arXiv subjects

Stefan Steinerberger

Publications and source records attributed to Stefan Steinerberger.

At least 127 records · Page 7Linked to original sources

arXiv2020

Using Expander Graphs to test whether samples are i.i.d

The purpose of this note is to point out that the theory of expander graphs leads to an interesting test whether $n$ real numbers $x_1, \dots, x_n$ could be $n$ independent samples of a random variable. To any distinct, real numbers $x_1, \dots, x_n$, we associate a 4-regular graph $G$ as follows: using $π$ to denote the permutation ordering the elements, $x_{π(1)} < x_{π(2)} < \dots < x_{π(n)}$, we build a graph on $\left\{1, \dots, n\right\}$ by connecting $i$ and $i+1$ (cyclically) and $π(i)$ and $π(i+1)$ (cyclically). If the numbers are i.i.d. samples, then a result of Friedman implies that $G$ is close to Ramanujan. This suggests a test for whether these numbers are i.i.d: compute the second largest (in absolute value) eigenvalue of the adjacency matrix. The larger $λ- 2\sqrt{3}$, the less likely it is for the numbers to be i.i.d. We explain why this is a reasonable test and give many examples.

math.ST↗

arXiv2020

The smoothest average: Dirichlet, Fejér and Chebyshev

We are interested in the ``smoothest'' averaging that can be achieved by convolving functions $f \in \ell^2(\mathbb{Z})$ with an averaging function $u$. More precisely, suppose $u:\{-n, \ldots, n\} \to \mathbb{R}$ is a symmetric function normalized to $\sum_{k=-n}^{n}u(k) = 1$. We show that every convolution operator is not-too-smooth, in the sense that $$\sup_{f \in \ell^2(\mathbb{Z})} \frac{\| \nabla (f*u)\|_{\ell^2(\mathbb{Z})}}{\|f\|_{\ell^2}}\geq \frac{2}{2n+1},$$ and we show that equality holds if and only if $u$ is constant on the interval $\{-n, \ldots, n\}$. In the setting where smoothness is measured by the $\ell^2$-norm of the discrete second derivative and we further restrict our attention to functions $u$ with nonnegative Fourier transform, we establish the inequality $$\sup_{f \in \ell^2(\mathbb{Z})} \frac{\| Δ(f*u)\|_{\ell^2(\mathbb{Z})}}{\|f\|_{\ell^2(\mathbb{Z})}} \geq \frac{4}{(n+1)^2},$$ with equality if and only if $u$ is the triangle function $u(k)=(n+1-|k|)/(n+1)^2$. We also discuss a continuous analogue and several open problems.

math.CA↗

arXiv2020

On Matrix Rearrangement Inequalities

Given two symmetric and positive semidefinite square matrices $A, B$, is it true that any matrix given as the product of $m$ copies of $A$ and $n$ copies of $B$ in a particular sequence must be dominated in the spectral norm by the ordered matrix product $A^m B^n$? For example, is $$ \| AABAABABB \| \leq \| AAAAABBBB \|\ ? $$ Drury has characterized precisely which disordered words have the property that an inequality of this type holds for all matrices $A,B$. However, the $1$-parameter family of counterexamples Drury constructs for these characterizations is comprised of $3 \times 3$ matrices, and thus as stated the characterization applies only for $N \times N$ matrices with $N \geq 3$. In contrast, we prove that for $2 \times 2$ matrices, the general rearrangement inequality holds for all disordered words. We also show that for larger $N \times N$ matrices, the general rearrangement inequality holds for all disordered words, for most $A,B$ (in a sense of full measure) that are sufficiently small perturbations of the identity.

math.FA↗

arXiv2020

A Semicircle Law for Derivatives of Random Polynomials

Let $x_1, \dots, x_n$ be $n$ independent and identically distributed random variables with mean zero, unit variance, and finite moments of all remaining orders. We study the random polynomial $p_n$ having roots at $x_1, \dots, x_n$. We prove that for $\ell \in \mathbb{N}$ fixed as $n \rightarrow \infty$, the $(n-\ell)-$th derivative of $p_n^{}$ behaves like a Hermite polynomial: for $x$ in a compact interval,$${n^{\ell/2}} \frac{\ell!}{n!} \cdot p_n^{(n-\ell)}\left( \frac{x}{\sqrt{n}}\right) \rightarrow He_{\ell}(x + γ_n),$$ where $He_{\ell}$ is the $\ell-$th probabilists' Hermite polynomial and $γ_n$ is a random variable converging to the standard $\mathcal{N}(0,1)$ Gaussian as $n \rightarrow \infty$. Thus, there is a universality phenomenon when differentiating a random polynomial many times: the remaining roots follow a Wigner semicircle distribution.

