperbolic area, with equality when $A$ and $B$ are concentric discs in the hyperbolic plane. We also prove horocyclic versions of the Prékopa-Leindler and Borell-Brascamp-Lieb inequalities. These inequalities slightly deviate from the metric measure space paradigm on curvature and Brunn-Minkowski type inequalities, where the structure of a metric space is imposed on the manifold, and the relevant curves are necessarily geodesics parameterized by arclength.

math.MG↗
arXiv2023

Long lines in subsets of large measure in high dimension

We show that for any set $A\subseteq [0,1]^n$ with $\text{Vol}(A)\ge 1/2$ there exists a line $\ell $ such that the one-dimensional Lebesgue measure of $\ell \cap A$ is at least $Ω( n^{1/4} )$. The exponent $1/4$ is tight. More generally, for a probability measure $μ$ on $\mathbb R ^n$ and $0<a<1$ define \begin{equation*} L(μ,a):= \inf_{A ; μ(A) = a} \sup _{\ell \text{ line}} \big| \ell \cap A\big| \end{equation*} where $|\cdot | $ stands for the one-dimensional Lebesgue measure. We study the asymptotic behavior of $L(μ,a)$ when $μ$ is a product measure and when $μ$ is the uniform measure on the $\ell _p$ ball. We observe a rather unified behavior in a large class of product measures. On the other hand, for $\ell_p$ balls with $1 \leq p \leq \infty$ we find that there are phase transitions of different types.

math.PR↗
arXiv2023

Linear extension operators for Sobolev spaces on radially-symmetric binary trees

Let $1 < p < \infty$ and suppose that we are given a function $f$ defined on the leaves of a weighted tree. We would like to extend $f$ to a function $F$ defined on the entire tree, so as to minimize the weighted $W^{1,p}$-Sobolev norm of the extension. An easy situation is when $p = 2$, where the harmonic extension operator provides such a function $F$. In this note we record our analysis of the particular case of a radially-symmetric binary tree, which is a complete, finite, binary tree with weights that depend only on the distance from the root. Neither the averaging operator nor the harmonic extension operator work here in general. Nevertheless, we prove the existence of a linear extension operator whose norm is bounded by a constant depending solely on $p$. This operator is a variant of the standard harmonic extension operator, and in fact it is harmonic extension with respect to a certain Markov kernel determined by $p$ and by the weights.

math.FA↗
arXiv2023

A weak version of the $\varepsilon$-Dvoretzky conjecture for normed spaces

We prove a weak version of the $\varepsilon$-Dvoretzky conjecture for normed spaces, showing the existence of a subspace of $\mathbb{R}^n$ of dimension at least $c \log n / |\log \varepsilon|$ in which the given norm is $\varepsilon$-close to a norm obeying a large discrete group of symmetries ("$1$-unconditional norm").

math.FA↗
arXiv2023

The lower bound for Koldobsky's slicing inequality via random rounding

We study the lower bound for Koldobsky's slicing inequality. We show that there exists a measure $μ$ and a symmetric convex body $K \subseteq \mathbb{R}^n$, such that for all $ξ\in S^{n-1}$ and all $t\in \mathbb{R},$ $$μ^+(K\cap(ξ^{\perp}+tξ))\leq \frac{c}{\sqrt{n}}μ(K)|K|^{-\frac{1}{n}}.$$ Our bound is optimal, up to the value of the universal constant. It improves slightly upon the results of the first named author and Koldobsky which included a doubly-logarithmic error. The proof is based on an efficient way of discretizing the unit sphere.

math.MG↗
arXiv2023

Logarithmic bounds for isoperimetry and slices of convex sets

We prove that the Bourgain slicing conjecture and the Kannan-Lovász-Simonovits (KLS) isoperimetric conjecture in $\mathbb{R}^n$ hold true up to a factor of $\sqrt{\log n}$. A new ingredient used in the proof is an improved log-concave Lichnerowicz inequality.

math.FA↗
arXiv2022

Local tail bounds for polynomials on the discrete cube

Let $P$ be a polynomial of degree $d$ in independent Bernoulli random variables which has zero mean and unit variance. The Bonami hypercontractivity bound implies that the probability that $|P| > t$ decays exponentially in $t^{2/d}$. Confirming a conjecture of Keller and Klein, we prove a local version of this bound, providing an upper bound on the difference between the $e^{-r}$ and the $e^{-r-1}$ quantiles of $P$.

math.PR↗
arXiv2021

Spectral monotonicity under Gaussian convolution

We show that the Poincaré constant of a log-concave measure in Euclidean space is monotone increasing along the heat flow. In fact, the entire spectrum of the associated Laplace operator is monotone decreasing. Two proofs of these results are given. The first proof analyzes a curvature term of a certain time-dependent diffusion, and the second proof constructs a contracting transport map following the approach of Kim and Milman.

