arXiv subjects

Stefan Steinerberger

Publications and source records attributed to Stefan Steinerberger.

At least 199 records · Page 11Linked to original sources

arXiv2017

Stochastic Neighbor Embedding separates well-separated clusters

Stochastic Neighbor Embedding and its variants are widely used dimensionality reduction techniques -- despite their popularity, no theoretical results are known. We prove that the optimal SNE embedding of well-separated clusters from high dimensions to any Euclidean space R^d manages to successfully separate the clusters in a quantitative way. The result also applies to a larger family of methods including a variant of t-SNE.

stat.ML↗

arXiv2017

A Spectral Gap Estimate and Applications

We consider the Schrödinger operator $$-\frac{d^2}{d x^2} + V \qquad \mbox{on an interval}~~[a,b]~\mbox{with Dirichlet boundary conditions},$$ where $V$ is bounded from below and prove a lower bound on the first eigenvalue $λ_1$ in terms of sublevel estimates: if $ w_V(y) = |I_y|,\text{ where } I_y := \left\{ x \in [a,b]: V(x) \leq y \right\},$ then $$ λ_1 \geq \frac{1}{250} \min_{y > \min V}{\left(\frac{1}{w_V(y)^2} + y\right)}.$$ The result is sharp up to a universal constant if $\left\{ x \in [a,b]: V(x) \leq y \right\}$ is an interval for the value of $y$ solving the minimization problem. An immediate application is as follows: let $Ω\subset \mathbb{R}^2$ be a convex domain with inradius $ρ$ and diameter $D$ and let $u:Ω\rightarrow \mathbb{R}$ be the first eigenfunction of the Laplacian $-Δ$ on $Ω$ with Dirichlet boundary conditions on $\partial Ω$. We prove $$ \| u \|_{L^{\infty}} \lesssim \frac{1}{ρ^{}} \left( \fracρ{D} \right)^{1/6} \|u\|_{L^2},$$ which answers a question of van den Berg in the special case of two dimensions.

math.SP↗

arXiv2017

On the location of maximal of solutions of Schrödinger's equation

We prove an inequality with applications to solutions of the Schrödinger equation. There is a universal constant $c>0$, such that if $Ω\subset \mathbb{R}^2$ is simply connected, $u:Ω\rightarrow \mathbb{R}$ vanishes on the boundary $\partial Ω$, and $|u|$ assumes a maximum in $x_0 \in Ω$, then $$ \inf_{y \in \partial Ω}{ \| x_0 - y\|} \geq c \left\| \frac{Δu}{u} \right\|^{-1/2}_{L^{\infty}(Ω)}.$$ It was conjectured by Pólya \& Szegő (and proven, independently, by Makai and Hayman) that a membrane vibrating at frequency $λ$ contains a disk of size $\sim λ^{-1/2}$. Our inequality implies a refined result: the point on the membrane that achieves the maximal amplitude is at distance $\sim λ^{-1/2}$ from the boundary. We also give an extension to higher dimensions (generalizing results of Lieb and Georgiev \& Mukherjee): if $u$ solves $-Δu = Vu$ on $Ω\subset \mathbb{R}^n$ with Dirichlet boundary conditions, then the ball $B$ with radius $\sim \|V\|_{L^{\infty}(Ω)}^{-1/2}$ centered at the point in which $|u|$ assumes a maximum is almost fully contained in $Ω$ in the sense that $|B \cap Ω| \geq 0.99 |B|.$

math.AP↗

arXiv2016

On the Diffusion Geometry of Graph Laplacians and Applications

We study directed, weighted graphs $G=(V,E)$ and consider the (not necessarily symmetric) averaging operator $$ (\mathcal{L}u)(i) = -\sum_{j \sim_{} i}{p_{ij} (u(j) - u(i))},$$ where $p_{ij}$ are normalized edge weights. Given a vertex $i \in V$, we define the diffusion distance to a set $B \subset V$ as the smallest number of steps $d_{B}(i) \in \mathbb{N}$ required for half of all random walks started in $i$ and moving randomly with respect to the weights $p_{ij}$ to visit $B$ within $d_{B}(i)$ steps. Our main result is that the eigenfunctions interact nicely with this notion of distance. In particular, if $u$ satisfies $\mathcal{L}u = λu$ on $V$ and $$ B = \left\{ i \in V: - \varepsilon \leq u(i) \leq \varepsilon \right\} \neq \emptyset,$$ then, for all $i \in V$, $$ d_{B}(i) \log{\left( \frac{1}{|1-λ|} \right) } \geq \log{\left( \frac{ |u(i)| }{\|u\|_{L^{\infty}}} \right)} - \log{\left(\frac{1}{2} + \varepsilon\right)}.$$ $d_B(i)$ is a remarkably good approximation of $|u|$ in the sense of having very high correlation. The result implies that the classical one-dimensional spectral embedding preserves particular aspects of geometry in the presence of clustered data. We also give a continuous variant of the result which has a connection to the hot spots conjecture.

