arXiv subjects

Stefan Steinerberger

Publications and source records attributed to Stefan Steinerberger.

At least 73 records · Page 4Linked to original sources

arXiv2023

An elementary proof of a lower bound for the inverse of the star discrepancy

A central problem in discrepancy theory is the challenge of evenly distributing points $\left\{x_1, \dots, x_n \right\}$ in $[0,1]^d$. Suppose a set is so regular that for some $\varepsilon> 0$ and all $y \in [0,1]^d$ the sub-region $[0,y] = [0,y_1] \times \dots \times [0,y_d]$ contains a number of points nearly proportional to its volume and $$\forall~y \in [0,1]^d \qquad \left| \frac{1}{n} \# \left\{1 \leq i \leq n: x_i \in [0,y] \right\} - \mbox{vol}([0,y]) \right| \leq \varepsilon,$$ how large does $n$ have to be depending on $d$ and $\varepsilon$? We give an elementary proof of the currently best known result, due to Hinrichs, showing that $n \gtrsim d \cdot \varepsilon^{-1}$.

math.CO↗

arXiv2023

Local sign changes of polynomials

The trigonometric monomial $\cos(\left\langle k, x \right\rangle)$ on $\mathbb{T}^d$, a harmonic polynomial $p: \mathbb{S}^{d-1} \rightarrow \mathbb{R}$ of degree $k$ and a Laplacian eigenfunction $-Δf = k^2 f$ have root in each ball of radius $\sim \|k\|^{-1}$ or $\sim k^{-1}$, respectively. We extend this to linear combinations and show that for any trigonometric polynomials on $\mathbb{T}^d$, any polynomial $p \in \mathbb{R}[x_1, \dots, x_d]$ restricted to $\mathbb{S}^{d-1}$ and any linear combination of global Laplacian eigenfunctions on $ \mathbb{R}^d$ with $d \in \left\{2,3\right\}$ the same property holds for any ball whose radius is given by the sum of the inverse constituent frequencies. We also refine the fact that an eigenfunction $- Δϕ= λϕ$ in $Ω\subset \mathbb{R}^n$ has a root in each $B(x, α_n λ^{-1/2})$ ball: the positive and negative mass in each $B(x,β_n λ^{-1/2})$ ball cancel when integrated against $\|x-y\|^{2-n}$.

math.CA↗

arXiv2023

Some Remarks on the Erdős Distinct Subset Sums Problem

Let $\left\{a_1, \dots, a_n\right\} \subset \mathbb{N}$ be a set of positive integers, $a_n$ denoting the largest element, so that for any two of the $2^n$ subsets the sum of all elements is distinct. Erdős asked whether this implies $a_n \geq c \cdot 2^n$ for some universal $c>0$. We prove, slightly extending a result of Elkies, that for any $a_1, \dots, a_n \in \mathbb{R}_{>0}$ $$ \int_{\mathbb{R}} \left( \frac{\sin{ x}}{ x} \right)^2 \prod_{i=1}^{n} \cos{( a_i x)^2} dx \geq \fracπ{2^{n}}$$ with equality if and only if all subset sums are $1-$separated. This leads to a new proof of the currently best lower bound $a_n \geq \sqrt{2/πn} \cdot 2^n$. The main new insight is that having distinct subset sums and $a_n$ small requires the random variable $X = \pm a_1 \pm a_2 \pm \dots \pm a_n$ to be close to Gaussian in a precise sense.

math.NT↗

arXiv2022

Rearrangement Inequalities on the Lattice Graph

The Polya-Szegő inequality in $\mathbb{R}^n$ states that, given a non-negative function $f:\mathbb{R}^{n} \rightarrow \mathbb{R}_{}$, its spherically symmetric decreasing rearrangement $f^*:\mathbb{R}^{n} \rightarrow \mathbb{R}_{}$ is `smoother' in the sense of $\| \nabla f^*\|_{L^p} \leq \| \nabla f\|_{L^p}$ for all $1 \leq p \leq \infty$. We study analogues on the lattice grid graph $\mathbb{Z}^2$. The spiral rearrangement is known to satisfy the Polya-Szegő inequality for $p=1$, the Wang-Wang rearrangement satisfies it for $p=\infty$ and no rearrangement can satisfy it for $p=2$. We develop a robust approach to show that both these rearrangements satisfy the Polya-Szegő inequality up to a constant for all $1 \leq p \leq \infty$. In particular, the Wang-Wang rearrangement satisfies $\| \nabla f^*\|_{L^p} \leq 2^{1/p} \| \nabla f\|_{L^p}$ for all $1 \leq p \leq \infty$. We also show the existence of (many) rearrangements on $\mathbb{Z}^d$ such that $\| \nabla f^*\|_{L^p} \leq c_d \cdot \| \nabla f\|_{L^p}$ for all $1 \leq p \leq \infty$.

