arXiv subjects

Stefan Steinerberger

Publications and source records attributed to Stefan Steinerberger.

At least 181 records · Page 10Linked to original sources

arXiv2018

Detecting localized eigenstates of linear operators

We describe a way of detecting the location of localized eigenvectors of a linear system $Ax = λx$ for eigenvalues $λ$ with $|λ|$ comparatively large. We define the family of functions $f_α: \left\{1.2. \dots, n\right\} \rightarrow \mathbb{R}_{}$ $$ f_α(k) = \log \left( \| A^α e_k \|_{\ell^2} \right),$$ where $α\geq 0$ is a parameter and $e_k = (0,0,\dots, 0,1,0, \dots, 0)$ is the $k-$th standard basis vector. We prove that eigenvectors associated to eigenvalues with large absolute value localize around local maxima of $f_α$: the metastable states in the power iteration method (slowing down its convergence) can be used to predict localization. We present a fast randomized algorithm and discuss different examples: a random band matrix, discretizations of the local operator $-Δ+ V$ and the nonlocal operator $(-Δ)^{3/4} + V$.

math.NA↗

arXiv2018

Riesz Energy on the Torus: Regularity of Minimizers

We study sets of $N$ points on the $d-$dimensional torus $\mathbb{T}^d$ minimizing interaction functionals of the type \[ \sum_{i, j =1 \atop i \neq j}^{N}{ f(x_i - x_j)}. \] The main result states that for a class of functions $f$ that behave like Riesz energies $f(x) \sim \|x\|^{-s}$ for $0< s < d$, the minimizing configuration of points has optimal regularity w.r.t. a Fourier-analytic regularity measure that arises in the study of irregularities of distribution. A particular consequence is that they are optimal quadrature points in the space of trigonometric polynomials up to a certain degree. The proof extends to other settings and also covers less singular functions such as $f(x) = \exp\bigl(- N^{\frac{2}{d}} \|x\|^2 \bigr)$.

math-ph↗

arXiv2018

A Sharp Estimate for Probability Distributions

We consider absolutely continuous probability distributions $f(x)dx$ on $\mathbb{R}_{\geq 0}$. A result of Feldheim and Feldheim shows, among other things, that if the distribution is not compactly supported, then there exist $z > 0$ such that most events in $\left\{X + Y \geq 2z\right\}$ are comprised of a 'small' term satisfying $\min(X,Y) \leq z$ and a 'large' term satisfying $\max(X,Y) \geq z$ (as opposed to two 'large' terms that are both larger than $z$) $$ \limsup_{z \rightarrow \infty}~ \mathbb{P}\left( \min(X,Y) \leq z | X+Y \geq 2z\right) = 1.$$ The result fails if the distribution is compactly supported. We prove $$\sup_{z > 0 } ~\mathbb{P}\left( \min(X,Y) \leq z | X+Y \geq 2z\right) \geq \frac{1}{24 + 8\log_2{( med(X) \|f\|_{L^{\infty}})}},$$ where $med(X)$ denotes the median. Interestingly, the logarithm is necessary and the result is sharp up to constants; we also discuss some open problems.

math.PR↗

arXiv2018

Triangles capturing many lattice points

We study a combinatorial problem that recently arose in the context of shape optimization: among all triangles with vertices $(0,0)$, $(x,0)$, and $(0,y)$ and fixed area, which one encloses the most lattice points from $\mathbb{Z}_{>0}^2$? Moreover, does its shape necessarily converge to the isosceles triangle $(x=y)$ as the area becomes large? Laugesen and Liu suggested that, in contrast to similar problems, there might not be a limiting shape. We prove that the limiting set is indeed nontrivial and contains infinitely many elements. We also show that there exist `bad' areas where no triangle is particularly good at capturing lattice points and show that there exists an infinite set of slopes $y/x$ such that any associated triangle captures more lattice points than any other fixed triangle for infinitely many (and arbitrarily large) areas; this set of slopes is a fractal subset of $[1/3, 3]$ and has Minkowski dimension at most $3/4$.

