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Stefan Steinerberger

Publications and source records attributed to Stefan Steinerberger.

At least 55 records · Page 3Linked to original sources
arXiv2024

Randomly Pivoted Partial Cholesky: Random How?

We consider the problem of finding good low rank approximations of symmetric, positive-definite $A \in \mathbb{R}^{n \times n}$. Chen-Epperly-Tropp-Webber showed, among many other things, that the randomly pivoted partial Cholesky algorithm that chooses the $i-$th row with probability proportional to the diagonal entry $A_{ii}$ leads to a universal contraction of the trace norm (the Schatten 1-norm) in expectation for each step. We show that if one chooses the $i-$th row with likelihood proportional to $A_{ii}^2$ one obtains the same result in the Frobenius norm (the Schatten 2-norm). Implications for the greedy pivoting rule and pivot selection strategies are discussed.

math.NA↗
arXiv2024

Sums of square roots that are close to an integer

Let $k \in \mathbb{N}$ and suppose we are given $k$ integers $1 \leq a_1, \dots, a_k \leq n$. If $\sqrt{a_1} + \dots + \sqrt{a_k}$ is not an integer, how close can it be to one? When $k=1$, the distance to the nearest integer is $\gtrsim n^{-1/2}$. Angluin-Eisenstat observed the bound $\gtrsim n^{-3/2}$ when $k=2$. We prove there is a universal $c>0$ such that, for all $k \geq 2$, there exists a $c_k > 0$ and $k$ integers in $\left\{1,2,\dots, n\right\}$ with $$ 0 <\|\sqrt{a_1} + \dots + \sqrt{a_k} \| \leq c_k\cdot n^{-c \cdot k^{1/3}},$$ where $\| \cdot \|$ denotes the distance to the nearest integer. This is a case of the square-root sum problem in numerical analysis where the usual cancellation constructions do not apply: even for $k=3$, constructing explicit examples of integers whose square root sum is nearly an integer appears to be nontrivial.

math.NT↗
arXiv2024

A Note on Approximate Hadamard Matrices

A Hadamard matrix is a scaled orthogonal matrix with $\pm 1$ entries. Such matrices exist in certain dimensions: the Hadamard conjecture is that such a matrix always exists when $n$ is a multiple of 4. A conjecture attributed to Ryser is that no circulant Hadamard matrices exist when $n > 4$. Recently, Dong and Rudelson proved the existence of approximate Hadamard matrices in all dimensions: there exist universal $0< c < C < \infty$ so that for all $n \geq 1$, there is a matrix $A \in \left\{-1,1\right\}^{n \times n}$ satisfying, for all $x \in \mathbb{R}^n$, $$ c \sqrt{n} \|x\|_2 \leq \|Ax\|_2 \leq C \sqrt{n} \|x\|_2.$$ We observe that, as a consequence of the existence of flat Littlewood polynomials, circulant approximate Hadamard matrices exist for all $n \geq 1$.

math.CO↗
arXiv2024

Bad Science Matrices

Inspired by the bad scientist who keeps repeating an experiment 20 times to get a single outcome with $p < 0.05$, we consider matrices $A \in \mathbb{R}^{n \times n}$ whose rows are normalized in $\ell^2$ and for which $2^{-n}\sum_{x \in \left\{-1,1\right\}^n} \|Ax\|_{\ell^{\infty}}$ is large. They correspond to affine transformations of the discrete unit cube to points with, on average, at least one large coordinate. Such matrices can be seen as a collection of fair tests on a fair coin where at least one outcome is typically atypical. We prove that, as $n \rightarrow \infty$, the quantity can scale as $$ \max_{A \in \mathbb{R}^{n \times n}} \frac{1}{2^{n}}\sum_{x \in \left\{-1,1\right\}^n} \|Ax\|_{\ell^{\infty}} = (1+o(1)) \cdot \sqrt{2\log{n}}.$$ We also present candidate maximizers up to dimension $n \leq 8$ which appear to be highly structured and have nice closed-form solutions.

