arXiv subjects

Stefan Steinerberger

Publications and source records attributed to Stefan Steinerberger.

At least 91 records · Page 5Linked to original sources

arXiv2022

The Boundary of a Graph and its Isoperimetric Inequality

We define, for any graph $G=(V,E)$, a boundary $\partial G \subseteq V$. The definition coincides with what one would expected for the discretization of (sufficiently nice) Euclidean domains and contains all vertices from the Chartrand-Erwin-Johns-Zhang boundary. Moreover, it satisfies an isoperimetric principle stating that graphs with many vertices have a large boundary unless they contain long paths: we show that for graphs with maximal degree $Δ$ $$ | \partial G| \geq \frac{1}{2Δ} \frac{|V|}{\mbox{diam}(G)}.$$ For graphs discretizing Euclidean domains, one has $\mbox{diam}(G) \sim |V|^{1/d}$ and recovers the scaling of the classical Euclidean isoperimetric principle.

math.CO↗

arXiv2021

Fundamental component enhancement via adaptive nonlinear activation functions

In many real world oscillatory signals, the fundamental component of a signal $f(t)$ might be weak or does not exist. This makes it difficult to estimate the instantaneous frequency of the signal. Traditionally, researchers apply the rectification trick, working with $|f(t)|$ or $\mbox{ReLu}(f(t))$ instead, to enhance the fundamental component. This raises an interesting question: what type of nonlinear function $g:\mathbb{R} \rightarrow \mathbb{R}$ has the property that $g(f(t))$ has a more pronounced fundamental frequency? $g(t) = |t|$ and $g(t) = \mbox{ReLu}(t)$ seem to work well in practice; we propose a variant of $g(t) = 1/(1-|t|)$ and provide a theoretical guarantee. Several simulated signals and real signals are analyzed to demonstrate the performance of the proposed solution.

math.OC↗

arXiv2021

Quantum Entanglement and the Growth of Laplacian Eigenfunctions

We study the growth of Laplacian eigenfunctions $ -Δϕ_k = λ_k ϕ_k$ on compact manifolds $(M,g)$. Hörmander proved sharp polynomial bounds on $\| ϕ_k\|_{L^{\infty}}$ which are attained on the sphere. On a `generic' manifold, the behavior seems to be different: both numerics and Berry's random wave model suggest $\| ϕ_k\|_{L^{\infty}} \lesssim \sqrt{\log{λ_k}}$ as the typical behavior. We propose a mechanism, centered around an $L^1-$analogue of the spectral projector, for explaining the slow growth in the generic case: for $ϕ_{n+1}(x_0)$ to be large, it is necessary that either (1) several of the first $n$ eigenfunctions were large in $x_0$ or (2) that $ϕ_{n+1}$ is strongly correlated with a suitable linear combination of the first $n$ eigenfunctions on most of the manifold or (3) both. An interesting byproduct is quantum entanglement for Laplacian eigenfunctions: the existence of two distinct points $x,y \in M$ such that the sequences $(ϕ_k(x))_{k=1}^{\infty}$ and $(ϕ_k(y))_{k=1}^{\infty}$ do not behave like independent random variables. The existence of such points is not to be expected for generic manifolds but common for the classical manifolds and subtly intertwined with eigenfunction concentration.

math.SP↗

arXiv2021

On Concavity of Solutions of the Nonlinear Poisson Equation

We consider the nonlinear Poisson equation $-Δu = f(u)$ in domains $Ω\subset \mathbb{R}^n$ with Dirichlet boundary conditions on $\partial Ω$. We show (for monotonically increasing concave $f$ with small Lipschitz constant) that if $D^2 u$ is negative semi-definite on the boundary, then $u$ is concave. A conjecture of Saint Venant from 1856 (proven by Polya in 1948) is that among all domains $Ω$ of fixed measure, the solution of $-Δu =1$ assumes its largest maximum when $Ω$ is a ball. We extend this to $-Δu =f(u)$ for monotonically increasing $f$ with small Lipschitz constant.

