arXiv subjects

Stefan Steinerberger

Publications and source records attributed to Stefan Steinerberger.

At least 145 records · Page 8Linked to original sources

arXiv2019

Spectral Limitations of Quadrature Rules and Generalized Spherical Designs

We study manifolds $M$ equipped with a quadrature rule $$ \int_{M}{ϕ(x) dx} \simeq \sum_{i=1}^{n}{a_i ϕ(x_i)}.$$ We show that $n-$point quadrature rules with nonnegative weights on a compact $d-$dimensional manifold cannot integrate more than at most the first $c_{d}n + o(n)$ Laplacian eigenfunctions exactly. The constants $c_d$ are explicitly computed and $c_2 = 4$. The result is new even on $\mathbb{S}^2$ where it generalizes results on spherical designs.

math.SP↗

arXiv2019

Generalized Designs on Graphs: Sampling, Spectra, Symmetries

Spherical Designs are finite sets of points on the sphere $\mathbb{S}^{d}$ with the property that the average of certain (low-degree) polynomials in these points coincides with the global average of the polynomial on $\mathbb{S}^{d}$. They are evenly distributed and often exhibit a great degree of regularity and symmetry. We point out that a spectral definition of spherical designs easily transfers to finite graphs -- these 'graphical designs' are subsets of vertices that are evenly spaced and capture the symmetries of the underlying graph (should they exist). Our main result states that good graphical designs either consist of many vertices or their neighborhoods have exponential volume growth. We show several examples, describe ways to find them and discuss problems.

math.CO↗

arXiv2019

Hot Spots in Convex Domains are in the Tips (up to an Inradius)

Let $Ω\subset \mathbb{R}^2$ be a bounded, convex domain and let $-Δϕ_1 = μ_1 ϕ_1$ be the first nontrivial Laplacian eigenfunction with Neumann boundary conditions. The Hot Spots conjecture claims that the maximum and minimum are attained at the boundary. We show that they are attained far away from one another: if $x_1, x_2 \in Ω$ satisfy $\|x_1 - x_2\| = \mbox{diam}(Ω)$, then every maximum and minimum is assumed within distance $c\cdot \mbox{inrad}(Ω)$ of $x_1$ and $x_2$, where $c$ is a universal constant (which is the optimal scaling up to the value of $c$).

math.AP↗

arXiv2019

A Wasserstein Inequality and Minimal Green Energy on Compact Manifolds

Let $M$ be a smooth, compact $d-$dimensional manifold, $d \geq 3,$ without boundary and let $G: M \times M \rightarrow \mathbb{R} \cup \left\{\infty\right\}$ denote the Green's function of the Laplacian $-Δ$ (normalized to have mean value 0). We prove a bound on the cost of transporting Dirac measures in $\left\{x_1, \dots, x_n\right\} \subset M$ to the normalized volume measure $dx$ in terms of the Green's function of the Laplacian $$ W_2\left( \frac{1}{n} \sum_{k=1}^{n}{δ_{x_k}}, dx\right) \lesssim_M \frac{1}{n^{1/d}} + \frac{1}{n} \left| \sum_{k, \ell=1 \atop k \neq \ell}^{n}G(x_k, x_{\ell})\right|^{1/2}.$$ We obtain the same result for the Coulomb kernel $G(x,y) = 1/\|x-y\|^{d-2}$ on the sphere $\mathbb{S}^d$, for $d \geq 3$, where we show that $$ W_2\left(\frac{1}{n} \sum_{k=1}^{n}{ δ_{x_k}}, dx\right) \lesssim \frac{1}{n^{1/d}} + \frac{1}{n} \left| \sum_{k, \ell=1 \atop k \neq \ell}^{n}{\left(\frac{1}{\|x_k - x_{\ell}\|^{d-2}} - c_d \right)} \right|^{\frac{1}{2}},$$ where $c_d$ is the constant that normalizes the Coulomb kernel to have mean value 0. We use this to show that minimizers of the discrete Green energy on compact manifolds have optimal rate of convergence $W_2\left( \frac{1}{n} \sum_{k=1}^{n}{δ_{x_k}}, dx\right) \lesssim n^{-1/d}$. The second inequality implies the same result for minimizers of the Coulomb energy on $\mathbb{S}^d$ which was recently proven by Marzo & Mas.