math.MG↗
arXiv2021

Chasing Nested Convex Bodies Nearly Optimally

The convex body chasing problem, introduced by Friedman and Linial, is a competitive analysis problem on any normed vector space. In convex body chasing, for each timestep $t\in\mathbb N$, a convex body $K_t\subseteq \mathbb R^d$ is given as a request, and the player picks a point $x_t\in K_t$. The player aims to ensure that the total distance $\sum_{t=0}^{T-1}||x_t-x_{t+1}||$ is within a bounded ratio of the smallest possible offline solution. In this work, we consider the nested version of the problem, in which the sequence $(K_t)$ must be decreasing. For Euclidean spaces, we consider a memoryless algorithm which moves to the so-called Steiner point, and show that in a certain sense it is exactly optimal among memoryless algorithms. For general finite dimensional normed spaces, we combine the Steiner point and our recent previous algorithm to obtain a new algorithm which is nearly optimal for all $\ell^p_d$ spaces with $p\geq 1$, closing a polynomial gap.

cs.DS↗
arXiv2021

A coordinate-free proof of the finiteness principle for the Whitney extension problem

We present a coordinate-free version of Fefferman's solution of Whitney's extension problem in the space $C^{m-1,1}(\mathbb{R}^n)$. While the original argument relies on an elaborate induction on collections of partial derivatives, our proof uses the language of ideals and translation-invariant subspaces in the ring of polynomials. We emphasize the role of compactness in the proof, first in the familiar sense of topological compactness, but also in the sense of finiteness theorems arising in logic and semialgebraic geometry.

math.CA↗
arXiv2021

Rigidity of Riemannian embeddings of discrete metric spaces

Let $M$ be a complete, connected Riemannian surface and suppose that $\mathcal{S} \subset M$ is a discrete subset. What can we learn about $M$ from the knowledge of all distances in the surface between pairs of points of $\mathcal{S}$? We prove that if the distances in $\mathcal{S}$ correspond to the distances in a $2$-dimensional lattice, or more generally in an arbitrary net in $\mathbb{R}^2$, then $M$ is isometric to the Euclidean plane. We thus find that Riemannian embeddings of certain discrete metric spaces are rather rigid. A corollary is that a subset of $\mathbb{Z}^3$ that strictly contains $\mathbb{Z}^2 \times \{ 0 \}$ cannot be isometrically embedded in any complete Riemannian surface.

math.DG↗
arXiv2020

Discrete variants of Brunn-Minkowski type inequalities

We present an alternative, short proof of a recent discrete version of the Brunn-Minkowski inequality due to Lehec and the second named author. Our proof also yields the four functions theorem of Ahlswede and Daykin as well as some new variants.

math.MG↗
arXiv2019

Spectral gaps, symmetries and log-concave perturbations

We discuss situations where perturbing a probability measure on $\mathbb{R}^n$ does not deteriorate its Poincaré constant by much. A particular example is the symmetric exponential measure in $\mathbb{R}^n$, even log-concave perturbations of which have Poincaré constants that grow at most logarithmically with the dimension. This leads to estimates for the Poincaré constants of $(n/2)$-dimensional sections of the unit ball of $\ell_p^n$ for $1 \leq p \leq 2$, which are optimal up to logarithmic factors. We also consider symmetry properties of the eigenspace of the Laplace-type operator associated with a log-concave measure. Under symmetry assumptions we show that the dimension of this space is exactly $n$, and we exhibit a certain interlacing between the "odd" and "even" parts of the spectrum.

math.FA↗
arXiv2019

Unimodal value distribution of Laplace eigenfunctions and a monotonicity formula

Let $M$ be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by $σ$. Let $f: M \rightarrow \mathbb{R}$ be a non-constant eigenfunction of the Laplacian. The random wave conjecture suggests that in certain situations, the value distribution of $f$ under $σ$ is approximately Gaussian. Write $μ$ for the measure whose density with respect to $σ$ is $|\nabla f|^2$. We observe that the value distribution of $f$ under $μ$ admits a unimodal density attaining its maximum at the origin. Thus, in a sense, the zero set of an eigenfunction is the largest of all level sets. When $M$ is a manifold with boundary, the same holds for Laplace eigenfunctions satisfying either the Dirichlet or the Neumann boundary conditions. Additionally, we prove a monotonicity formula for level sets of solid spherical harmonics, essentially by viewing nodal sets of harmonic functions as weighted minimal hypersurfaces.

math.SP↗
arXiv2019

Complex interpolation of $\mathbb{R}$-norms, duality and foliations

The complex method of interpolation, going back to Calderón and Coifman et al., on the one hand, and the Alexander-Wermer-Slodkowski theorem on polynomial hulls with convex fibers, on the other hand, are generalized to a method of interpolation of real (finite-dimensional) Banach spaces and of convex functions. The underlying duality in this method is given by the Legendre transform. Our results can also be interpreted as new properties of solutions of the homogeneous complex Monge-Ampère equation.

math.CV↗