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Assaf Naor

Publications and source records attributed to Assaf Naor.

At least 19 recordsLinked to original sources
arXiv2026

De-H\"oldering factorization

We study a factorization notion for Lipschitz functions between metric spaces in which such a function is written as a composition of a H\"older function and a function that suitably ``undoes'' the H\"older regularity. We show simple ways to construct such ``de-H\"oldering'' factorizations. If the identity mapping on a metric space $\mathcal{M}$ admits a de-H\"oldering factorization through a metric space $\mathcal{Z}$ that has a conical geodesic bicombing, then the class of metric spaces from which one can extend $\mathcal{Z}$-valued Lipschitz functions is shown to be contained in the corresponding class of metric spaces for $\mathcal{M}$-valued Lipschitz functions. As a quick consequence of these abstract permanence properties, we deduce that every $L_1$-valued Lipschitz function from a subset of $\ell_2$ can be extended to a Lipschitz function that takes values in $L_1$ and is defined on all of $\ell_2$, answering a 1992 question of Ball. By work of Makarychev and Makarychev, this implies that every weighted graph has a vertex cut sparsifier of size $n$ and quality $O(\sqrt{\log n})$, improving Moitra's 2009 bound. We also show that for every metric space $\mathcal{Z}$ that has a conical geodesic bicombing, any metric transform of a metric space $\mathcal{M}$ has $\mathcal{Z}$-valued Lipschitz extension modulus at most a universal constant multiple of the $\mathcal{Z}$-valued Lipschitz extension modulus of $\mathcal{M}$ itself, improving the 2002 bound of Brudnyi and Shvartsman.

math.MG↗
arXiv2026

A threshold phenomenon for embeddings of Euclidean snowflakes and impossibility of dimension reduction

Fix $0<\theta\leqslant 1$. We prove that if $1\leqslant p \leqslant 2/\theta$, then the $\theta$-snowflake of $\ell_2^k$, namely, $\mathbb{R}^k$ equipped with the metric $((x,y)\in \mathbb{R}^k\times \mathbb{R}^k)\mapsto \|x-y\|_2^\theta$, embeds with distortion $O(1)$ into $\ell_p^m$ for some integer $m\lesssim_{p,\theta}k$, which is optimal as $k\to \infty$, as seen by comparing dimensions. However, for $p$ larger than the sharp threshold $2/\theta$ the following change in behavior occurs: If a $(1/\sqrt{k})$-dense subset of the Euclidean sphere $S^{k-1}$ embeds into $\ell_p^m$ with distortion $O(1)$, then necessarily $m\gtrsim_{p,\theta}( k/\log k)^{p\theta/2}$, which grows super-linearly in $k$ as $p\theta/2>1$, and this dimension bound is optimal as $k\to \infty$ up to lower order factors. We deduce from this statement that if $2<p<\infty$, then there exist arbitrarily large $n$-point subsets of $\ell_p$ with the property that if they embed with distortion $O(1)$ into $\ell_p^m$, then necessarily $m\gtrsim_p ((\log n)/(\log\log n)^2)^{p/2}$, thus demonstrating that the statement of the Johnson--Lindenstrauss dimension reduction lemma fails to hold for $\ell_p$

math.MG↗
arXiv2026

Planar lamplighter is not of negative type

The lamplighter group over the planar integer grid is proved to not be bi-Lipschitz equivalent to any metric space of negative type, so in particular it does not admit a bi-Lipschitz embedding into $L_1$. This shows the existence of finitely generated metabelian groups on which word metrics are never comparable up to constant factors to conditionally negative definite (CND) kernels, and that the property of admitting a word metric-comparable CND kernel is not preserved by wreath products.

math.MG↗
arXiv2025

An optimal algorithm for average distance in typical regular graphs

We design a deterministic algorithm that, given $n$ points in a \emph{typical} constant degree regular~graph, queries $O(n)$ distances to output a constant factor approximation to the average distance among those points, thus answering a question posed in~\cite{MN14}. Our algorithm uses the method of~\cite{MN14} to construct a sequence of constant degree graphs that are expanders with respect to certain nonpositively curved metric spaces, together with a new rigidity theorem for metric transforms of nonpositively curved metric spaces. The fact that our algorithm works for typical (uniformly random) constant degree regular graphs rather than for all constant degree graphs is unavoidable, thanks to the following impossibility result that we obtain: For every fixed $k\in \N$, the approximation factor of any algorithm for average distance that works for all constant degree graphs and queries $o(n^{1+1/k})$ distances must necessarily be at least $2(k+1)$. This matches the upper bound attained by the algorithm that was designed for general finite metric spaces in~\cite{BGS}. Thus, any algorithm for average distance in constant degree graphs whose approximation guarantee is less than $4$ must query $\Omega(n^2)$ distances, any such algorithm whose approximation guarantee is less than $6$ must query $\Omega(n^{3/2})$ distances, any such algorithm whose approximation guarantee less than $8$ must query $\Omega(n^{4/3})$ distances, and so forth, and furthermore there exist algorithms achieving those parameters.