math.AP↗

arXiv2021

An upper bound on the Hot Spots constant

Let $D \subset \mathbb{R}^d$ be a bounded, connected domain with smooth boundary and let $-Δu = μ_1 u$ be the first nontrivial eigenfunction of the Laplace operator with Neumann boundary conditions. We prove $$ \max_{x \in D} ~u(x) \leq 60 \cdot \max_{x \in \partial D} ~u(x)$$ and emphasize that this constant is uniform among all connected domains with smooth boundary in all dimensions. In particular, the Hot Spots Conjecture cannot fail by an arbitrary factor. The inequality also holds for other (Neumann-)eigenfunctions (possibly with a different constant) provided the eigenvalue is smaller than the first Dirichlet eigenvalue. An example of Kleefeld shows that the optimal constant is at least $1 + 10^{-3}$.

math.AP↗

arXiv2021

Effective Bounds for the Decay of Schrödinger Eigenfunctions and Agmon bubbles

We study solutions of $-Δu + V u = λu$ on $\mathbb{R}^n$. Such solutions localize in the `allowed' region $\left\{x \in \mathbb{R}^n: V(x) \leq λ\right\}$ and decay exponentially in the `forbidden' region $\left\{x \in \mathbb{R}^n: V(x) > λ\right\}$. One way of making this precise is Agmon's inequality implying decay estimates in terms of the Agmon metric. We prove a complementary decay estimate in terms of harmonic measure which can improve on Agmon's estimate, connect the Agmon metric to decay of harmonic measure and prove a sharp pointwise Agmon estimate.

math.AP↗

arXiv2021

Surrounding the solution of a Linear System of Equations from all sides

Suppose $A \in \mathbb{R}^{n \times n}$ is invertible and we are looking for the solution of $Ax = b$. Given an initial guess $x_1 \in \mathbb{R}$, we show that by reflecting through hyperplanes generated by the rows of $A$, we can generate an infinite sequence $(x_k)_{k=1}^{\infty}$ such that all elements have the same distance to the solution, i.e. $\|x_k - x\| = \|x_1 - x\|$. If the hyperplanes are chosen at random, averages over the sequence converge and $$ \mathbb{E} \left\| x - \frac{1}{m} \sum_{k=1}^{m}{ x_k} \right\| \leq \frac{1 + \|A\|_F \|A^{-1}\|}{\sqrt{m}} \cdot\|x-x_1\|.$$ The bound does not depend on the dimension of the matrix. This introduces a purely geometric way of attacking the problem: are there fast ways of estimating the location of the center of a sphere from knowing many points on the sphere? Our convergence rate (coinciding with that of the Random Kaczmarz method) comes from averaging, can one do better?

math.NA↗

arXiv2021

Polynomials with Zeros on the Unit Circle: Regularity of Leja Sequences

Let $z_1, \dots, z_m$ be $m$ distinct complex numbers, normalized to $|z_k| = 1$, and consider the polynomial $$ p_{m}(z) = \prod_{k=1}^{m}{(z-z_k)}.$$ We define a sequence of polynomials in a greedy fashion, $$ p_{N+1}(z) = p_{N}(z) \left(z - z^*\right)\qquad \mbox{where}~z^* = \arg\max_{|z|=1} |p_{N}(z)|,$$ and prove that, independently of the initial polynomial $p_m$, the roots of $p_{N}$ equidistribute in angle at rate at most $(\log{N})^2/N$. This even persists when sometimes adding `adversarial' points by hand. We rephrase the main result in terms of a dynamical system involving the inverse fractional Laplacian $(-Δ)^{-1/2}$ and conjecture that, when phrased in this language, the underlying regularity phenomenon might appear in a very general setting.

