arXiv subjects

Stefan Steinerberger

Publications and source records attributed to Stefan Steinerberger.

At least 19 recordsLinked to original sources

arXiv2026

An Efficient Triangulation of $\mathbb{R}P^5$

We present a $6$-dimensional centrally symmetric simplicial polytope for which the antipodal quotient of its boundary forms a $24$-vertex triangulation of the $5$-dimensional real projective space. This $6$-polytope is highly symmetric with an automorphism group of order $192$, and is of independent interest. We conjecture that our construction uses the fewest number of vertices among all triangulations of $\mathbb{R}P^5$. Our method also produces two triangulations of $\mathbb{R}P^6$ on $45$ and $49$ vertices; both improve the previously best known construction in dimension $6$ that used $53$ vertices.

math.CO↗

arXiv2026

Slow dispersion in Floquet-Dirac Hamiltonians

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{kraisler2025time} that there exists a time-periodically forced one-dimensional Dirac equation with unusually slow dispersive decay rate of $t^{-1/5}$. It is to be expected that such behavior is not generic and requires a very particular forcing term; we provide a more general ansatz and systematic procedure to construct such an equation with a dispersive decay rate no faster than $t^{-1/10}$. Our limitations are purely algebraic and it stands to reason that arbitrarily slow decay, $t^{-\varepsilon}$ for every $\varepsilon > 0$, should be achievable.

math.AP↗

arXiv2026

An Almgren monotonicity formula for discrete harmonic functions

The celebrated Almgren monotonicity formula for harmonic functions $u:\mathbb{R}^n \rightarrow \mathbb{R}$ says that its $L^2-$energy concentrated on a sphere of radius $r$, when measured in a suitable sense, is non-decreasing: if $u$ oscillates at a certain scale, it has even larger oscillations at a larger scale. We prove a discrete analogue of the Almgren monotonicity formula for harmonic functions on infinite combinatorial graphs $G=(V,E)$. Some applications are discussed.

math.AP↗

arXiv2026

A set of points on the sphere with small Riesz energy

We construct a set of points $\left\{x_1, \dots, x_n\right\} \subset \mathbb{S}^2$ such that $$ \sum_{i \neq j} \frac{1}{\|x_i - x_j\|^2} \leq \frac{n^2 \log{n}}{4} + cn^2 + O(n^{11/6} \log{n}),$$ where the constant $c \sim -0.085768\dots$ is given in closed form and matches the constant that was conjectured by Brauchart-Hardin-Saff to be optimal. The point set is motivated by the crystallization conjecture and consists of pieces of the hexagonal lattice projected onto the sphere in a tightly interlocked way.

math.CA↗

arXiv2026

Gradient descent with exponentially increasing stepsizes and restarts

Let $f:\mathbb{R}^d \rightarrow \mathbb{R}$. We consider gradient descent $x_{n+1} = x_n - τ_n \nabla f(x_n)$, where the stepsize $τ_n = τ\cdot e^{rn}$ is exponentially growing (with $τ> 0$ and $0 < r \ll 1$). This diverges for almost all initial values. We show that restarting the algorithm whenever $\|x_{n+1} - x_n\| \geq e^r\|x_n - x_{n-1}\|$ has good properties: it works very well in practice; we determine the limiting convergence rate in the case of convergence to a non-degenerate local minimum: it improves on classic gradient descent even though computational cost is comparable. The precise choice of $0 < r \ll 1$ does not matter much and the method is virtually independent of an initial stepsize $τ$ that is too small: while the convergence rate for gradient descent decays linearly as $τ\rightarrow 0$, it decays as $1/\log(1/τ)$ in this modified version; numerical examples illustrate the results.

math.OC↗

arXiv2026

A Remark on the Odd Area of Unit Disks

Let $F$ be a family of $n$ unit disks in $\mathbb{R}^2$ 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. The purpose of this note is to disprove the conjecture $\mbox{OA}(F) \geq π$ which was suggested in the literature and to present some examples.

math.MG↗

arXiv2026

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

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 better results in almost all cases, Telgarsky-Vattani note a typical improvement of $5\%$ -- $10\%$. We point out that a very minor variation of Hartigan's method leads to another $2\%$ -- $5\%$ improvement; the improvement tends to become larger when either dimension or $k$ increase.