math.SP↗

arXiv2016

An amusing sequence of functions

We consider the amusing sequence of functions $f_n: \mathbb{R} \rightarrow \mathbb{R}$ given by $$ f_n(x) = \sum_{k=1}^{n}{\frac{|\sin{(k πx)}|}{k}}.$$ Every rational point is eventually the location of a strict local minimum of $f_n$: more precisely, $f_n$ has a strict local minimum in all rational points $x=p/q \in \mathbb{Q}$ with $|q| \leq \sqrt{n}$.

math.HO↗

arXiv2016

Optimal Gabor frame bounds for separable lattices and estimates for Jacobi theta functions

We study sharp frame bounds of Gabor frames with the standard Gaussian window and prove that the square lattice optimizes both the lower and the upper frame bound among all rectangular lattices. This proves a conjecture of Floch, Alard & Berrou (as reformulated by Strohmer & Beaver). The proof is based on refined log-convexity/concavity estimates for the Jacobi theta functions $θ_3$ and $θ_4$.

math.FA↗

arXiv2016

Refined Heinz-Kato-Löwner inequalities

A version of the Cauchy-Schwarz inequality in operator theory is the following: for any two symmetric, positive definite matrices $A,B \in \mathbb{R}^{n \times n}$ and arbitrary $X \in \mathbb{R}^{n \times n}$ $$ \|AXB\| \leq \|A^2 X\|^{\frac{1}{2}} \|X B^2\|^{\frac{1}{2}}.$$ This inequality is classical and equivalent to the celebrated Heinz-Löwner, Heinz-Kato and Cordes inequalities. We characterize cases of equality: in particular, after factoring out the symmetry coming from multiplication with scalars $ \|A^2 X\| = 1 = \|X B^2\|$, the case of equality requires that $A$ and $B$ have a common eigenvalue $λ_i = μ_j$. We also derive improved estimates and show that if either $λ_i λ_j = μ_k^2$ or $λ_i^2 = μ_j μ_k$ does not have a solution, i.e. if $d > 0$ where \begin{align*} d &= \min_{1 \leq i,j,k \leq n} \{ | \log{ λ_i} + \log{ λ_j} - 2\log{ μ_k}|:λ_i, λ_j \in σ(A), μ_k \in σ(B) \} &+\min_{1 \leq i,j,k \leq n}\{ | 2\log{λ_i} - \log{ μ_j} - \log{μ_k } |:λ_i \in σ(A), μ_j, μ_k \in σ(B) \}, \end{align*} then there is an improved inequality $$ \|AXB\| \leq (1 - c_{n,d})\|A^2 X\|^{\frac{1}{2}} \|X B^2\|^{\frac{1}{2}}$$ for some $c_{n,d} > 0$ that only depends only on $n$ and $d$. We obtain similar results for the McIntosh inequality and the Cordes inequality and expect the method to have many further applications.

math.FA↗

arXiv2016

Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots

We study a recent result of Bourgain, Clozel and Kahane, a version of which states that a sufficiently nice function $f:\mathbb{R} \rightarrow \mathbb{R}$ that coincides with its Fourier transform and vanishes at the origin has a root in the interval $(c, \infty)$, where the optimal $c$ satisfies $0.41 \leq c \leq 0.64$. A similar result holds in higher dimensions. We improve the one-dimensional result to $0.45 \leq c \leq 0.594$, and the lower bound in higher dimensions. We also prove that extremizers exist, and have infinitely many double roots. With this purpose in mind, we establish a new structure statement about Hermite polynomials which relates their pointwise evaluation to linear flows on the torus, and applies to other families of orthogonal polynomials as well.