math.CA↗

arXiv2019

Poissonian Pair Correlation in Higher Dimensions

Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). A sequence $(x_n)$ is said to have Poissonian pair correlation if, for all $s>0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\}} = 2s.$$ It is known that this implies uniform distribution of the sequence $(x_n)$. Hinrichs, Kaltenböck, Larcher, Stockinger \& Ullrich extended this result to higher dimensions and showed that sequences $(x_n)$ in $[0,1]^d$ that satisfy, for all $s>0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: \|x_m - x_n\|_{\infty} \leq \frac{s}{N} \right\}} = (2s)^d.$$ are also uniformly distributed. We prove the same result for the extension by the Euclidean norm: if a sequence $(x_n)$ in $\mathbb{T}^d$ satisfies, for all $s > 0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: \|x_m - x_n\|_{2} \leq \frac{s}{N} \right\}} = ω_d s^d$$ where $ω_d$ is the volume of the unit ball, then $(x_n)$ is uniformly distributed. Our approach shows that Poissonian Pair Correlation implies an exponential sum estimate that resembles and implies the Weyl criterion.

math.CA↗

arXiv2019

Dynamically Defined Sequences with Small Discrepancy

We study the problem of constructing sequences $(x_n)_{n=1}^{\infty}$ on $[0,1]$ in such a way that $$ D_N^* = \sup_{0 \leq x \leq 1} \left| \frac{ \left\{1 \leq i \leq N: x_i \leq x \right\}}{N} - x \right|$$ is uniformly small. A result of Schmidt shows that necessarily $D_N^* \gtrsim (\log{N}) N^{-1}$ for infinitely many $N$ and there are several classical constructions attaining this growth. We describe a type of uniformly distributed sequence that seems to be completely novel: given $\left\{x_1, \dots, x_{N-1} \right\}$, we construct $x_N$ in a greedy manner $$ x_N = \arg\min_{\min_k |x-x_k| \geq N^{-10}} \sum_{k=1}^{N-1}{1-\log{(2\sin{(π|x-x_k|)})}}.$$ We prove that $D_N \lesssim (\log{N}) N^{-1/2}$ and conjecture that $D_N \lesssim (\log{N}) N^{-1}$. Numerical examples illustrate this conjecture in a very impressive manner. We also establish a discrepancy bound $D_N \lesssim (\log{N})^d N^{-1/2}$ for an analogous construction in higher dimensions and conjecture it to be $D_N \lesssim (\log{N})^d N^{-1}$.

math.CO↗

arXiv2019

Roots of trigonometric polynomials and the Erdős-Turán theorem

We prove, informally put, that it is not a coincidence that $\cos{(n θ)} + 1 \geq 0$ and that the roots of $z^n + 1 =0$ are uniformly distributed in angle -- a version of the statement holds for all trigonometric polynomials with `few' real roots. The Erdős-Turán theorem states that if $p(z) =\sum_{k=0}^{n}{a_k z^k}$ is suitably normalized and not too large for $|z|=1$, then its roots are clustered around $|z| = 1$ and equidistribute in angle at scale $\sim n^{-1/2}$. We establish a connection between the rate of equidistribution of roots in angle and the number of sign changes of the corresponding trigonometric polynomial $q(θ) = \Re \sum_{k=0}^{n}{a_k e^{i k θ}}$. If $q(θ)$ has $\lesssim n^δ$ roots for some $0 < δ< 1/2$, then the roots of $p(z)$ do not frequently cluster in angle at scale $\sim n^{-(1-δ)} \ll n^{-1/2}$.