cs.DS↗
arXiv2025

Approximate isoperimetry for convex polytopes

For all $n,\phi\in \mathbb{N}$ with $\phi\geqslant n+1$, the smallest possible isoperimetric quotient of an $n$-dimensional convex polytope that has $\phi$ facets is shown to be bounded from above and from below by positive universal constant multiples of $\max\big\{n/\sqrt{1+\log (\phi/n)},\sqrt{n}\big\}$. For all $n\in \mathbb{N}$ and $2n\leqslant \beta\in 2\mathbb{N}$, it is shown that every $n$-dimensional origin-symmetric convex polytope that has $\beta$ vertices admits an affine image whose isoperimetric quotient is at most a universal constant multiple of $\min\big\{\sqrt{\log(\beta/n)},n\big\}$, which is sharp. The weak isomorphic reverse isoperimetry conjecture is proved for $n$-dimensional convex polytopes that have $O(n)$ facets by demonstrating that any such polytope $K$ has an image $K'$ under a volume preserving matrix and a convex body $L\subseteq K'$ such that the isoperimetric quotient of $L$ is at most a universal constant multiple of $\sqrt{n}$, and also $\sqrt[n]{\mathrm{vol}_n(L)/\mathrm{vol}_n(K)}$ is at least a positive universal constant.

math.MG↗
arXiv2025

The separation modulus of unitarily invariant matrix norms

If $X=(M_n(\mathbb{R}),\|\cdot\|)$ is a unitarily invariant normed space on , then we prove (via exact computations for a Jacobi orthogonal random matrix ensemble) that the spectral gap of the Laplacian with Dirichlet boundary conditions on the unit ball $B_X$ of $X$ satisfies $\lambda(X)\asymp n^3 \|I\|^2$. This leads to a confirmation of the weak isomorphic reverse isoperimetry conjecture for $X$, namely, we demonstrate that there exists a convex body $L=L_X\subset B_X$ such that $\mathrm{vol}_{n^2}(L)^{1/n^2}\asymp \mathrm{vol}_{n^2}(B_{X})^{1/n^2}$, yet its isoperimetric quotient is at most a universal constant multiple of $n$. As a corollary (and motivation) of these results, we deduce that the separation modulus of $X$ satisfies $\mathsf{SEP}(X)\asymp \sqrt{n}\|I_n\|_X\mathrm{diam}(B_X)$, where $\mathrm{diam}(B_X)$ is the diameter of $B_X$ with respect to the standard Euclidean metric on $M_n(\mathbb{R})$. Assuming oracle access to norm evaluations in $X$, by combining this with a new deterministic algorithm for efficiently computing a $O(1)$-approximation of the diameter of convex bodies in $\mathbb{R}^n$ that are given by a weak membership oracle and are symmetric with respect to coordinate permutations and reflections about the axes, we obtain an oracle polynomial time algorithm whose output is guaranteed to be the separation modulus of $X$ up to positive universal constant factors. We also deduce an upper bound on the Lipschitz extension modulus of $X$ that improves over the previously best-known bound even in the special case when $X$ is $M_n(\mathbb{R})$ equipped with the $\ell_{2}^n\to \ell_{2}^n$ operator norm.

math.MG↗
arXiv2025

Euclidean embedding, randomized clustering, and Lipschitz extension for finite and doubling subsets of $L_p$ when $p>2$

Fix $p>2$. We prove that the Euclidean distortion of every $n$-point subset of $L_p$ is $p^3(\log n)^{\frac12+o(1)}$, thus, in particular, demonstrating that all $n$-point subsets of $L_p$ exhibit an asymptotic improvement over the $O(\log n)$ Euclidean distortion guarantee that Bourgain's embedding theorem provides for arbitrary $n$-point metric spaces. We also prove that the separation modulus of every $n$-point subset of $ L_p$ is $O(p^2\sqrt{\log n})$, which is sharp up to the dependence on $p$. We deduce from (a refinement of) this asymptotic evaluation of the finitary separation modulus of $ L_p$ that for any $n$-point subset $\mathcal{C}$ of $ L_p$, any Banach space $\mathbf{Z}$, and any $1$-Lipschitz function $f:\mathcal{C}\to \mathbf{Z}$, there exists a $O(p^2\sqrt{\log n})$-Lipschitz function $F:L_p\to \mathbf{Z}$ that extends $f$. We obtain analogous separation and extension statements for doubling subsets of $L_p$.