math.PR↗

arXiv2020

On Eigenvectors of Random Band Matrices with Large Band

We study random, symmetric $N \times N$ band matrices with a band of size $W$ and Bernoulli random variables as entries. This interpolates between nearest neighbour interaction $W = 1$ and Wigner matrices $W = N$. Eigenvectors are known to be localized for $W \ll N^{1/8}$, delocalized for $W \gg N^{4/5}$ and it is conjectured that the transition for the bulk occurs at $W \sim N^{1/2}$. Eigenvalues in the spectral edge change their behavior at $W \sim N^{5/6}$ but nothing is known about the associated eigenvectors. We show that up to $W \ll N^{5/7}$ any random matrix has with large probability some eigenvectors in the spectral edge, which either exhibit mass concentration or interact strongly on a small scale.

math.PR↗

arXiv2020

Fourier Uncertainty Principles, Scale Space Theory and the Smoothest Average

Let $f \in L^{2}(\mathbb{R}^n)$ and suppose we are interested in computing its average at a fixed scale. This is easy: we pick the density $u_{}$ of a probability distribution with mean 0 and some moment at the desired scale and compute the convolution $u_{} * f$. Is there a particularly natural choice for $u$? This question is studied in scale space theory and the Gaussian is a popular answer. We were interested whether a canonical choice for $u$ can arise from a new axiom: having fixed a scale, the average should oscillate as little as possible, i.e. $$ u_{} = \arg\min_{u_{}} \sup_{f \in L^2(\mathbb{R}^n)} \frac{\| \nabla (u_{} *f) \|_{L^2(\mathbb{R}^n)}}{\|f\|_{L^2(\mathbb{R}^n)}}.$$ This optimal function turns out to be a minimizer of an uncertainty principle: for $α> 0$ and $β> n/2$, there exists $c_{α, β,n} > 0$ such that for all $u \in L^1(\mathbb{R}^n)$ $$ \| |ξ|^β \cdot \widehat{u}\|^α_{L^{\infty}(\mathbb{R}^n)} \cdot \| |x|^α \cdot u \|^β_{L^1(\mathbb{R}^n)} \geq c_{α, β,n} \|u\|_{L^1(\mathbb{R}^n)}^{α+ β}.$$ For $β= 1$, any nonnegative extremizer of the inequality serves as the best averaging function in the sense above, $β\neq 1$ corresponds to other derivatives. For $(n, β)=(1,1)$ we use the Shannon-Whittaker formula to prove that the characteristic function $u(x) = χ_{[-1/2,1/2]}$ is a local minimizer among functions defined on $[-1/2,1/2]$ for $α\in \left\{2,3,4,5,6\right\}$. We provide a sufficient condition for general $α$ in terms of a sign pattern for the hypergeometric function $_1F_2$.

math.CA↗

arXiv2020

Synchronization of Kuramoto Oscillators in Dense Networks

We study synchronization properties of systems of Kuramoto oscillators. The problem can also be understood as a question about the properties of an energy landscape created by a graph. More formally, let $G=(V,E)$ be a connected graph and $(a_{ij})_{i,j=1}^{n}$ denotes its adjacency matrix. Let the function $f:\mathbb{T}^n \rightarrow \mathbb{R}$ be given by $$ f(θ_1, \dots, θ_n) = \sum_{i,j=1}^{n}{ a_{ij} \cos{(θ_i - θ_j)}}.$$ This function has a global maximum when $θ_i = θ$ for all $1\leq i \leq n$. It is known that if every vertex is connected to at least $μ(n-1)$ other vertices for $μ$ sufficiently large, then every local maximum is global. Taylor proved this for $μ\geq 0.9395$ and Ling, Xu \& Bandeira improved this to $μ\geq 0.7929$. We give a slight improvement to $μ\geq 0.7889$. Townsend, Stillman \& Strogatz suggested that the critical value might be $μ_c = 0.75$.