math.CA↗

arXiv2016

Extracting Geography from Trade Data

Understanding international trade is a fundamental problem in economics -- one standard approach is via what is commonly called the "gravity equation", which predicts the total amount of trade $F_ij$ between two countries $i$ and $j$ as $$ F_{ij} = G \frac{M_i M_j}{D_{ij}},$$ where $G$ is a constant, $M_i, M_j$ denote the "economic mass" (often simply the gross domestic product) and $D_{ij}$ the "distance" between countries $i$ and $j$, where "distance" is a complex notion that includes geographical, historical, linguistic and sociological components. We take the \textit{inverse} route and ask ourselves to which extent it is possible to reconstruct meaningful information about countries simply from knowing the bilateral trade volumes $F_{ij}$: indeed, we show that a remarkable amount of geopolitical information can be extracted. The main tool is a spectral decomposition of the Graph Laplacian as a tool to perform nonlinear dimensionality reduction. This may have further applications in economic analysis and provides a data-based approach to "trade distance".

q-fin.TR↗

arXiv2016

Spectral Echolocation via the Wave Embedding

Spectral embedding uses eigenfunctions of the discrete Laplacian on a weighted graph to obtain coordinates for an embedding of an abstract data set into Euclidean space. We propose a new pre-processing step of first using the eigenfunctions to simulate a low-frequency wave moving over the data and using both position as well as change in time of the wave to obtain a refined metric to which classical methods of dimensionality reduction can then applied. This is motivated by the behavior of waves, symmetries of the wave equation and the hunting technique of bats. It is shown to be effective in practice and also works for other partial differential equations -- the method yields improved results even for the classical heat equation.

stat.ML↗

arXiv2016

Well-distributed great circles on S^2

Let $C_1, \dots, C_n$ denote the $1/n-$neighborhood of $n$ great circles on $\mathbb{S}^2$. We are interested in how much these areas have to overlap and prove the sharp bounds $$ \sum_{i, j = 1 \atop i \neq j}^{n}{|C_i \cap C_j|^s} \gtrsim_s \begin{cases} n^{2 - 2s} \qquad &\mbox{if}~0 \leq s < 2 \\ n^{-2} \log{n} \qquad &\mbox{if}~s = 2\\ n^{1- 3s/2} \qquad &\mbox{if}~s > 2. \end{cases} .$$ For $s=1$ there are arrangements for which the sum of mutual overlap is uniformly bounded (for the analogous problem in $\mathbb{R}^2$ the lower bound is $\gtrsim \log{n}$) and there are strong connections to minimal energy configurations of $n$ charged electrons on $\mathbb{S}^2$ (the J. J. Thomson problem).

math.MG↗

arXiv2016

A Hidden Signal in the Ulam sequence

The Ulam sequence is defined as $a_1 =1, a_2 = 2$ and $a_n$ being the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives $$1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, \dots$$ Ulam remarked that understanding the sequence, which has been described as 'quite erratic', seems difficult and indeed nothing is known. We report the empirical discovery of a surprising global rigidity phenomenon: there seems to exist a real $α\sim 2.5714474995\dots$ such that $$\left\{αa_n: n\in \mathbb{N}\right\} \quad \mbox{mod}~2π\quad \mbox{generates an absolutely continuous \textit{non-uniform} measure}$$ supported on a subset of $\mathbb{T}$. Indeed, for the first $10^7$ elements of Ulam's sequence, $$ \cos{\left( 2.5714474995~ a_n\right)} < 0 \qquad \mbox{for all}~a_n \notin \left\{2, 3, 47, 69\right\}.$$ The same phenomenon arises for some other initial conditions $a_1, a_2$: the distribution functions look very different from each other and have curious shapes. A similar but more subtle phenomenon seems to arise in Lagarias' variant of MacMahon's 'primes of measurement' sequence.

math.CO↗

arXiv2016

Carrier frequencies, holomorphy and unwinding

We prove that functions of intrinsic-mode type (a classical models for signals) behave essentially like holomorphic functions: adding a pure carrier frequency $e^{int}$ ensures that the anti-holomorphic part is much smaller than the holomorphic part $ \| P_{-}(f)\|_{L^2} \ll \|P_{+}(f)\|_{L^2}.$ This enables us to use techniques from complex analysis, in particular the \textit{unwinding series}. We study its stability and convergence properties and show that the unwinding series can stabilize and show that the unwinding series can provide a high resolution time-frequency representation, which is robust to noise.

math.NA↗

arXiv2016

Nonlinear phase unwinding of functions

We study a natural nonlinear analogue of Fourier series. Iterative Blaschke factorization allows one to formally write any holomorphic function $F$ as a series which successively unravels or unwinds the oscillation of the function $$ F = a_1 B_1 + a_2 B_1 B_2 + a_3 B_1 B_2 B_3 + \dots$$ where $a_i \in \mathbb{C}$ and $B_i$ is a Blaschke product. Numerical experiments point towards rapid convergence of the formal series but the actual mechanism by which this is happening has yet to be explained. We derive a family of inequalities and use them to prove convergence for a large number of function spaces: for example, we have convergence in $L^2$ for functions in the Dirichlet space $\mathcal{D}$. Furthermore, we present a numerically efficient way to expand a function without explicit calculations of the Blaschke zeroes going back to Guido and Mary Weiss.