math.CA↗

arXiv2021

Randomized Kaczmarz converges along small singular vectors

Randomized Kaczmarz is a simple iterative method for finding solutions of linear systems $Ax = b$. We point out that the arising sequence $(x_k)_{k=1}^{\infty}$ tends to converge to the solution $x$ in an interesting way: generically, as $k \rightarrow \infty$, $x_k - x$ tends to the singular vector of $A$ corresponding to the smallest singular value. This has interesting consequences: in particular, the error analysis of Strohmer \& Vershynin is optimal. It also quantifies the `pre-convergence' phenomenon where the method initially seems to converge faster. This fact also allows for a fast computation of vectors $x$ for which the Rayleigh quotient $\|Ax\|/\|x\|$ is small: solve $Ax = 0$ via Randomized Kaczmarz.

math.NA↗

arXiv2021

A Weighted Randomized Kaczmarz Method for Solving Linear Systems

The Kaczmarz method for solving a linear system $Ax = b$ interprets such a system as a collection of equations $\left\langle a_i, x\right\rangle = b_i$, where $a_i$ is the $i-$th row of $A$, then picks such an equation and corrects $x_{k+1} = x_k + λa_i$ where $λ$ is chosen so that the $i-$th equation is satisfied. Convergence rates are difficult to establish. Assuming the rows to be normalized, $\|a_i\|_{\ell^2}=1$, Strohmer \& Vershynin established that if the order of equations is chosen at random, $\mathbb{E}~ \|x_k - x\|_{\ell^2}$ converges exponentially. We prove that if the $i-$th row is selected with likelihood proportional to $\left|\left\langle a_i, x_k \right\rangle - b_i\right|^{p}$, where $0<p<\infty$, then $\mathbb{E}~\|x_k - x\|_{\ell^2}$ converges faster than the purely random method. As $p \rightarrow \infty$, the method de-randomizes and explains, among other things, why the maximal correction method works well. We empirically observe that the method computes approximations of small singular vectors of $A$ as a byproduct.

math.NA↗

arXiv2021

Finding Structure in Sequences of Real Numbers via Graph Theory: a Problem List

We investigate a method of generating a graph $G=(V,E)$ out of an ordered list of $n$ distinct real numbers $a_1, \dots, a_n$. These graphs can be used to test for the presence of interesting structure in the sequence. We describe sequences exhibiting intricate hidden structure that was discovered this way. Our list includes sequences of Deutsch, Erdős, Freud & Hegyvari, Recaman, Quet, Zabolotskiy and Zizka. Since our observations are mostly empirical, each sequence in the list is an open problem.

math.CO↗

arXiv2021

A common variable minimax theorem for graphs

Let $\mathcal{G} = \{G_1 = (V, E_1), \dots, G_m = (V, E_m)\}$ be a collection of $m$ graphs defined on a common set of vertices $V$ but with different edge sets $E_1, \dots, E_m$. Informally, a function $f :V \rightarrow \mathbb{R}$ is smooth with respect to $G_k = (V,E_k)$ if $f(u) \sim f(v)$ whenever $(u, v) \in E_k$. We study the problem of understanding whether there exists a nonconstant function that is smooth with respect to all graphs in $\mathcal{G}$, simultaneously, and how to find it if it exists.

math.SP↗

arXiv2021

Quantile-Based Random Kaczmarz for corrupted linear systems of equations

We consider linear systems $Ax = b$ where $A \in \mathbb{R}^{m \times n}$ consists of normalized rows, $\|a_i\|_{\ell^2} = 1$, and where up to $βm$ entries of $b$ have been corrupted (possibly by arbitrarily large numbers). Haddock, Needell, Rebrova and Swartworth propose a quantile-based Random Kaczmarz method and show that for certain random matrices $A$ it converges with high likelihood to the true solution. We prove a deterministic version by constructing, for any matrix $A$, a number $β_A$ such that there is convergence for all perturbations with $β< β_A$. Assuming a random matrix heuristic, this proves convergence for tall Gaussian matrices with up to $\sim 0.5\%$ corruption (a number that can likely be improved).