math.OC↗

arXiv2020

A Spectral Approach to the Shortest Path Problem

Let $G=(V,E)$ be a simple, connected graph. One is often interested in a short path between two vertices $u,v$. We propose a spectral algorithm: construct the function $ϕ:V \rightarrow \mathbb{R}_{\geq 0}$ $$ ϕ= \arg\min_{f:V \rightarrow \mathbb{R} \atop f(u) = 0, f \not\equiv 0} \frac{\sum_{(w_1, w_2) \in E}{(f(w_1)-f(w_2))^2}}{\sum_{w \in V}{f(w)^2}}.$$ $ϕ$ can also be understood as the smallest eigenvector of the Laplacian Matrix $L=D-A$ after the $u-$th row and column have been removed. We start in the point $v$ and construct a path from $v$ to $u$: at each step, we move to the neighbor for which $ϕ$ is the smallest. This algorithm provably terminates and results in a short path from $v$ to $u$, often the shortest. The efficiency of this method is due to a discrete analogue of a phenomenon in Partial Differential Equations that is not well understood. We prove optimality for trees and discuss a number of open questions.

math.CO↗

arXiv2020

On Vickrey's Income Averaging

We consider a small set of axioms for income averaging -- recursivity, continuity, and the boundary condition for the present. These properties yield a unique averaging function that is the density of the reflected Brownian motion with a drift started at the current income and moving over the past incomes. When averaging is done over the short past, the weighting function is asymptotically converging to a Gaussian. When averaging is done over the long horizon, the weighing function converges to the exponential distribution. For all intermediate averaging scales, we derive an explicit solution that interpolates between the two.

econ.TH↗

arXiv2020

Positive-definite Functions, Exponential Sums and the Greedy Algorithm: a curious Phenomenon

We describe a curious dynamical system that results in sequences of real numbers in $[0,1]$ with seemingly remarkable properties. Let the function $f:\mathbb{T} \rightarrow \mathbb{R}$ satisfy $\hat{f}(k) \geq c|k|^{-2}$ and define a sequence via $$ x_n = \arg\min_x \sum_{k=1}^{n-1}{f(x-x_k)}.$$ Such sequences $(x_n)_{n=1}^{\infty}$ seem to be astonishingly regularly distributed in various ways (satisfying favorable exponential sum estimates; every interval $J \subset [0,1]$ contains $\sim |J|n$ elements). We prove $$ W_2\left( \frac{1}{n} \sum_{k=1}^{n}{δ_{x_k}}, dx\right) \leq \frac{c}{\sqrt{n}},$$ where $W_2$ is the 2-Wasserstein distance. Much stronger results seem to be true and it seems like an interesting problem to understand this dynamical system better. We obtain optimal results in dimension $d \geq 3$: using $G(x,y)$ to denote the Green's function of the Laplacian on a compact manifold, we show that $$ x_n = \arg\min_{x \in M} \sum_{k=1}^{n-1}{G(x,x_k)} \quad \mbox{satisfies} \quad W_2\left( \frac{1}{n} \sum_{k=1}^{n}{δ_{x_k}}, dx\right) \lesssim \frac{1}{n^{1/d}}.$$

math.CA↗

arXiv2020

Spectral Clustering Revisited: Information Hidden in the Fiedler Vector

We are interested in the clustering problem on graphs: it is known that if there are two underlying clusters, then the signs of the eigenvector corresponding to the second largest eigenvalue of the adjacency matrix can reliably reconstruct the two clusters. We argue that the vertices for which the eigenvector has the largest and the smallest entries, respectively, are unusually strongly connected to their own cluster and more reliably classified than the rest. This can be regarded as a discrete version of the Hot Spots conjecture and should be useful in applications. We give a rigorous proof for the stochastic block model and several examples.

math.PR↗

arXiv2020

Regularized Potentials of Schrödinger Operators and a Local Landscape Function

We study localization properties of low-lying eigenfunctions $$(-Δ+V) ϕ= λϕ\qquad \mbox{in}~Ω$$ for rapidly varying potentials $V$ in bounded domains $Ω\subset \mathbb{R}^d$. Filoche & Mayboroda introduced the landscape function $(-Δ+ V)u=1$ and showed that the function $u$ has remarkable properties: localized eigenfunctions prefer to localize in the local maxima of $u$. Arnold, David, Filoche, Jerison \& Mayboroda showed that $1/u$ arises naturally as the potential in a related equation. Motivated by these questions, we introduce a one-parameter family of regularized potentials $V_t$ that arise from convolving $V$ with the radial kernel $$ V_t(x) = V * \left( \frac{1}{t} \int_0^t \frac{ \exp\left( - \|\cdot\|^2/ (4s) \right)}{(4 πs )^{d/2}} ds \right).$$ We prove that for eigenfunctions $(-Δ+V) ϕ= λϕ$ this regularization $V_t$ is, in a precise sense, the canonical effective potential on small scales. The landscape function $u$ respects the same type of regularization. This allows allows us to derive landscape-type functions out of solutions of the equation $(-Δ+ V)u = f$ for a general right-hand side $f:Ω\rightarrow \mathbb{R}_{>0}$.