math.FA↗
arXiv2024

Nonlinear recursions on the reals and a problem of Graham

We study sequences $(x_n)_{n=1}^{\infty}$ of reals given by $x_{n+1} = f(x)$ where $$f(x) = x - \sum_{i=1}^{m} \frac{α_i}{x - β_i},$$ where $α_1, \dots, α_m \in \mathbb{R}_{>0}$ and $β_1, \dots, β_m \in \mathbb{R}$ are arbitrary. A special case is $x_{n+1} = x_n - 1/x_n$ due to Ronald Graham for which Chamberland \& Martelli showed that the dynamics is chaotic (topologically conjugate to the doubling map). We prove that the general nonlinear recursion, despite being potentially chaotic, is effective at ensuring that most iterates end up close to one of the poles $β_i$ relatively quickly. More precisely, for a positive proportion of initial values $x \in \mathbb{R}$, the sequence gets very close (distance $\lesssim |x|^{-1}$) to one of the poles $β_i$ within a relatively small ($\lesssim x^2$) number of iteration steps.

math.DS↗
arXiv2024

Maximal polarization for periodic configurations on the real line

We prove that among all 1-periodic configurations $Γ$ of points on the real line $\mathbb{R}$ the quantities $$ \min_{x \in \mathbb{R}} \sum_{γ\in Γ} e^{- πα(x - γ)^2} \quad \text{and} \quad \max_{x \in \mathbb{R}} \sum_{γ\in Γ} e^{- πα(x - γ)^2}$$ are maximized and minimized, respectively, if and only if the points are equispaced and whenever the number of points $n$ per period is sufficiently large (depending on $α$). This solves the polarization problem for periodic configurations with a Gaussian weight on $\mathbb{R}$ for large $n$. The first result is shown using Fourier series. The second result follows from work of Cohn and Kumar on universal optimality and holds for all $n$ (independent of $α$).

math.CA↗
arXiv2023

Rigidity of Kantorovich solutions in discrete Optimal Transport

We study optimal transport plans from $m$ equally weighted points (with weights $1/m$) to $n$ equally weighted points (with weights $1/n$). The Birkhoff-von Neumann Theorem implies that if $m=n$, then the optimal transport plan can be realized by a bijective map: the mass from each $x_i$ is sent to a unique $y_j$. This is impossible when $m \neq n$, however, a certain degree of rigidity prevails. We prove, assuming w.l.o.g. $m < n$, that for generic transport costs the optimal transport plan sends mass from each source $x_i$ to $n/m \leq \mbox{different targets} \leq n/m + m-1$. Moreover, the average target receives mass from $\leq 1 + m/\sqrt{n}$ sources. Stronger results might be true: in experiments, one observes that each source tends to distribute its mass over roughly $n/m +c$ different targets where $c$ appears to be rather small.

math.OC↗
arXiv2023

On the connection between uniqueness from samples and stability in Gabor phase retrieval

Gabor phase retrieval is the problem of reconstructing a signal from only the magnitudes of its Gabor transform. Previous findings suggest a possible link between unique solvability of the discrete problem (recovery from measurements on a lattice) and stability of the continuous problem (recovery from measurements on an open subset of $\mathbb{R}^2$). In this paper, we close this gap by proving that such a link cannot be made. More precisely, we establish the existence of functions which break uniqueness from samples without affecting stability of the continuous problem. Furthermore, we prove the novel result that counterexamples to unique recovery from samples are dense in $L^2(\mathbb{R})$. Finally, we develop an intuitive argument on the connection between directions of instability in phase retrieval and certain Laplacian eigenfunctions associated to small eigenvalues.

math.FA↗
arXiv2023

Discrete Rearrangements and the Polya-Szego Inequality on Graphs

For any $f: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ the symmetric decreasing rearrangement $f^*$ satisfies the Polya-Szegő inequality $\| \nabla f^*\|_{L^p} \leq \| \nabla f\|_{L^p}$. The goal of this paper is to establish analogous results in the discrete setting for graphs satisfying suitable conditions. We prove that if the edge-isoperimetric problem on a graph has a sequence of nested minimizers, then this sequence gives rise to a rearrangement satisfying the Polya-Szegő inequality in $L^1$. This shows, for example, that a specific rearrangement on the grid graph $\mathbb{Z}^2$, going around the origin in a spiral-like manner, satisfies $\| \nabla f^*\|_{L^1} \leq \| \nabla f\|_{L^1}$. The $L^{\infty}-$case is implied by an optimal ordering condition in vertex-isoperimetry. We use these ideas to prove that the canonical rearrangement on the infinite $d-$regular tree satisfies the Polya-Szegő inequality for all $1 \leq p \leq \infty$.

