Stefan Steinerberger

activity

20232026

collaborators

25 papers

math.OC2026

Gradient descent with exponentially increasing stepsizes and restarts

François Clément, Stefan Steinerberger

Let $f : R^{d} \to R$ . We consider gradient descent $x_{n + 1} = x_{n} - τ_{n} \nabla f (x_{n})$ , where the stepsize $τ_{n} = τ \cdot e^{r n}$ is exponentially growing (wi…

math.MG2026

A Remark on the Odd Area of Unit Disks

Stefan Steinerberger

Let $F$ be a family of $n$ unit disks in $R^{2}$ with $n$ being odd. We use $\mbox{OA}(F)$ to denote the area of the set of points that is covered by an odd number of disks.…

cs.LG2026

An effective variant of the Hartigan $k$ -means algorithm

François Clément, Stefan Steinerberger

The k-means problem is perhaps the classical clustering problem and often synonymous with Lloyd's algorithm (1957). It has become clear that Hartigan's algorithm (1975) gives bette…

math.CA2026

Buffon Discrepancy and the Steinhaus Longimeter

Stefan Steinerberger

Let $Ω \subset R^{2}$ be a convex set. We study the problem of distributing a one-dimensional set $S$ with total length $L$ so that for any line $ℓ$ in $R^{2}$ the…

math.AP2026

Slow dispersion in Floquet-Dirac Hamiltonians

Anthony Bloch, Amir Sagiv, Stefan Steinerberger

We study dispersive decay for non-autonomous Hamiltonian systems. While the general theory for dispersion in such non-autonomous systems is largely open, it was shown \cite{kraisle…

math.MG2026

Distance Equilibrium Measures and Curvature in Metric Spaces

Stefan Steinerberger

Let $(X, d)$ be a compact metric space. We consider the behavior of probability measures $μ$ with the property that $$ \int_{X} d(x, y) dμ(y) \qquad \mbox{is independent of}~x \in X…

math.CA2025

Many critical points for discrete Riesz energy on $T^{2}$

François Clément, Stefan Steinerberger

It is widely believed that the energy functional $E_{p} : (S^{2})^{n} \to R$ $E_{p} = i \neq = j i , j = 1 \sum n \frac{1}{∥ x _{i} - x _{j} ∥ ^{p}}$ has a number of cr…

math.CO2025

Balanced Stick Breaking

François Clément, Stefan Steinerberger

Consider an infinite sequence $(x_{k})_{k = 1}^{\infty}$ on the unit circle $S^{1}$ . We may interpret the first $n$ elements $(x_{k})_{k = 1}^{n}$ as places where the `circular stic…

math.OC2025

Robust Online Sampling from Possibly Moving Target Distributions

François Clément, Stefan Steinerberger

We suppose we are given a list of points $x_{1}, \dots, x_{n} \in R$ , a target probability measure $μ$ and are asked to add additional points $x_{n + 1}, \dots, x_{n + m}$ so that…

eess.SP2025

Time-Frequency Filtering Meets Graph Clustering

Marcelo A. Colominas, Stefan Steinerberger, Hau-Tieng Wu

We show that the problem of identifying different signal components from a time-frequency representation can be equivalently phrased as a graph clustering problem: given a graph $G…

math.DS2025

Superpolynomial convergence in the Riemann Rearrangement Theorem

Stefan Steinerberger

Let $x \in R$ be arbitrary and consider the `greedy' approximation of $x$ by signed harmonic sums: given $a_{n} = \sum_{k \leq n} ε_{k} / k$ with $\varepsilon_k \in \l…

math.CA2025

Potential Theory and the Boundary of Combinatorial Graphs

Stefan Steinerberger

Let $G = (V, E)$ be a finite, connected graph. We investigate a notion of boundary $\partial G \subseteq V$ and argue that it is well behaved from the point of view of potential theor…

math.NT2025

On an iterated arithmetic function problem of Erdos and Graham

Stefan Steinerberger

Erdős and Graham define $g (n) = n + ϕ (n)$ and the iterated application $g_{k} (n) = g (g_{k - 1} (n))$ . They ask for solutions of $g_{k + r} (n) = 2 g_{k} (n)$ and observe $g_{k+2}(10) = 2 g_…

math.CO2025

Many antipodes implies many neighbors

Stefan Steinerberger

Suppose ${x_{1}, \dots, x_{n}} \subset R^{2}$ is a set of $n$ points in the plane with diameter $\leq 1$ , meaning $∥ x_{i} - x_{j} ∥ \leq 1$ for all $1 \leq i, j \leq n$ …