math.CO↗

arXiv2018

General and Refined Montgomery Lemmata

Montgomery's Lemma on the torus $\mathbb{T}^d$ states that a sum of $N$ Dirac masses cannot be orthogonal to many low-frequency trigonometric functions in a quantified way. We provide an extension to general manifolds that also allows for positive weights: let $(M,g)$ be a smooth compact $d-$dimensional manifold without boundary, let $(ϕ_k)_{k=0}^{\infty}$ denote the Laplacian eigenfunctions, let $\left\{ x_1, \dots, x_N\right\} \subset M$ be a set of points and $\left\{a_1, \dots, a_N\right\} \subset \mathbb{R}_{\geq 0}$ be a sequence of nonnegative weights. Then $$\sum_{k=0}^{X}{ \left| \sum_{n=1}^{N}{ a_n ϕ_k(x_n)} \right|^2} \gtrsim_{(M,g)} \left(\sum_{i=1}^{N}{a_i^2} \right) \frac{ X}{(\log{X})^{\frac{d}{2}}}.$$ This result is sharp up to the logarithmic factor. Furthermore, we prove a refined spherical version of Montgomery's Lemma, and provide applications to estimates of discrepancy and discrete energies of $N$ points on the sphere $\mathbb{S}^{d}$.

math.CA↗

arXiv2017

Efficient Algorithms for t-distributed Stochastic Neighborhood Embedding

t-distributed Stochastic Neighborhood Embedding (t-SNE) is a method for dimensionality reduction and visualization that has become widely popular in recent years. Efficient implementations of t-SNE are available, but they scale poorly to datasets with hundreds of thousands to millions of high dimensional data-points. We present Fast Fourier Transform-accelerated Interpolation-based t-SNE (FIt-SNE), which dramatically accelerates the computation of t-SNE. The most time-consuming step of t-SNE is a convolution that we accelerate by interpolating onto an equispaced grid and subsequently using the fast Fourier transform to perform the convolution. We also optimize the computation of input similarities in high dimensions using multi-threaded approximate nearest neighbors. We further present a modification to t-SNE called "late exaggeration," which allows for easier identification of clusters in t-SNE embeddings. Finally, for datasets that cannot be loaded into the memory, we present out-of-core randomized principal component analysis (oocPCA), so that the top principal components of a dataset can be computed without ever fully loading the matrix, hence allowing for t-SNE of large datasets to be computed on resource-limited machines.

cs.LG↗

arXiv2017

On the Spectral Resolution of Products of Laplacian Eigenfunctions

We study products of eigenfunctions of the Laplacian $-Δϕ_λ = λϕ_λ$ on compact manifolds. If $ϕ_μ, ϕ_λ$ are two eigenfunctions and $μ\leq λ$, then one would perhaps expect their product $ϕ_μϕ_λ$ to be mostly a linear combination of eigenfunctions with eigenvalue close to $λ$. This can faily quite dramatically: on $\mathbb{T}^2$, we see that $$ 2\sin{(n x)} \sin{((n+1) x)} = \cos{(x)} - \cos{( (2n+1) x)} $$ has half of its $L^2-$mass at eigenvalue 1. Conversely, the product $$ \sin{(n x)} \sin{(m y)} \qquad \mbox{lives at eigenvalue} \quad \max{\left\{m^2,n^2\right\}} \leq m^2 + n^2 \leq 2\max{\left\{m^2,n^2\right\}}$$ and the heuristic is valid. We show that the main reason is that in the first example 'the waves point in the same direction': if the heuristic fails and multiplication carries $L^2-$mass to lower frequencies, then $ϕ_μ$ and $ϕ_λ$ are strongly correlated at scale $ \sim λ^{-1/2}$ (the shorter wavelength) $$ \left\| \int_{M}{ p(t,x,y)( ϕ_λ(y) - ϕ_λ(x))( ϕ_μ(y) - ϕ_μ(x)) dy} \right\|_{L^2_x} \gtrsim \| ϕ_μϕ_λ\|_{L^2},$$ where $p(t,x,y)$ is the classical heat kernel and $t \sim λ^{-1}$. This turns out to be a fairly fundamental principle and is even valid for the Hadamard product of eigenvectors of a Graph Laplacian.