math.CO↗
arXiv2023

Random Growth via Gradient Flow Aggregation

We introduce Gradient Flow Aggregation (GFA), a random growth model. Given a set of existing particles $\left\{x_1, \dots, x_n\right\} \subset \mathbb{R}^2$, a new particle arrives from a random direction at $\infty$ and flows in direction $\nabla E$ where $$ E(x) = \sum_{i=1}^{n} \frac{1}{\|x-x_i\|^α} \qquad \mbox{where} ~0 < α< \infty.$$ The case $α= 0$ will refer to the logarithmic energy $- \sum\log \|x-x_i\|$. Particles stop once they are at distance 1 of one of the existing particles at which point they are added to the set and remain fixed for all time. We prove, under a non-degeneracy assumption, a Beurling-type estimate which, via Kesten's method, can be used to deduce sub-ballistic growth for $0 \leq α< 1$ $$\mbox{diam}(\left\{x_1, \dots, x_n\right\}) \leq c_α \cdot n^{\frac{3 α+1}{2α+ 2}}.$$ This is optimal when $α=0$. The case $α= 0$ leads to a `round' full-dimensional tree. The larger the value of $α$ the sparser the tree. Some instances of the higher-dimensional setting are also discussed.

math.PR↗
arXiv2023

On localization of eigenfunctions of the magnetic Laplacian

Let $Ω\subset \mathbb{R}^d$ and consider the magnetic Laplace operator given by $ H(A) = \left(- i\nabla - A(x)\right)^2$, where $A:Ω\rightarrow \mathbb{R}^d$, subject to Dirichlet eigenfunction. This operator can, for certain vector fields $A$, have eigenfunctions $H(A) ψ= λψ$ that are highly localized in a small region of $Ω$. The main goal of this paper is to show that if $|ψ|$ assumes its maximum in $x_0 \in Ω$, then $A$ behaves `almost' like a conservative vector field in a $1/\sqrtλ-$neighborhood of $x_0$ in a precise sense: we expect localization in regions where $\left|\mbox{curl} A \right|$ is small. The result is illustrated with numerical examples.

math.AP↗
arXiv2023

Single radius spherical cap discrepancy via gegenbadly approximable numbers

A celebrated result of Beck shows that for any set of $N$ points on $\mathbb{S}^d$ there always exists a spherical cap $B \subset \mathbb{S}^d$ such that number of points in the cap deviates from the expected value $σ(B) \cdot N$ by at least $N^{1/2 - 1/2d}$, where $σ$ is the normalized surface measure. We refine the result and show that, when $d \not\equiv 1 ~(\mbox{mod}~4)$, there exists a (small and very specific) set of real numbers such that for every $r>0$ from the set one is always guaranteed to find a spherical cap $C_r$ with the given radius $r$ for which the result holds. The main new ingredient is a generalization of the notion of badly approximable numbers to the setting of Gegenbauer polynomials: these are fixed numbers $ x \in (-1,1)$ such that the sequence of Gegenbauer polynomials $(C_n^λ(x))_{n=1}^{\infty}$ avoids being close to 0 in a precise quantitative sense.

math.CA↗
arXiv2023

A random line intersects $\mathbb{S}^2$ in two probabilistically independent locations

We consider random lines in $\mathbb{R}^3$ (random with respect to the kinematic measure) and how they intersect $\mathbb{S}^2$. It is known that the entry point and the exit point behave like \textit{independent} uniformly distributed random variables. We give a new proof using bilinear integral geometry and use this approach to show that this property is extremely rare: if $K \subset \mathbb{R}^n$ is a bounded, convex domain with smooth boundary with this property (i.e., the intersection points with a random line are independent), then $n=3$ and $K$ is a ball.