math.FA↗

arXiv2022

The first eigenvector of a distance matrix is nearly constant

Let $x_1, \dots, x_n$ be points in a metric space and define the distance matrix $D \in \mathbb{R}^{n \times n}$ by ${D}_{ij} = d(x_i, x_j)$. The Perron-Frobenius Theorem implies that there is an eigenvector $v \in \mathbb{R}^n_{}$ with non-negative entries associated to the largest eigenvalue. We prove that this eigenvector is nearly constant in the sense that the inner product with the constant vector $\mathbb{1} \in \mathbb{R}^n$ is large $$ \left\langle v, \mathbb{1} \right\rangle \geq \frac{1}{\sqrt{2}} \cdot \| v\|_{\ell^2} \cdot \|\mathbb{1} \|_{\ell^2}$$ and that each entry satisfies $v_i \geq \|v\|_{\ell^2}/\sqrt{4n}$. Both inequalities are sharp.

math.FA↗

arXiv2022

Guessing cards with complete feedback

We consider the following game that has been used as a way of testing claims of extrasensory perception (ESP). One is given a deck of $mn$ cards comprised of $n$ distinct types each of which appears exactly $m$ times: this deck is shuffled and then cards are discarded from the deck one at a time from top to bottom. At each step, a player (whose psychic powers are being tested) tries to guess the type of the card currently on top, which is then revealed to the player before being discarded. We study the expected number $S_{n,m}$ of correct predictions a player can make: one could always guess the exact same type of card which shows that one can achieve $S_{n,m}>m$. We prove that the optimal (non-psychic) strategy is just slightly better than that and find the first order correction when $n, m$ grows at suitable rates. This is very different from the case where $m$ is fixed and $n$ is large (He & Ottolini) and similar to the case of fixed $n$ and $m$ is large (Graham & Diaconis). The case $m=n$ answers a question of Diaconis.

math.PR↗

arXiv2022

An inequality characterizing convex domains

A property of smooth convex domains $Ω\subset \mathbb{R}^n$ is that if two points on the boundary $x, y \in \partial Ω$ are close to each other, then their normal vectors $n(x), n(y)$ point roughly in the same direction and this direction is almost orthogonal to $x-y$ (for `nearby' $x$ and $y$). We prove there exists a constant $c_n > 0$ such that if $Ω\subset \mathbb{R}^n$ is a bounded domain with $C^1-$boundary $\partial Ω$, then $$ \int_{\partial Ω\times \partial Ω} \frac{\left|\left\langle n(x), y - x \right\rangle \left\langle y - x, n(y) \right\rangle \right| }{\|x - y\|^{n+1}}~d σ(x) dσ(y) \geq c_n |\partial Ω|$$ and equality occurs if and only if the domain $Ω$ is convex.

math.CA↗

arXiv2022

Magnetic Schrödinger operators and landscape functions

We study localization properties of low-lying eigenfunctions of magnetic Schrödinger operators $$\frac{1}{2} \left(- i\nabla - A(x)\right)^2 ϕ+ V(x) ϕ= λϕ,$$ where $V:Ω\rightarrow \mathbb{R}_{\geq 0}$ is a given potential and $A:Ω\rightarrow \mathbb{R}^d$ induces a magnetic field. We extend the Filoche-Mayboroda inequality and prove a refined inequality in the magnetic setting which can predict the points where low-energy eigenfunctions are localized. This result is new even in the case of vanishing magnetic field. Numerical examples illustrate the results.