math.CA↗

arXiv2019

Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions

Let $Ω\subset \mathbb{R}^n$ be a convex domain and let $f:Ω\rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $Δf \geq 0$). Then $$ \frac{1}{|Ω|} \int_Ω{f dx} \leq \frac{c_n}{ |\partial Ω| } \int_{\partial Ω}{ f dσ},$$ where $c_n \leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ Ω_2 \subset Ω_1 \subset \mathbb{R}^n$: $$ \frac{|\partial Ω_1|}{|Ω_1|} \frac{| Ω_2|}{|\partial Ω_2|} \leq n.$$

math.CA↗

arXiv2019

A Nonlocal Functional Promoting Low-Discrepancy Point Sets

Let $X = \left\{x_1, \dots, x_N\right\} \subset \mathbb{T}^d \cong [0,1]^d$ be a set of $N$ points in the $d-$dimensional torus that we want to arrange as regularly possible. The purpose of this paper is to introduce a curious energy functional $$ E(X) = \sum_{1 \leq m,n \leq N \atop m \neq n} \prod_{k=1}^{d}{ (1 - \log{\left(2 \sin{ \left( π|x_{m,k} - y_{m,k} |\right)} \right)})}$$ and to suggest that moving a set $X$ into the direction $-\nabla E(X)$ may have the effect of increasing regularity of the set in the sense of decreasing discrepancy. We numerically demonstrate the effect for Halton, Hammersley, Kronecker, Niederreiter and Sobol sets. Lattices in $d=2$ are critical points of the energy functional, some (possibly all) are strict local minima.

math.OC↗

arXiv2019

Transport and Interface: an Uncertainty Principle for the Wasserstein distance

Let $f: [0,1]^d \rightarrow \mathbb{R}$ be a continuous function with zero mean and interpret $f_{+} = \max(f, 0)$ and $f_{-} = -\min(f, 0)$ as the densities of two measures. We prove that if the cost of transport from $f_{+}$ to $f_{-}$ is small (in terms of the Wasserstein distance $W^1$), then the nodal set $\left\{x \in (0,1)^d: f(x) = 0 \right\}$ has to be large (`if it is always easy to buy milk, there must be many supermarkets'). More precisely, we show that $$ W_1(f_+, f_-) \cdot \mathcal{H}^{d-1}\left\{x \in (0,1)^d: f(x) = 0 \right\} \gtrsim_{d} \left( \frac{\|f\|_{L^1}}{\|f\|_{L^{\infty}}} \right)^{4 - \frac1d} \|f\|_{L^1} \, .$$ We apply this ``uncertainty principle" to the metric Sturm-Liouville theory in higher dimensions to show that a linear combination of eigenfunctions of an elliptic operator cannot have an arbitrarily small zero set.

math.CA↗

arXiv2019

Varadhan Asymptotics for the Heat Kernel on Finite Graphs

Let $G$ be a simple, finite graph and let $p_t(x,y)$ denote the heat kernel on $G$. The purpose of this short note is to show that for $t \rightarrow 0^+$ $$ p_t(x,y) = \# \left\{\mbox{paths of length}~d(x,y)~\mbox{between}~x~\mbox{and}~y\right\} \frac{t^{d(x,y)}}{d(x,y)!} + \mathcal{O}(t^{d(x,y)+1}),$$ where $d(x,y)$ is the usual Graph distance. This is the discrete analogue of the classical Varadhan asymptotic for the heat kernel on manifolds and refines a result of Keller, Lenz, Münch, Schmidt and Telcs. The asymptotic behavior encapsulates additional geometric information: if the Graph is bipartite, then the next term in the expansion is negative.

math.AP↗

arXiv2019

A Dimension-Free Hermite-Hadamard Inequality via Gradient Estimates for the Torsion Function