math.PR↗
arXiv2023

Polarization and Greedy Energy on the Sphere

We investigate the behavior of a greedy sequence on the sphere $\mathbb{S}^d$ defined so that at each step the point that minimizes the Riesz $s$-energy is added to the existing set of points. We show that for $0<s<d$, the greedy sequence achieves optimal second-order behavior for the Riesz $s$-energy (up to constants). In order to obtain this result, we prove that the second-order term of the maximal polarization with Riesz $s$-kernels is of order $N^{s/d}$ in the same range $0<s<d$. Furthermore, using the Stolarsky principle relating the $L^2$-discrepancy of a point set with the pairwise sum of distances (Riesz energy with $s=-1$), we also obtain a simple upper bound on the $L^2$-spherical cap discrepancy of the greedy sequence and give numerical examples that indicate that the true discrepancy is much lower.

math.CA↗
arXiv2023

Concentration of Hitting Times in Erdös-Rényi graphs

We consider Erdős-Rényi graphs $G(n,p)$ for $0 < p < 1$ fixed and $n \rightarrow \infty$ and study the expected number of steps, $H_{wv}$, that a random walk started in $w$ needs to first arrive in $v$. A natural guess is that an Erdős-Rényi random graph is so homogeneous that it does not really distinguish between vertices and $H_{wv} = (1+o(1)) n$. Löwe-Terveer established a CLT for the Mean Starting Hitting Time suggesting $H_{w v} = n \pm \mathcal{O}(\sqrt{n})$. We prove the existence of a strong concentration phenomenon: $H_{w v}$ is given, up to a very small error of size $\lesssim \sqrt{\log{n}}/\sqrt{n}$, by an explicit simple formula involving only the total number of edges $|E|$, the degree of $v$ and the distance $d(v,w)$.

math.PR↗
arXiv2023

Spectrahedral Geometry of Graph Sparsifiers

We propose an approach to graph sparsification based on the idea of preserving the smallest $k$ eigenvalues and eigenvectors of the Graph Laplacian. This is motivated by the fact that small eigenvalues and their associated eigenvectors tend to be more informative of the global structure and geometry of the graph than larger eigenvalues and their eigenvectors. The set of all weighted subgraphs of a graph $G$ that have the same first $k$ eigenvalues (and eigenvectors) as $G$ is the intersection of a polyhedron with a cone of positive semidefinite matrices. We discuss the geometry of these sets and deduce the natural scale of $k$. Various families of graphs illustrate our construction.

cs.DM↗
arXiv2023

Quadratic Crofton and sets that see themselves as little as possible

Let $Ω\subset \mathbb{R}^2$ and let $\mathcal{L} \subset Ω$ be a one-dimensional set with finite length $L =|\mathcal{L}|$. We are interested in minimizers of an energy functional that measures the size of a set projected onto itself in all directions: we are thus asking for sets that see themselves as little as possible (suitably interpreted). Obvious minimizers of the functional are subsets of a straight line but this is only possible for $L \leq \mbox{diam}(Ω)$. The problem has an equivalent formulation: the expected number of intersections between a random line and $\mathcal{L}$ depends only on the length of $\mathcal{L}$ (Crofton's formula). We are interested in sets $\mathcal{L}$ that minimize the variance of the expected number of intersections. We solve the problem for convex $Ω$ and slightly less than half of all values of $L$: there, a minimizing set is the union of copies of the boundary and a line segment.

math.CA↗
arXiv2023

Graph curvature via resistance distance

Let $G=(V,E)$ be a finite, combinatorial graph. We define a notion of curvature on the vertices $V$ via the inverse of the resistance distance matrix. We prove that this notion of curvature has a number of desirable properties. Graphs with curvature bounded from below by $K>0$ have diameter bounded from above. The Laplacian $L=D-A$ satisfies a Lichnerowicz estimate, there is a spectral gap $λ_2 \geq 2K$. We obtain matching two-sided bounds on the maximal commute time between any two vertices in terms of $|E| \cdot |V|^{-1} \cdot K^{-1}$. Moreover, we derive quantitative rates for the mixing time of the corresponding Markov chain and prove a general equilibrium result.

math.CO↗