math.AP↗

arXiv2017

The Universal Aesthetics of Mathematics

The unique and beautiful character of certain mathematical results and proofs is often considered one of the most gratifying aspects of engaging with mathematics. We study whether this perception of mathematical arguments having an intrinsic 'character' is subjective or universal -- this was done by having test subjects with varying degrees of mathematical experience match mathematical arguments with paintings and music: 'does this proof feel more like Bach or Schubert?' The results suggest that such a universal connection indeed exists.

math.HO↗

arXiv2017

A Local Faber-Krahn inequality and Applications to Schrödinger's Equation

We prove a local Faber-Krahn inequality for solutions $u$ to the Dirichlet problem for $Δ+ V$ on an arbitrary domain $Ω$ in $\mathbb{R}^n$. Suppose a solution $u$ assumes a global maximum at some point $x_0 \in Ω$ and $u(x_0)>0$. Let $T(x_0)$ be the smallest time at which a Brownian motion, started at $x_0$, has exited the domain $Ω$ with probability $\ge 1/2$. For nice (e.g., convex) domains, $T(x_0) \asymp d(x_0,\partialΩ)^2$ but we make no assumption on the geometry of the domain. Our main result is that there exists a ball $B$ of radius $\asymp T(x_0)^{1/2}$ such that $$ \| V \|_{L^{\frac{n}{2}, 1}(Ω\cap B)} \ge c_n > 0, $$ provided that $n \ge 3$. In the case $n = 2$, the above estimate fails and we obtain a substitute result. The Laplacian may be replaced by a uniformly elliptic operator in divergence form. This result both unifies and strenghtens a series of earlier results.

math.AP↗

arXiv2017

Randomized Near Neighbor Graphs, Giant Components, and Applications in Data Science

If we pick $n$ random points uniformly in $[0,1]^d$ and connect each point to its $k-$nearest neighbors, then it is well known that there exists a giant connected component with high probability. We prove that in $[0,1]^d$ it suffices to connect every point to $ c_{d,1} \log{\log{n}}$ points chosen randomly among its $ c_{d,2} \log{n}-$nearest neighbors to ensure a giant component of size $n - o(n)$ with high probability. This construction yields a much sparser random graph with $\sim n \log\log{n}$ instead of $\sim n \log{n}$ edges that has comparable connectivity properties. This result has nontrivial implications for problems in data science where an affinity matrix is constructed: instead of picking the $k-$nearest neighbors, one can often pick $k' \ll k$ random points out of the $k-$nearest neighbors without sacrificing efficiency. This can massively simplify and accelerate computation, we illustrate this with several numerical examples.

math.CO↗

arXiv2017

Poissonian Pair Correlation and Discrepancy

A sequence $(x_n)_{n=1}^{\infty}$ on the torus $\mathbb{T} \cong [0,1]$ is said to exhibit Poissonian pair correlation if the local gaps behave like the gaps of a Poisson random variable, i.e. $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\}} = 2s \qquad \mbox{almost surely.}$$ We show that being close to Poissonian pair correlation for few values of $s$ is enough to deduce global regularity statements: if, for some~$0 < δ< 1/2$, a set of points $\left\{x_1, \dots, x_N \right\}$ satisfies $$ \frac{1}{N}\# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\} \leq (1+δ)2s \qquad \mbox{for all} \hspace{6pt} 1 \leq s \leq (8/δ)\sqrt{\log{N}},$$ then the discrepancy $D_N$ of the set satisfies $D_N \lesssim δ^{1/3} + N^{-1/3}δ^{-1/2}$. We also show that distribution properties are reflected in the global deviation from the Poissonian pair correlation $$ N^2 D_N^5 \lesssim \frac{2}{N}\int_{0}^{N/2} \left| \frac{1}{N}\# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\} - 2s \right|^2 ds \lesssim N^2 D_N^2,$$ where the lower is bound is conditioned on $D_N \gtrsim N^{-1/3}$. The proofs use a connection between exponential sums, the heat kernel on $\mathbb{T}$ and spatial localization. Exponential sum estimates are obtained as a byproduct. We also describe a connection to diaphony and several open problems.