math.AP↗

arXiv2020

Three Convolution Inequalities on the Real Line with Connections to Additive Combinatorics

We discuss three convolution inequalities that are connected to additive combinatorics. Cloninger and the second author showed that for nonnegative $f \in L^1(-1/4, 1/4)$, $$ \max_{-1/2 \leq t \leq 1/2} \int_{\mathbb{R}}{f(t-x) f(x) dx} \geq 1.28 \left( \int_{-1/4}^{1/4}{f(x) dx}\right)^2$$ which is related to $g-$Sidon sets (1.28 cannot be replaced by 1.52). We prove a dual statement, related to difference bases, and show that for $f \in L^1(\mathbb{R})$, $$ \min_{0 \leq t \leq 1}\int_{\mathbb{R}}{f(x) f(x+t) dx} \leq 0.42 \|f\|_{L^1}^2,$$ where the constant 1/2 is trivial, 0.42 cannot be replaced by 0.37. This suggests a natural conjecture about the asymptotic structure of $g-$difference bases. Finally, we show for all functions $f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$, $$ \int_{-\frac{1}{2}}^{\frac{1}{2}}{ \int_{\mathbb{R}}{f(x) f(x+t) dx}dt} \leq 0.91 \|f\|_{L^1}\|f\|_{L^2}$$

math.CA↗

arXiv2020

Non-Convex Planar Harmonic Maps

We formulate a novel characterization of a family of invertible maps between two-dimensional domains. Our work follows two classic results: The Radó-Kneser-Choquet (RKC) theorem, which establishes the invertibility of harmonic maps into a convex planer domain; and Tutte's embedding theorem for planar graphs - RKC's discrete counterpart - which proves the invertibility of piecewise linear maps of triangulated domains satisfying a discrete-harmonic principle, into a convex planar polygon. In both theorems, the convexity of the target domain is essential for ensuring invertibility. We extend these characterizations, in both the continuous and discrete cases, by replacing convexity with a less restrictive condition. In the continuous case, Alessandrini and Nesi provide a characterization of invertible harmonic maps into non-convex domains with a smooth boundary by adding additional conditions on orientation preservation along the boundary. We extend their results by defining a condition on the normal derivatives along the boundary, which we call the cone condition; this condition is tractable and geometrically intuitive, encoding a weak notion of local invertibility. The cone condition enables us to extend Alessandrini and Nesi to the case of harmonic maps into non-convex domains with a piecewise-smooth boundary. In the discrete case, we use an analog of the cone condition to characterize invertible discrete-harmonic piecewise-linear maps of triangulations. This gives an analog of our continuous results and characterizes invertible discrete-harmonic maps in terms of the orientation of triangles incident on the boundary.

math.AP↗

arXiv2019

A Metric Sturm-Liouville theory in Two Dimensions

A central result of Sturm-Liouville theory (also called the Sturm-Hurwitz Theorem) states that if $ϕ_k$ is a sequence of eigenfunctions of a second order differential operator on the interval $I \subset \mathbb{R}$, then any linear combination satisfies a uniform bound on the roots $$ \# \left\{x \in I:\sum_{k \geq n}{ a_k ϕ_k(x)} = 0 \right\} \geq n-1.$$ We provide a sharp (up to logarithmic factors) generalization to two dimensions: let $(M,g)$ be a compact two-dimensional manifold (with or without boundary), let $(ϕ_k)$ denote the sequence of eigenfunctions of a uniformly elliptic operator $-\mbox{div}(a(\cdot) \nabla)$ (with Dirichlet or Neumann boundary conditions). Then, for any linear combination of eigenfunctions above a certain index $n$, $$ f = \sum_{k \geq n}{a_k ϕ_k} ~ \mbox{we have} \quad \mathcal{H}^1 \left\{ x: f(x) = 0\right\} \gtrsim_{} \frac{\sqrt{n}}{\sqrt{\log{n}}} \log \left(n \frac{\|f\|_{L^2(M)}}{\|f\|_{L^1(M)}} \right)^{-1/2} \frac{\|f\|_{L^1(M)}}{\| f \|_{L^{\infty}(M)}} .$$ Examples on $M=\mathbb{T}^2$ and $M=\mathbb{S}^2$ shows that this is optimal up to the logarithmic factors. The proof is using optimal transport and a new inequality for the Wasserstein metric $W_p$: if $f(x)dx$ and $g(x)dx$ are two absolutely continuous measures on a two-dimensional domain $M$ with continuous densities and the same total mass, then, for all $1 \leq p <\infty$, $$ W_p(f(x)dx, g(x) dx) \cdot \mathcal{H}^1 \left\{x \in M: f(x) = g(x) \right\} \gtrsim_{M,p} \frac{\|f-g\|_{L^1(M)}^{1+1/p}}{\|f-g\|_{L^{\infty}(M)}}.$$