math.AP↗

arXiv2022

Curvature on Graphs via Equilibrium Measures

We introduce a notion of curvature on finite, combinatorial graphs. It can be easily computed by solving a linear system of equations. We show that graphs with curvature bounded below by $K>0$ have diameter bounded by $\mbox{diam}(G) \leq 2/K$ (a Bonnet-Myers theorem), that $\mbox{diam}(G) = 2/K$ implies that $G$ has constant curvature (a Cheng theorem) and that there is a spectral gap $λ_1 \geq K/(2n)$ (a Lichnerowicz theorem). It is computed for several families of graphs and often coincides with Ollivier curvature or Lin-Lu-Yau curvature. The von Neumann minimax theorem features prominently in the proofs.

math.CO↗

arXiv2022

May the force be with you

Modern methods in dimensionality reduction are dominated by nonlinear attraction-repulsion force-based methods (this includes t-SNE, UMAP, ForceAtlas2, LargeVis, and many more). The purpose of this paper is to demonstrate that all such methods, by design, come with an additional feature that is being automatically computed along the way, namely the vector field associated with these forces. We show how this vector field gives additional high-quality information and propose a general refinement strategy based on ideas from Morse theory. The efficiency of these ideas is illustrated specifically using t-SNE on synthetic and real-life data sets.

cs.LG↗

arXiv2022

Eigenvector Phase Retrieval: Recovering eigenvectors from the absolute value of their entries

We consider the eigenvalue problem $Ax = λx$ where $A \in \mathbb{R}^{n \times n}$ and the eigenvalue is also real $λ\in \mathbb{R}$. If we are given $A$, $λ$ and, additionally, the absolute value of the entries of $x$ (the vector $(|x_i|)_{i=1}^n$), is there a fast way to recover $x$? In particular, can this be done quicker than computing $x$ from scratch? This may be understood as a special case of the phase retrieval problem. We present a randomized algorithm which provably converges in expectation whenever $λ$ is a simple eigenvalue. The problem should become easier when $|λ|$ is large and we discuss another algorithm for that case as well.

math.FA↗

arXiv2022

On Combinatorial Properties of Greedy Wasserstein Minimization

We discuss a phenomenon where Optimal Transport leads to a remarkable amount of combinatorial regularity. Consider infinite sequences $(x_k)_{k=1}^{\infty}$ in $[0,1]$ constructed in a greedy manner: given $x_1, \dots, x_n$, the new point $x_{n+1}$ is chosen so as to minimize the Wasserstein distance $W_2$ between the empirical measure of the $n+1$ points and the Lebesgue measure, $$x_{n+1} = \arg\min_x ~W_2\left( \frac{1}{n+1} \sum_{k=1}^{n} δ_{x_k} + \frac{δ_{x}}{n+1}, dx\right).$$ This leads to fascinating sequences (for example: $x_{n+1} = (2k+1)/(2n+2)$ for some $k \in \mathbb{Z}$) which coincide with sequences recently introduced by Ralph Kritzinger in a different setting. Numerically, the regularity of these sequences rival the best known constructions from Combinatorics or Number Theory. We prove a regularity result below the square root barrier.

math.CO↗

arXiv2022

Approximate Solutions of Linear Systems at a Universal Rate

Let $A \in \mathbb{R}^{n \times n}$ be invertible, $x \in \mathbb{R}^n$ unknown and $b =Ax $ given. We are interested in approximate solutions: vectors $y \in \mathbb{R}^n$ such that $\|Ay - b\|$ is small. We prove that for all $0< \varepsilon <1 $ there is a composition of $k$ orthogonal projections onto the $n$ hyperplanes generated by the rows of $A$, where $$k \leq 2 \log\left(\frac{1}{\varepsilon} \right) \frac{ n}{ \varepsilon^{2}}$$ which maps the origin to a vector $y\in \mathbb{R}^n$ satisfying $\| A y - Ax\| \leq \varepsilon \cdot \|A\| \cdot \| x\|$. We note that this upper bound on $k$ is independent of the matrix $A$. This procedure is stable in the sense that $\|y\| \leq 2\|x\|$. The existence proof is based on a probabilistically refined analysis of the Random Kaczmarz method which seems to achieve this rate when solving for $A x = b$ with high likelihood.