math.NT↗

arXiv2017

Topological Bounds for Fourier Coefficients and Applications to Torsion

Let $Ω\subset \mathbb{R}^2$ be a bounded convex domain in the plane and consider \begin{align*} -Δu &=1 \qquad \mbox{in}~Ω\\ u &= 0 \qquad \mbox{on}~\partial Ω. \end{align*} If $u$ assumes its maximum in $x_0 \in Ω$, then the eccentricity of level sets close to the maximum is determined by the Hessian $D^2u(x_0)$. We prove that $D^2u(x_0)$ is negative definite and give a quantitative bound on the spectral gap $$ λ_{\max}\left(D^2u(x_0)\right) \leq - c_1\exp\left( -c_2\frac{diam(Ω)}{inrad(Ω)} \right)$$ for universal $c_1, c_2$ This is sharp up to constants. The proof is based on a new lower bound for Fourier coefficients whose proof has a topological component: if $f:\mathbb{T} \rightarrow \mathbb{R}$ is continuous and has $n$ sign changes, then $$ \sum_{k=0}^{n/2}{ \left| \left\langle f, \sin{kx} \right\rangle \right| + \left| \left\langle f, \cos{kx} \right\rangle \right| } \gtrsim_n \frac{ | f\|^{n+1}_{L^1(\mathbb{T})}}{ \| f\|^{n }_{L^{\infty}(\mathbb{T})}}.$$ This statement immediately implies estimates on higher derivatives of harmonic functions $u$ in the unit ball: if $u$ is very flat in the origin, then the boundary function $u(\cos{t}, \sin{t}):\mathbb{T} \rightarrow \mathbb{R}$ has to have either large amplitude or many roots. It also implies that the solution of the heat equation starting with $f:\mathbb{T} \rightarrow \mathbb{R}$ cannot decay faster than $\sim\exp(-(\# \mbox{sign changes})^2 t/4)$.

math.CA↗

arXiv2017

Exponential Sums and Riesz energies

We bound an exponential sum that appears in the study of irregularities of distribution (the low-frequency Fourier energy of the sum of several Dirac measures) by geometric quantities: a special case is that for all $\left\{ x_1, \dots, x_N\right\} \subset \mathbb{T}^2$, $X \geq 1$ and a universal $c>0$ $$ \sum_{i,j=1}^{N}{ \frac{X^2}{1 + X^4 \|x_i -x_j\|^4}} \lesssim \sum_{k \in \mathbb{Z}^2 \atop \|k\| \leq X}{ \left| \sum_{n=1}^{N}{ e^{2 πi \left\langle k, x_n \right\rangle}}\right|^2} \lesssim \sum_{i,j=1}^{N}{ X^2 e^{-c X^2\|x_i -x_j\|^2}}.$$ Since this exponential sum is intimately tied to rather subtle distribution properties of the points, we obtain nonlocal structural statements for near-minimizers of the Riesz-type energy. In the regime $X \gtrsim N^{1/2}$ both upper and lower bound match for maximally-separated point sets satisfying $\|x_i -x_j\| \gtrsim N^{-1/2}$.