math.AP↗

arXiv2019

A Remark on the Arcsine Distribution and the Hilbert Transform

It is known that if $(p_n)_{n \in \mathbb{N}}$ is a sequence of orthogonal polynomials in $L^2([-1,1], w(x)dx)$, then the roots are distributed according to an arcsine distribution $π^{-1} (1-x^2)^{-1}dx$ for a wide variety of weights $w(x)$. We connect this to a result of the Hilbert transform due to Tricomi: if $f(x)(1-x^2)^{1/4} \in L^2(-1,1)$ and its Hilbert transform $Hf$ vanishes on $(-1,1)$, then the function $f$ is a multiple of the arcsine distribution $$ f(x) = \frac{c}{\sqrt{1-x^2}}χ_{(-1,1)} \qquad \mbox{where}~c~\in \mathbb{R}.$$ We also prove a localized Parseval-type identity that seems to be new: if $f(x)(1-x^2)^{1/4} \in L^2(-1,1)$ and $f(x) \sqrt{1-x^2}$ has mean value 0 on $(-1,1)$, then $$ \int_{-1}^{1}{ (Hf)(x)^2 \sqrt{1-x^2} dx} = \int_{-1}^{1}{ f(x)^2 \sqrt{1-x^2} dx}.$$

math.CA↗

arXiv2019

On Sublevel Set Estimates and the Laplacian

Carbery proved that if $u:\mathbb{R}^n \rightarrow \mathbb{R}$ is a positive, strictly convex function satisfying $\det D^2u \geq 1$, then we have the estimate $$ \left| \left\{x \in \mathbb{R}^n: u(x) \leq s \right\} \right| \lesssim_n s^{n/2}$$ and this is optimal. We give a short proof that also implies other results. Our main result is an estimate for the sublevel set of functions $u:[0,1]^2 \rightarrow \mathbb{R}$ satisfying $1 \leq Δu \leq c$ for some universal constant $c$: for any $α> 0$, we have $$ \left| \left\{x \in [0,1]^2 : |u(x)| \leq \varepsilon\right\}\right| \lesssim_{c} \sqrt{\varepsilon} + \varepsilon^{α- \frac12} \int_{[0,1]^2}{\frac{|\nabla u|}{|u|^α} dx}.$$ For 'typical' functions, we expect the integral to be finite for $α< 1$. While Carbery-Christ-Wright have shown that no sublevel set estimates independent of $u$ exist, this result shows that for 'typical' functions satisfying $Δu \geq 1$, we expect the sublevel set to be $\lesssim \varepsilon^{1/2-}$. We do not know whether this is sharp or whether similar statements are true in higher dimensions.

math.CA↗

arXiv2019

Optimal Trapping of Brownian Motion: A Nonlinear Analogue of the Torsion Function

We study the problem of maximizing the expected lifetime of drift diffusion in a bounded domain. More formally, we consider the PDE \[ - Δu + b(x) \cdot \nabla u = 1 \qquad \mbox{in}~Ω\] subject to Dirichlet boundary conditions for $\|b\|_{L^{\infty}}$ fixed. We show that, in any given $C^2-$domain $Ω$, the vector field maximizing the expected lifetime is (nonlinearly) coupled to the solution and satisfies $b = -\|b\|_{L^{\infty}} \nabla u/ |\nabla u|$ which reduces the problem to the study of the nonlinear PDE \[ -Δu - b \cdot \left| \nabla u \right| = 1,\] where $b = \|b\|_{L^{\infty}}$ is a constant. We believe that this PDE is a natural and interesting nonlinear analogue of the torsion function. We prove that, for fixed volume, $\| \nabla u\|_{L^1}$ and $\|Δu\|_{L^1}$ are maximized if $Ω$ is the ball (the ball is also known to maximize $\|u\|_{L^p}$ for $p \geq 1$ from a result of Hamel \& Russ).

math.AP↗