cs.LG↗

arXiv2026

Complex Interpolation of Matrices with an application to Multi-Manifold Learning

Given two symmetric positive-definite matrices $A, B \in \mathbb{R}^{n \times n}$, we study the spectral properties of the interpolation $A^{1-x} B^x$ for $0 \leq x \leq 1$. The presence of `common structures' in $A$ and $B$, eigenvectors pointing in a similar direction, can be investigated using this interpolation perspective. Generically, exact log-linearity of the operator norm $\|A^{1-x} B^x\|$ is equivalent to the existence of a shared eigenvector in the original matrices; stability bounds show that approximate log-linearity forces principal singular vectors to align with leading eigenvectors of both matrices. These results give rise to and provide theoretical justification for a multi-manifold learning framework that identifies common and distinct latent structures in multiview data.

cs.LG↗

arXiv2026

Buffon Discrepancy and the Steinhaus Longimeter

Let $Ω\subset \mathbb{R}^2$ be a convex set. We study the problem of distributing a one-dimensional set $S$ with total length $L$ so that for any line $\ell$ in $\mathbb{R}^2$ the number of intersections $\#(\ell \cap S)$ is proportional to the length $\mathcal{H}^1(\ell \cap Ω)$ as much as possible; we use the term Buffon discrepancy for the largest error. A construction of Steinhaus can be generalized to prove the existence of sets with Buffon discrepancy $\lesssim L^{1/3}$. We also show that the unit disk $\mathbb{D}$ admits a set with uniformly bounded Buffon discrepancy as $L \rightarrow \infty$.

math.CA↗

arXiv2026

Distance Equilibrium Measures and Curvature in Metric Spaces

Let $(X,d)$ be a compact metric space. We consider the behavior of probability measures $μ$ with the property that $$ \int_{X} d(x, y) dμ(y) \qquad \mbox{is independent of}~x \in X.$$ It appears that such measures, when they exist, encode a `curvature-type' quantity. We investigate this in the special case where $X$ is a closed, convex curve in $\mathbb{R}^2$ and $d = \| \cdot \|_2$ is the Euclidean distance: even a single point with small curvature implies non-existence of such a measure. Conversely, such a measure $μ$ exists for all curves whose curvature is sufficiently close to constant. Curvature is usually defined by second derivatives; this one is defined via an integral equation which makes sense in much rougher spaces. Connections to curvature on graphs, the Gross-Stadje Theorem and magnitude are discussed.

math.MG↗

arXiv2025

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

It is widely believed that the energy functional $E_p:(\mathbb{S}^2)^n \rightarrow \mathbb{R}$ $$ E_p = \sum_{i,j=1 \atop i \neq j}^{n} \frac{1}{\|x_i-x_j\|^p}$$ has a number of critical points, $\nabla E(x) = 0$, that grows exponentially in $n$. Despite having been extensively tested and being physically well motivated, no rigorous result in this direction exists. We prove a version of this result on the two-dimensional flat torus $\mathbb{T}^2$ and show that there are infinitely many $n \in \mathbb{N}$ such that the number of critical points of $E_p: (\mathbb{T}^2)^n \rightarrow \mathbb{R}$ is at least $\exp(c \sqrt{n})$ provided $p \geq 5 \log{n}$. We also investigate the special cases $n=3,4,5$ which turn out to be surprisingly interesting.

math.CA↗

arXiv2025

From pinned billiard balls to partial differential equations

We discuss the propagation of kinetic energy through billiard balls fixed in place along a one-dimensional segment. The number of billiard balls is assumed to be large but finite and we assume kinetic energy propagates following the usual collision laws of physics. Assuming an underlying stochastic mean-field for the expectation and the variance of the kinetic energy, we derive a coupled system of nonlinear partial difference equations. Our results are illustrated by numerical simulations.

math-ph↗

arXiv2025

Balanced Stick Breaking

Consider an infinite sequence $(x_k)_{k=1}^{\infty}$ on the unit circle $\mathbb{S}^1$. We may interpret the first $n$ elements $(x_k)_{k=1}^{n}$ as places where the `circular stick' $\mathbb{S}^1$ is broken into a total of $n+1$ pieces. It is clear that they cannot all be the same length all the time. de Bruijn and Erdős (1949) show that the ratio of the largest to the smallest has to be arbitrarily close to 2 infinitely many times which is sharp. They also consider the problem of balancing the length of $r$ consecutive intervals and prove $$ \frac{\max \mbox{length of}~r~\mbox{consecutive intervals}}{\min \mbox{length of}~r~\mbox{consecutive intervals}} \geq 1 + \frac{1}{r}.$$ We prove that this ratio can be as small as $1 + c \log{r}/ r$. This is done by means of refined discrepancy estimates for the van der Corput sequence over very short intervals and proves a conjecture of Brethouwer.