math.FA↗
arXiv2024

Random zero sets with local growth guarantees

We prove that if $(\mathcal{M},d)$ is an $n$-point metric space that embeds quasisymmetrically into a Hilbert space, then for every $\tau>0$ there is a random subset $\mathcal{Z}$ of $\mathcal{M}$ such that for any pair of points $x,y\in \mathcal{M}$ with $d(x,y)\ge \tau$, the probability that both $x\in \mathcal{Z}$ and $d(y,\mathcal{Z})\ge \beta\tau/\sqrt{1+\log (|B(y,\kappa \beta \tau)|/|B(y,\beta \tau)|)}$ is $\Omega(1)$, where $\kappa>1$ is a universal constant and $\beta>0$ depends only on the modulus of the quasisymmetric embedding. The proof relies on a refinement of the Arora--Rao--Vazirani rounding technique. Among the applications of this result is that the largest possible Euclidean distortion of an $n$-point subset of $\ell_1$ is $\Theta(\sqrt{\log n})$, and the integrality gap of the Goemans--Linial semidefinite program for the Sparsest Cut problem on inputs of size $n$ is $\Theta(\sqrt{\log n})$. Multiple further applications are given.

math.MG↗
arXiv2023

An integer parallelotope with small surface area

We prove that for any $n\in \mathbb{N}$ there is a convex body $K\subseteq \mathbb{R}^n$ whose surface area is at most $n^{\frac12+o(1)}$, yet the translates of $K$ by the integer lattice $\mathbb{Z}^n$ tile $\mathbb{R}^n$.

math.MG↗
arXiv2022

Cayley graphs that have a quantum ergodic eigenbasis

We investigate which finite Cayley graphs admit a quantum ergodic eigenbasis, proving that this holds for any Cayley graph on a group of size $n$ for which the sum of the dimensions of its irreducible representations is $o(n)$, yet there exist Cayley graphs that do not have any quantum ergodic eigenbasis.

math.SP↗
arXiv2021

Extension, separation and isomorphic reverse isoperimetry

The Lipschitz extension modulus $e(M)$ of a metric space $M$ is the infimum over $L\ge 1$ such that for any Banach space $Z$ and any $C\subset M$, any 1-Lipschitz function $f:C\to Z$ can be extended to an $L$-Lipschitz function $F:M\to Z$. Johnson, Lindenstrauss and Schechtman proved that if $X$ is an $n$-dimensional normed space, then $e(X)=O(n)$. In the reverse direction, we prove that every $n$-dimensional normed space $X$ satisfies $e(X)\ge n^c$, where $c>0$ is a universal constant. Our core technical contribution is a geometric structural result on stochastic clustering of finite dimensional normed spaces which implies upper bounds on their Lipschitz extension moduli using an extension method of Lee and the author. The separation modulus of a metric space $(M,d_M)$ is the infimum over $\sigma>0$ such that for any $\Delta>0$ there is a distribution over random partitions of $M$ into clusters of diameter at most $\Delta$ such that for every $x,y\in M$ the probability that they belong to different clusters is at most $\sigma d_M(x,y)/\Delta$. We obtain upper and lower bounds on the separation moduli of finite dimensional normed spaces that relate them to well-studied volumetric invariants. Using these connections, we find the growth rate of the separation moduli of various normed spaces. We formulate a conjecture on isomorphic reverse isoperimetry that can be used with our volumetric bounds on the separation modulus to obtain many more asymptotic evaluations of the separation moduli of normed spaces. Our estimates on the separation modulus imply improved bounds on the Lipschitz extension moduli of various classical spaces. In particular, we deduce an improved bound on $e(\ell_p^n)$ when $p>2$ that resolves a conjecture of Brudnyi and Brudnyi, and prove that $e(\ell_\infty^n)\asymp{\sqrt{n}}$, which is the first time that the order of $e(X)$ has been evaluated for any normed space $X$.

math.MG↗
arXiv2021

FKN, first proof, rewritten

About twenty years ago we wrote a paper, "Boolean Functions whose Fourier Transform is Concentrated on the First Two Levels", \cite{FKN}. In it we offered several proofs of the statement that Boolean functions $f(x_1,x_2,\dots,x_n)$, whose Fourier coefficients are concentrated on the lowest two levels are close to a constant function or to a function of the form $f=x_k$ or $f=1-x_k$. Returning to the paper lately, we noticed that the presentation of the first proof is rather cumbersome, and includes several typos. In this note we rewrite that proof, as a service to the public.