math.CA↗

arXiv2016

Stability Estimates for Truncated Fourier and Laplace Transforms

We prove sharp stability estimates for the Truncated Laplace Transform and Truncated Fourier Transform. The argument combines an approach recently introduced by Alaifari, Pierce and the second author for the truncated Hilbert transform with classical results of Bertero, Grünbaum, Landau, Pollak and Slepian. In particular, we prove there is a universal constant $c >0$ such that for all $f \in L^2(\mathbb{R})$ with compact support in $[-1,1]$ normalized to $\|f\|_{L^2[-1,1]} = 1$ $$ \int_{-1}^{1}{|\widehat{f}(ξ)|^2dξ} \gtrsim \left(c\left\|f_x \right\|_{L^2[-1,1]} \right)^{- c\left\|f_x \right\|_{L^2[-1,1]}}$$ The inequality is sharp in the sense that there is an infinite sequence of orthonormal counterexamples if $c$ is chosen too small. The question whether and to which extent similar inequalities hold for generic families of integral operators remains open.

math.CA↗

arXiv2016

Fast Escape in Incompressible Vector Fields

Swimmers caught in a rip current flowing away from the shore are advised to swim orthogonally to the current to escape it. We describe a mathematical principle in a similar spirit. More precisely, we consider flows $γ$ in the plane induced by incompressible vector fields $\textbf{v}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ satisfying $ c_1 < \|v\| < c_2.$ The length $\ell$ a flow curve $\dot γ(t) = \textbf{v}(γ(t))$ until $γ$ leaves a disk of radius 1 centered at the initial position can be as long as $\ell \sim c_2/c_1$. The same is true for the orthogonal flow $\textbf{v}^{\perp} = (-\textbf{v}_2, \textbf{v}_1)$. We show that a combination does strictly better: there always exists a curve flowing first along $\textbf{v}^{\perp}$ and then along $\textbf{v}$ which escapes the unit disk before reaching the length $ \sqrt{4πc_2 / c_1}$. Moreover, if the escape length of $\textbf{v}$ is uniformly $\sim c_2/c_1$, then the escape length of $\textbf{v}^{\perp}$ is uniformly $\sim 1$ (allowing for a fast escape from the current). We also prove an elementary quantitative Poincaré-Bendixson theorem that seems to be new.

math.CA↗

arXiv2016

On Suprema of Autoconvolutions with an Application to Sidon sets

Let $f$ be a nonnegative function supported on $(-1/4, 1/4)$. We show $$ \sup_{x \in \mathbb{R}}{\int_{\mathbb{R}}{f(t)f(x-t)dt}} \geq 1.28\left(\int_{-1/4}^{1/4}{f(x)dx} \right)^2,$$ where 1.28 improves on a series of earlier results. The inequality arises naturally in additive combinatorics in the study of Sidon sets. We derive a relaxation of the problem that reduces to a finite number of cases and yields slightly stronger results. Our approach should be able to prove lower bounds that are arbitrary close to the sharp result. Currently, the bottleneck in our approach is runtime: new ideas might be able to significantly speed up the computation.

math.CO↗

arXiv2016

Directional Poincare inequalities along mixing flows

We provide a refinement of the Poincaré inequality on the torus $\mathbb{T}^d$: there exists a Lebesgue-null set $\mathcal{B} \subset \mathbb{T}^d$ of directions such that for every $α\in \mathcal{B}$ there is a $c_α > 0$ with $$ \|\nabla f\|_{L^2(\mathbb{T}^d)}^{d-1} \| \left\langle \nabla f, α\right\rangle\|_{L^2(\mathbb{T}^d)} \geq c_α\|f\|_{L^2(\mathbb{T}^d)}^{d} \qquad\mbox{for all}~f\in H^1(\mathbb{T}^d)~\mbox{with mean 0.}$$ The derivative $\left\langle \nabla f, α\right\rangle$ does not detect any oscillation in directions orthogonal to $α$, however, for certain $α$ the geodesic flow in direction $α$ is sufficiently ergodic to compensate for that defect. On the two-dimensional torus $\mathbb{T}^2$ the inequality holds for $α= (1, \sqrt{2})$ but fails for $α= (1,e)$. Similar results should hold at a great level of generality on very general domains.

math.CA↗