arXiv subjects

Abraham Rabinowitz

Publications and source records attributed to Abraham Rabinowitz.

3 recordsLinked to original sources

arXiv2023

ICML 2023 Topological Deep Learning Challenge : Design and Results

This paper presents the computational challenge on topological deep learning that was hosted within the ICML 2023 Workshop on Topology and Geometry in Machine Learning. The competition asked participants to provide open-source implementations of topological neural networks from the literature by contributing to the python packages TopoNetX (data processing) and TopoModelX (deep learning). The challenge attracted twenty-eight qualifying submissions in its two-month duration. This paper describes the design of the challenge and summarizes its main findings.

cs.LG↗

arXiv2022

Scaling asymptotics for Szegő kernels on Grauert tubes

Let $M_τ$ be the Grauert tube of radius $τ$ of a closed, real analytic manifold $M$. Associated to the Grauert tube boundary is the orthogonal projection $Π_τ\colon L^2(\partial M_τ) \to H^2(\partial M_τ)$, called the Szegő projector. Let $D_{\sqrtρ}$ denote the Hamilton vector field of the Grauert tube function $\sqrtρ$ acting as a differential operator. We prove scaling asymptotics for the spectral localization kernel of the Toeplitz operator $Π_τD_{\sqrtρ} Π_τ$. We also prove scaling asymptotics for the tempered spectral projections kernel $P_{χ, λ}(z,w) = \sum_{λ_j \le λ} e^{-2τλ_j} ϕ_{λ_j}^\mathbb{C}(z) \overline{ϕ_{λ_j}^\mathbb{C}(w)}$, where $ϕ_{λ_j}^\mathbb{C}$ are analytic extensions to the Grauert tube of Laplace eigenfunctions on $M$.

math.SP↗

arXiv2022

Szegő kernel asymptotics and concentration of Husimi Distributions of eigenfunctions

We work on the boundary $\partial M_τ$ of a Grauert tube of a closed, real analytic Riemannian manifold $M$. The Toeplitz operator $Π_τD_{\sqrtρ} Π_τ$ associated to the Reeb vector field is a positive, self-adjoint, elliptic operator on $H^2(\partial M_τ)$. We compute $λ\to \infty$ asymptotics under parabolic rescaling in a neighborhood of the geodesic (Reeb) flow $G^{t}_τ = \exp tΞ_{\sqrtρ}$ for the spectral projection kernel $Π_{χ, λ}$ associated to $Π_τD_{\sqrtρ} Π_τ$. We also compute scaling asymptotics for tempered sums of Husimi distributions (analytic continuations) on $\partial M_τ$ of Laplace eigenfunctions on $M$. Both asymptotic formulae can be expressed in terms of the metaplectic representation of the linearization of the geodesic flow $G^{t}_τ$ on Bargmann--Fock space. As a corollary, we obtain sharp $L^p \to L^{q}$ norm estimates for $Π_{χ, λ}$ and sharp $L^p$ estimates for Husimi distributions.

math.SP↗