math.CO↗
arXiv2020

Impossibility of almost extension

Let $({\mathbf X},\|\cdot\|_{\mathbf X}), ({\mathbf Y},\|\cdot\|_{\mathbf Y})$ be normed spaces with ${\mathrm{dim}}({\mathbf X})=n$. Bourgain's almost extension theorem asserts that for any ${\varepsilon}>0$, if ${\mathcal{N}}$ is an ${\varepsilon}$-net of the unit sphere of ${\mathbf X}$ and $f:{\mathcal{N}}\to {\mathbf Y}$ is $1$-Lipschitz, then there exists an $O(1)$-Lipschitz $F:{\mathbf X}\to {\mathbf Y}$ such that $\|F(a)-f(a)\|_{\mathbf Y}\lesssim n{\varepsilon}$ for all $a\in \mathcal{N}$. We prove that this is optimal up to lower order factors, i.e., sometimes $\max_{a\in {\mathcal{N}}} \|F(a)-f(a)\|_{\mathbf Y}\gtrsim n^{1-o(1)}{\varepsilon}$ for every $O(1)$-Lipschitz $F:{\mathbf X}\to {\mathbf Y}$. This improves Bourgain's lower bound of $\max_{a\in {\mathcal{N}}} \|F(a)-f(a)\|_{\mathbf Y}\gtrsim n^{c}{\varepsilon}$ for some $0<c<\frac12$. If ${\mathbf X}=\ell_2^n$, then the approximation in the almost extension theorem can be improved to $\max_{a\in {\mathcal{N}}} \|F(a)-f(a)\|_{\mathbf Y}\lesssim \sqrt{n}{\varepsilon}$. We prove that this is sharp, i.e., sometimes $\max_{a\in {\mathcal{N}}} \|F(a)-f(a)\|_{\mathbf Y}\gtrsim \sqrt{n}{\varepsilon}$ for every $O(1)$-Lipschitz $F:\ell_2^n\to {\mathbf Y}$.

math.FA↗
arXiv2020

Foliated corona decompositions

We prove that the $L_4$ norm of the vertical perimeter of any measurable subset of the $3$-dimensional Heisenberg group $\mathbb{H}$ is at most a universal constant multiple of the (Heisenberg) perimeter of the subset. We show that this isoperimetric-type inequality is optimal in the sense that there are sets for which it fails to hold with the $L_4$ norm replaced by the $L_q$ norm for any $q<4$. This is in contrast to the $5$-dimensional setting, where the above result holds with the $L_4$ norm replaced by the $L_2$ norm. The proof of the aforementioned isoperimetric inequality introduces a new structural methodology for understanding the geometry of surfaces in $\mathbb{H}$. In previous work (2017) we showed how to obtain a hierarchical decomposition of Ahlfors-regular surfaces into pieces that are approximately intrinsic Lipschitz graphs. Here we prove that any such graph admits a foliated corona decomposition, which is a family of nested partitions into pieces that are close to ruled surfaces. Apart from the intrinsic geometric and analytic significance of these results, which settle questions posed by Cheeger-Kleiner-Naor (2009) and Lafforgue-Naor (2012), they have several noteworthy implications, including the fact that the $L_1$ distortion of a word-ball of radius $n\ge 2$ in the discrete $3$-dimensional Heisenberg group is bounded above and below by universal constant multiples of $\sqrt[4]{\log n}$; this is in contrast to higher dimensional Heisenberg groups, where our previous work showed that the distortion of a word-ball of radius $n\ge 2$ is of order $\sqrt{\log n}$.

math.MG↗
arXiv2019

Krivine diffusions attain the Goemans--Williamson approximation ratio

Answering a question of Abbasi-Zadeh, Bansal, Guruganesh, Nikolov, Schwartz and Singh (2018), we prove the existence of a slowed-down sticky Brownian motion whose induced rounding for MAXCUT attains the Goemans--Williamson approximation ratio. This is an especially simple particular case of the general rounding framework of Krivine diffusions that we investigate elsewhere.

cs.DS↗
arXiv2019

Concentration of Markov chains with bounded moments

Let $\{W_t\}_{t=1}^{\infty}$ be a finite state stationary Markov chain, and suppose that $f$ is a real-valued function on the state space. If $f$ is bounded, then Gillman's expander Chernoff bound (1993) provides concentration estimates for the random variable $f(W_1)+\cdots+f(W_n)$ that depend on the spectral gap of the Markov chain and the assumed bound on $f$. Here we obtain analogous inequalities assuming only that the $q$'th moment of $f$ is bounded for some $q \geq 2$. Our proof relies on reasoning that differs substantially from the proofs of Gillman's theorem that are available in the literature, and it generalizes to yield dimension-independent bounds for mappings $f$ that take values in an $L_p(μ)$ for some $p\ge 2$, thus answering (even in the Hilbertian special case $p=2$) a question of Kargin (2007).

math.PR↗