math.NA↗

arXiv2021

A Graph Decomposition motivated by the Geometry of Randomized Rounding

We introduce a graph decomposition which exists for all simple, connected graphs $G=(V,E)$. The decomposition $V = A \cup B \cup C$ is such that each vertex in $A$ has more neighbors in $B$ than in $A$ and vice versa. $C$ is `balanced': each $v \in C$ has the same number of neighbours in $A$ and $B$. These decompositions arise naturally from the behavior of an associated dynamical system (`Randomized Rounding') on $(\mathbb{S}^1)^{|V|}$. Connections to judicious partitions and the \textsc{MaxCut} problem (in particular the Burer-Monteiro-Zhang heuristic) are being discussed.

math.CO↗

arXiv2021

How well-conditioned can the eigenvalue problem be?

The condition number for eigenvalue computations is a well--studied quantity. But how small can we expect it to be? Namely, which is a perfectly conditioned matrix w.r.t. eigenvalue computations? In this note we answer this question with exact first order asymptotic.

math.NA↗

arXiv2021

On the Logarithmic Energy of Points on S^2

We revisit a classical question: how large is the minimal logarithmic energy of $n$ points on $\mathbb{S}^2$ $$ \mathcal{E}_{\log}(n) = \min_{x_1, \dots, x_n \in \mathbb{S}^2} \quad \sum_{i,j =1 \atop i \neq j}^{n}{ \log{\frac{1}{\|x_i-x_j\|}} } ? $$ Betermin & Sandier (building on work of Sandier & Serfaty) showed that $$\mathcal{E}_{\log}(n) = \left( \frac{1}{2} - \log{2} \right)n^2 - \frac{n \log{n}}{2} + c_{\log} \cdot n + o(n),$$ where the constant $c_{\log}$ is characterized by a certain renormalized minimization problem. Brauchart, Hardin \& Saff conjectured a closed form expression for $c_{\log}$ ($\sim -0.05$) assuming analytic continuation. We describe a simple renormalization approach that results in a purely local problem involving superpositions of Gaussians. In particular, if the hexagonal lattice minimizes Gaussians energy, this would prove that $c_{\log}$ indeed coincides with the conjectured value. We also improve the lower bound from $c_{\log} \geq -0.223$ to $c_{\log} \geq -0.095$.

math.CA↗

arXiv2021

A 0.502$\cdot$MaxCut Approximation using Quadratic Programming

We study the MaxCut problem for graphs $G=(V,E)$. The problem is NP-hard, there are two main approximation algorithms with theoretical guarantees: (1) the Goemans \& Williamson algorithm uses semi-definite programming to provide a 0.878MaxCut approximation (which, if the Unique Games Conjecture is true, is the best that can be done in polynomial time) and (2) Trevisan proposed an algorithm using spectral graph theory from which a 0.614MaxCut approximation can be obtained. We discuss a new approach using a specific quadratic program and prove that its solution can be used to obtain at least a 0.502MaxCut approximation. The algorithm seems to perform well in practice.

cs.DS↗

arXiv2021

Towards Optimal Gradient Bounds for the Torsion Function in the Plane

Let $Ω\subset \mathbb{R}^2$ be a bounded, convex domain and let $u$ be the solution of $-Δu = 1$ vanishing on the boundary $\partial Ω$. The estimate $$ \| \nabla u\|_{L^{\infty}(Ω)} \leq c |Ω|^{1/2}$$ is classical. We use the P-functional, the stability theory of the torsion function and Brownian motion to establish the estimate for a universal $c < (2π)^{-1/2}$. We also give a numerical construction showing that the optimal constant satisfies $c \geq 0.358$. The problem is important in different settings: (1) as the maximum shear stress in Saint Venant Elasticity Theory, (2) as an optimal control problem for the constrained maximization of the lifetime of Brownian motion started close to the boundary and (3) and optimal Hermite-Hadamard inequalities for subharmonic functions on convex domains.

math.AP↗