Let $Ω\subset \mathbb{R}^n$ be a convex domain and let $f:Ω\rightarrow \mathbb{R}$ be a subharmonic function, $Δf \geq 0$, which satisfies $f \geq 0$ on the boundary $\partial Ω$. Then $$ \int_Ω{f ~dx} \leq |Ω|^{\frac{1}{n}} \int_{\partial Ω}{f ~dσ}.$$ Our proof is based on a new gradient estimate for the torsion function, $Δu = -1$ with Dirichlet boundary conditions, which is of independent interest.

math.CA↗

arXiv2019

Tax Mechanisms and Gradient Flows

We demonstrate how a static optimal income taxation problem can be analyzed using dynamical methods. Specifically, we show that the taxation problem is intimately connected to the heat equation. Our first result is a new property of the optimal tax which we call the fairness principle. The optimal tax at any income is invariant under a family of properly adjusted Gaussian averages (the heat kernel) of the optimal taxes at other incomes. That is, the optimal tax at a given income is equal to the weighted by the heat kernels average of optimal taxes at other incomes and income densities. Moreover, this averaging happens at every scale tightly linked to each other providing a unified weighting scheme at all income ranges. The fairness principle arises not due to equality considerations but rather it represents an efficient way to smooth the burden of taxes and generated revenues across incomes. Just as nature wants to distribute heat evenly, the optimal way for a government to raise revenues is to distribute the tax burden and raised revenues evenly among individuals. We then construct a gradient flow of taxes -- a dynamic process changing the existing tax system in the direction of the increase in tax revenues -- and show that it takes the form of a heat equation. The fairness principle holds also for the short-term asymptotics of the gradient flow, where the averaging is done over the current taxes. The gradient flow we consider can be viewed as a continuous process of a reform of the nonlinear income tax schedule and thus unifies the variational approach to taxation and optimal taxation. We present several other characteristics of the gradient flow focusing on its smoothing properties.

econ.TH↗

arXiv2019

On Zeroes of Random Polynomials and Applications to Unwinding

Let $μ$ be a probability measure in $\mathbb{C}$ with a continuous and compactly supported density function, let $z_1, \dots, z_n$ be independent random variables, $z_i \sim μ$, and consider the random polynomial $$ p_n(z) = \prod_{k=1}^{n}{(z - z_k)}.$$ We determine the asymptotic distribution of $\left\{z \in \mathbb{C}: p_n(z) = p_n(0)\right\}$. In particular, if $μ$ is radial around the origin, then those solutions are also distributed according to $μ$ as $n \rightarrow \infty$. Generally, the distribution of the solutions will reproduce parts of $μ$ and condense another part on curves. We use these insights to study the behavior of the Blaschke unwinding series on random data.

math.PR↗

arXiv2019

Heavy-tailed kernels reveal a finer cluster structure in t-SNE visualisations

T-distributed stochastic neighbour embedding (t-SNE) is a widely used data visualisation technique. It differs from its predecessor SNE by the low-dimensional similarity kernel: the Gaussian kernel was replaced by the heavy-tailed Cauchy kernel, solving the "crowding problem" of SNE. Here, we develop an efficient implementation of t-SNE for a $t$-distribution kernel with an arbitrary degree of freedom $ν$, with $ν\to\infty$ corresponding to SNE and $ν=1$ corresponding to the standard t-SNE. Using theoretical analysis and toy examples, we show that $ν<1$ can further reduce the crowding problem and reveal finer cluster structure that is invisible in standard t-SNE. We further demonstrate the striking effect of heavier-tailed kernels on large real-life data sets such as MNIST, single-cell RNA-sequencing data, and the HathiTrust library. We use domain knowledge to confirm that the revealed clusters are meaningful. Overall, we argue that modifying the tail heaviness of the t-SNE kernel can yield additional insight into the cluster structure of the data.