math.NT↗

arXiv2017

A Variation on the Donsker-Varadhan Inequality for the Principial Eigenvalue

The purpose of this short note is to give a variation on the classical Donsker-Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain $Ω$ by the largest mean first exit time of the associated drift-diffusion process via $$λ_1 \geq \frac{1}{\sup_{x \in Ω} \mathbb{E}_x τ_{Ω^c}}.$$ Instead of looking at the mean of the first exit time, we study quantiles: let $d_{p, \partial Ω}:Ω\rightarrow \mathbb{R}_{\geq 0}$ be the smallest time $t$ such that the likelihood of exiting within that time is $p$, then $$λ_1 \geq \frac{\log{(1/p)}}{\sup_{x \in Ω} d_{p,\partial Ω}(x)}.$$ Moreover, as $p \rightarrow 0$, this lower bound converges to $λ_1$.

math.SP↗

arXiv2017

Stability, Fairness and Random Walks in the Bargaining Problem

We study the classical bargaining problem and its two canonical solutions, (Nash and Kalai-Smorodinsky), from a novel point of view: we ask for stability of the solution if both players are able distort the underlying bargaining process by reference to a third party (e.g. a court). By exploring the simplest case, where decisions of the third party are made randomly we obtain a stable solution, where players do not have any incentive to refer to such a third party. While neither the Nash nor the Kalai-Smorodinsky solution are able to ensure stability in case reference to a third party is possible, we found that the Kalai-Smorodinsky solution seems to always dominate the stable allocation which constitutes novel support in favor of the latter.

cs.GT↗

arXiv2017

Clustering with t-SNE, provably

t-distributed Stochastic Neighborhood Embedding (t-SNE), a clustering and visualization method proposed by van der Maaten & Hinton in 2008, has rapidly become a standard tool in a number of natural sciences. Despite its overwhelming success, there is a distinct lack of mathematical foundations and the inner workings of the algorithm are not well understood. The purpose of this paper is to prove that t-SNE is able to recover well-separated clusters; more precisely, we prove that t-SNE in the `early exaggeration' phase, an optimization technique proposed by van der Maaten & Hinton (2008) and van der Maaten (2014), can be rigorously analyzed. As a byproduct, the proof suggests novel ways for setting the exaggeration parameter $α$ and step size $h$. Numerical examples illustrate the effectiveness of these rules: in particular, the quality of embedding of topological structures (e.g. the swiss roll) improves. We also discuss a connection to spectral clustering methods.

cs.LG↗

arXiv2017

The Geometry of Nodal Sets and Outlier Detection

Let $(M,g)$ be a compact manifold and let $-Δϕ_k = λ_k ϕ_k$ be the sequence of Laplacian eigenfunctions. We present a curious new phenomenon which, so far, we only managed to understand in a few highly specialized cases: the family of functions $f_N:M \rightarrow \mathbb{R}_{\geq 0}$ $$ f_N(x) = \sum_{k \leq N}{ \frac{1}{\sqrt{λ_k}} \frac{|ϕ_k(x)|}{\|ϕ_k\|_{L^{\infty}(M)}}}$$ seems strangely suited for the detection of anomalous points on the manifold. It may be heuristically interpreted as the sum over distances to the nearest nodal line and potentially hints at a new phenomenon in spectral geometry. We give rigorous statements on the unit square $[0,1]^2$ (where minima localize in $\mathbb{Q}^2$) and on Paley graphs (where $f_N$ recovers the geometry of quadratic residues of the underlying finite field $\mathbb{F}_p$). Numerical examples show that the phenomenon seems to arise on fairly generic manifolds.

math.SP↗

arXiv2017

Optimal Jittered Sampling for two Points in the Unit Square

Jittered Sampling is a refinement of the classical Monte Carlo sampling method. Instead of picking $n$ points randomly from $[0,1]^2$, one partitions the unit square into $n$ regions of equal measure and then chooses a point randomly from each partition. Currently, no good rules for how to partition the space are available. In this paper, we present a solution for the special case of subdividing the unit square by a decreasing function into two regions so as to minimize the expected squared $\mathcal{L}_2-$discrepancy. The optimal partitions are given by a \textit{highly} nonlinear integral equation for which we determine an approximate solution. In particular, there is a break of symmetry and the optimal partition is not into two sets of equal measure. We hope this stimulates further interest in the construction of good partitions.

math.NA↗