math.NT2025

On the largest singular vector of the Redheffer matrix

François Clément, Stefan Steinerberger

The Redheffer matrix $A_{n} \in R^{n \times n}$ is defined by setting $A_{ij} = 1$ if $j = 1$ or $i$ divides $j$ and 0 otherwise. One of its many interesting properties is tha…

math.CO2025

Small gaps in the Ulam sequence

François Clément, Stefan Steinerberger

The Ulam sequence, described by Stanislaw Ulam in the 1960s, starts $1, 2$ and then iteratively adds the smallest integer that can be uniquely written as the sum of two distinct ear…

math.PR2025

Regularizing random points by deleting a few

Dmitriy Bilyk, Stefan Steinerberger

It is well understood that if one is given a set $X \subset [0, 1]$ of $n$ independent uniformly distributed random variables, then $$ \sup_{0 \leq x \leq 1} \left| \frac{\# X \cap…

math.MG2025

A Stability Version of the Jones Opaque Set Inequality

Stefan Steinerberger

Let $Ω \subset R^{2}$ be a bounded, convex set. A set $O \subset R^{2}$ is an opaque set (for $Ω$ ) if every line that intersects $Ω$ also intersects $O$ . What is the m…

math.CA2024

Failure of Orthogonality of Rounded Fourier Bases

François Clément, Stefan Steinerberger

The purpose of this note is to prove estimates for $$ \left| \sum_{k=1}^{n} \mbox{sign} \left( \cos \left( \frac{2πa}{n} k \right) \right) \mbox{sign} \left( \cos \left( \frac{2πb}…

math.DS2024

A curious dynamical system in the plane

Stefan Steinerberger, Tony Zeng

For any irrational $α > 0$ and any initial value $z_{- 1} \in C$ , we define a sequence of complex numbers $(z_{n})_{n = 0}^{\infty}$ as follows: $z_{n}$ is $z_{n - 1} + e^{2 π i α n}$ …

math.CO2024

On differences of two harmonic numbers

Jeck Lim, Stefan Steinerberger

We prove that the existence of infinitely many $(m_{k}, n_{k}) \in N^{2}$ such that the difference of harmonic numbers $H_{m_{k}} - H_{n_{k}}$ approximates 1 well $$ \lim_{k \righta…

math.FA2024

A lower bound for the Balan--Jiang matrix problem

Afonso S. Bandeira, Dustin G. Mixon, Stefan Steinerberger

We prove the existence of a positive semidefinite matrix $A \in R^{n \times n}$ such that any decomposition into rank-1 matrices has to have factors with a large $ℓ^{1} -$ …

math.SP2024

Universal lower bounds for Dirichlet eigenvalues

Stefan Steinerberger

Let $Ω \subset R^{d}$ be a bounded domain and let $λ_{1}, λ_{2}, \dots$ denote the sequence of eigenvalues of the Laplacian subject to Dirichlet boundary conditions. We consider…

math.CO2024

On a problem involving unit fractions

Stefan Steinerberger

Erdős and Graham proposed to determine the number of subsets $S \subseteq {1, 2, \dots, n}$ with $\sum_{s \in S} 1/ s = 1$ and asked, among other things, whether that numb…

Gradient descent with exponentially increasing stepsizes and restarts

A Remark on the Odd Area of Unit Disks

An effective variant of the Hartigan k-means algorithm

Buffon Discrepancy and the Steinhaus Longimeter

Slow dispersion in Floquet-Dirac Hamiltonians

Distance Equilibrium Measures and Curvature in Metric Spaces

Many critical points for discrete Riesz energy on T2

Balanced Stick Breaking

Robust Online Sampling from Possibly Moving Target Distributions

Time-Frequency Filtering Meets Graph Clustering

Superpolynomial convergence in the Riemann Rearrangement Theorem

Potential Theory and the Boundary of Combinatorial Graphs

On an iterated arithmetic function problem of Erdos and Graham

Many antipodes implies many neighbors

On the largest singular vector of the Redheffer matrix

Small gaps in the Ulam sequence

Regularizing random points by deleting a few

A Stability Version of the Jones Opaque Set Inequality

Failure of Orthogonality of Rounded Fourier Bases

A curious dynamical system in the plane

On differences of two harmonic numbers

A lower bound for the Balan--Jiang matrix problem

Universal lower bounds for Dirichlet eigenvalues

On a problem involving unit fractions

An effective variant of the Hartigan $k$ -means algorithm

Many critical points for discrete Riesz energy on $T^{2}$