cs.LG↗

arXiv2019

Quadrature Points via Heat Kernel Repulsion

We discuss the classical problem of how to pick $N$ weighted points on a $d-$dimensional manifold so as to obtain a reasonable quadrature rule $$ \frac{1}{|M|}\int_{M}{f(x) dx} \simeq \frac{1}{N} \sum_{n=1}^{N}{a_i f(x_i)}.$$ This problem, naturally, has a long history; the purpose of our paper is to propose selecting points and weights so as to minimize the energy functional $$ \sum_{i,j =1}^{N}{ a_i a_j \exp\left(-\frac{d(x_i,x_j)^2}{4t}\right) } \rightarrow \min, \quad \mbox{where}~t \sim N^{-2/d},$$ $d(x,y)$ is the geodesic distance and $d$ is the dimension of the manifold. This yields point sets that are theoretically guaranteed, via spectral theoretic properties of the Laplacian $-Δ$, to have good properties. One nice aspect is that the energy functional is universal and independent of the underlying manifold; we show several numerical examples.

math.NA↗

arXiv2019

Quantitative Homogenization and Convergence of Moving Averages

We study homogenization it its most basic form $$-\left(a\left(\frac{x}{\varepsilon}\right) u_{\varepsilon}'(x)\right)' = f(x) \quad \mbox{for} ~x \in (0,1),$$ where $a(\cdot)$ is a positive $1-$periodic continuous function, $f$ is smooth and $u_{\varepsilon}$ is subjected to Dirichlet boundary conditions. Classically, there is a homogenized equation with $a(\cdot)$ replaced by a constant coefficient $\overline{a} > 0$ whose solution $u$ satisfies $\|u-u_{\varepsilon}\|_{L^{\infty}} \lesssim \varepsilon$. We show that local averages can result in faster convergence: for example, if $a(x) = a(1-x)$, then for $x \in (\varepsilon, 1-\varepsilon)$ $$ \left| \frac{1}{\varepsilon} \int_{x-\varepsilon/2}^{x+\varepsilon/2}{ u_{\varepsilon}(y) dy} - u(x) \right| \lesssim_{a, f} \varepsilon^2.$$ If the condition on $a(\cdot)$ is not satisfied, then subtracting an explicitly given linear function (depending on $a(\cdot),f$) results in the same bound. We also describe another approach to quantitative homogenization problems and illustrate it on the same example.

math.AP↗

arXiv2019

An Extremal Property of the Hexagonal Lattice

We describe an extremal property of the hexagonal lattice $Λ\subset \mathbb{R}^2$. Let $p$ denote the circumcenter of its fundamental triangle (a so-called deep hole) and let $A_r$ denote the set of lattice points that are at distance $r$ from $p$ \begin{equation} A_r = \left\{ λ\in Λ: \| λ- p \| = r\right\}. \end{equation} If $Γ$ is a small perturbation of $Λ$ in the space of lattices with fixed density and $C_r$ denotes the set of points in $A_r$ shifted to the new lattice, then \begin{equation} \sum_{μ\in C_r}{ \| p - μ\|} - \sum_{λ\in A_r}{ \| p - λ\|} \gtrsim r \, |A_r| \, d(Λ, Γ)^2, \end{equation} where $d(Λ, Γ)$ denotes the distance between the lattices: the hexagonal lattice has the property that `far away points are closer than they are for nearby lattices'. This has implications in the calculus of variations: assume \begin{equation} g_Γ(z) = \sum_{γ\in Γ} f( \|z - γ\|) \quad \mbox{ satisfies } \quad \min_{z \in \mathbb{R}^2} g_Λ(z) = g_Λ(p). \end{equation} For a certain class of compactly supported functions $f$, the hexagonal lattice $Λ$ is then a strict local maximizer of \begin{equation} \max_Γ \min_{z \in \mathbb{R}^2} \sum_{γ\in Γ}{f( \|z - γ\| )}, \end{equation} where the maximum runs over all lattices of fixed density.

math.MG↗