math.CO↗

arXiv2025

Robust Online Sampling from Possibly Moving Target Distributions

We suppose we are given a list of points $x_1, \dots, x_n \in \mathbb{R}$, a target probability measure $μ$ and are asked to add additional points $x_{n+1}, \dots, x_{n+m}$ so that $x_1, \dots, x_{n+m}$ is as close as possible to the distribution of $μ$; additionally, we want this to be true uniformly for all $m$. We propose a simple method that achieves this goal. It selects new points in regions where the existing set is lacking points and avoids regions that are already overly crowded. If we replace $μ$ by another measure $μ_2$ in the middle of the computation, the method dynamically adjusts and allows us to keep the original sampling points. $x_{n+1}$ can be computed in $\mathcal{O}(n)$ steps and we obtain state-of-the-art results. It appears to be an interesting dynamical system in its own right; we analyze a continuous mean-field version that reflects much of the same behavior.

math.OC↗

arXiv2025

Time-Frequency Filtering Meets Graph Clustering

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 $G=(V,E)$ one aims to identify `clusters', subgraphs that are strongly connected and have relatively few connections between them. The graph clustering problem is well studied, we show how these ideas can suggest (many) new ways to identify signal components. Numerical experiments illustrate the ideas.

eess.SP↗

arXiv2025

Greedy Matching in Optimal Transport with concave cost

We consider the optimal transport problem between a set of $n$ red points and a set of $n$ blue points subject to a concave cost function such as $c(x,y) = \|x-y\|^{p}$ for $0< p < 1$. Our focus is on a particularly simple matching algorithm: match the closest red and blue point, remove them both and repeat. We prove that it provides good results in any metric space $(X,d)$ when the cost function is $c(x,y) = d(x,y)^{p}$ with $0 < p < 1/2$. Empirically, the algorithm produces results that are remarkably close to optimal -- especially as the cost function gets more concave; this suggests that greedy matching may be a good toy model for Optimal Transport for very concave transport cost.

math.CA↗

arXiv2025

Superpolynomial convergence in the Riemann Rearrangement Theorem

Let $x \in \mathbb{R}$ be arbitrary and consider the `greedy' approximation of $x$ by signed harmonic sums: given $a_n = \sum_{k \leq n} \varepsilon_k/k$ with $\varepsilon_k \in \left\{-1,1\right\}$, we set $\varepsilon_{n+1} = 1$ if $a_n \leq x$ and $\varepsilon_{n+1} = -1$ otherwise. Bettin-Molteni-Sanna showed (Adv. Math. 2020) that this procedure has remarkable approximation properties: for almost all $x \in \mathbb{R}$ one has superpolynomial convergence in the sense that for every $k \in \mathbb{N}$ there are infinitely many $n \in \mathbb{N}$ with $|a_n - x| \leq n^{-k}$. We extend this result from $\pm 1 \pm 1/2 \pm 1/3 \dots \pm 1/n$ to moment sequences, i.e. sequences defined as the moments of a measure $μ$ supported on $[0,1]$.

math.DS↗

arXiv2025

Potential Theory and the Boundary of Combinatorial Graphs

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 theory. This is done by proving a number of discrete analogous of classical results for compact domains $Ω\subset \mathbb{R}^d$. These include (1) an analogue of Pólya's result that a random walk in $Ω$ typically hits the boundary $\partial Ω$ within $\lesssim \mbox{diam}(Ω)^2$ units of time, (2) an analogue of the Faber-Krahn inequality, (3) an analogue of the Hardy inequality, (4) an analogue of the Alexandrov-Bakelman-Pucci estimate, (5) a stability estimate for hot spots and (6) a Theorem of Björck stating that probability measures $μ$ that maximize $\int_{Ω\times Ω} \|x-y\|^α dμ(x) dμ(y)$ are fully supported in the boundary.

math.CA↗