math.NA↗

arXiv2022

An Agmon estimate for Schrödinger operators on Graphs

The Agmon estimate shows that eigenfunctions of Schrödinger operators, $ -Δϕ+ V ϕ= E ϕ$, decay exponentially in the `classically forbidden' region where the potential exceeds the energy level $\left\{x: V(x) > E \right\}$. Moreover, the size of $|ϕ(x)|$ is bounded in terms of a weighted (Agmon) distance between $x$ and the allowed region. We derive such a statement on graphs when $-Δ$ is replaced by the Graph Laplacian $L = D-A$: we identify an explicit Agmon metric and prove a pointwise decay estimate in terms of the Agmon distance.

math.SP↗

arXiv2022

Random Walks, Equidistribution and Graphical Designs

Let $G=(V,E)$ be a $d$-regular graph on $n$ vertices and let $μ_0$ be a probability measure on $V$. The act of moving to a randomly chosen neighbor leads to a sequence of probability measures supported on $V$ given by $μ_{k+1} = A D^{-1} μ_k$, where $A$ is the adjacency matrix and $D$ is the diagonal matrix of vertex degrees of $G$. Ordering the eigenvalues of $ A D^{-1}$ as $1 = λ_1 \geq |λ_2| \geq \dots \geq |λ_n| \geq 0$, it is well-known that the graphs for which $|λ_2|$ is small are those in which the random walk process converges quickly to the uniform distribution: for all initial probability measures $μ_0$ and all $k \geq 0$, $$ \sum_{v \in V} \left| μ_k(v) - \frac{1}{n} \right|^2 \leq λ_2^{2k}.$$ One could wonder whether this rate can be improved for specific initial probability measures $μ_0$. We show that if $G$ is regular, then for any $1 \leq \ell \leq n$, there exists a probability measure $μ_0$ supported on at most $\ell$ vertices so that $$ \sum_{v \in V} \left| μ_k(v) - \frac{1}{n} \right|^2 \leq λ_{\ell+1}^{2k}.$$ The result has applications in the graph sampling problem: we show that these measures have good sampling properties for reconstructing global averages.

math.CO↗

arXiv2022

Intrinsic Sparsity of Kantorovich Solutions

Let $X,Y$ be two finite sets of points having $\#X = m$ and $\#Y = n$ points with $μ= (1/m) \sum_{i=1}^{m} δ_{x_i}$ and $ν= (1/n) \sum_{j=1}^{n} δ_{y_j}$ being the associated uniform probability measures. A result of Birkhoff implies that if $m = n$, then the Kantorovich problem has a solution which also solves the Monge problem: optimal transport can be realized with a bijection $π: X \rightarrow Y$. This is impossible when $m \neq n$. We observe that when $m \neq n$, there exists a solution of the Kantorovich problem such that the mass of each point in $X$ is moved to at most $n/\gcd(m,n)$ different points in $Y$ and that, conversely, each point in $Y$ receives mass from at most $m/\gcd(m,n)$ points in $X$.

math.OC↗

arXiv2022

The product of two high-frequency Graph Laplacian eigenfunctions is smooth

In the continuous setting, we expect the product of two oscillating functions to oscillate even more (generically). On a graph $G=(V,E)$, there are only $|V|$ eigenvectors of the Laplacian $L=D-A$, so one oscillates `the most'. The purpose of this short note is to point out an interesting phenomenon: if $ϕ_1, ϕ_2$ are delocalized eigenvectors of $L$ corresponding to large eigenvalues, then their (pointwise) product $ϕ_1 \cdot ϕ_2$ is smooth (in the sense of small Dirichlet energy): highly oscillatory functions have largely matching oscillation patterns.

math.SP↗

arXiv2022

Sums of Distances on Graphs and Embeddings into Euclidean Space

Let $G=(V,E)$ be a finite, connected graph. We consider a greedy selection of vertices: given a list of vertices $x_1, \dots, x_k$, take $x_{k+1}$ to be any vertex maximizing the sum of distances to the existing vertices and iterate: we keep adding the `most remote' vertex. The frequency with which the vertices of the graph appear in this sequence converges to a set of probability measures with nice properties. The support of these measures is, generically, given by a rather small number of vertices $m \ll |V|$. We prove that this suggests that the graph $G$ is at most '$m$-dimensional' by exhibiting an explicit $1-$Lipschitz embedding $ϕ: G \rightarrow \ell^1(\mathbb{R}^m)$ with good properties.

math.CO↗