On a quantum-classical correspondence: from graphs to manifolds

TL;DR

This paper connects graph Laplacians on manifolds to semiclassical pseudodifferential operators, showing how discrete graph-based dynamics approximate continuous geodesic flows with high probability, bridging graph theory and geometric analysis.

Contribution

It establishes conditions under which graph Laplacians are semiclassical pseudodifferential operators and demonstrates their ability to approximate geodesic flows on manifolds using discrete data.

Findings

Graph Laplacians can be viewed as semiclassical pseudodifferential operators under certain conditions.

The geodesic flow on a manifold can be approximated by matrix dynamics derived from graph Laplacians.

High-probability bounds are provided for the approximation accuracy between discrete and continuous dynamics.

Abstract

We establish conditions for which graph Laplacians $\Delta_{\lambda,\epsilon}$ on compact, boundaryless, smooth submanifolds $\mathcal{M}$ of Euclidean space are semiclassical pseudodifferential operators ($\Psi$DOs): essentially, that the graph Laplacian's kernel bandwidth ($\textit{bias term}$) $\sqrt{\epsilon}$ decays faster than the semiclassical parameter $h$, $\textit{i.e.}$, $h \gg \sqrt{\epsilon}$ and we compute the symbol. Coupling this with Egorov's theorem and coherent states $\psi_h$ localized at $(x_0, \xi_0) \in T^*\mathcal{M}$, we show that with $U_{\lambda,\epsilon}^t := e^{-i t \sqrt{\Delta}_{\lambda,\epsilon}}$ spectrally defined, the (co-)geodesic flow $\Gamma^t$ on $T^*\mathcal{M}$ is approximated by $\langle U_{\lambda,\epsilon}^{-t} \operatorname{Op}_h(a) U_{\lambda,\epsilon}^t \psi_h, \psi_h \rangle = a \circ \Gamma^t(x_0, \xi_0) + O(h)$. Then, we turn to the discrete setting: for $\Delta_{\lambda,\epsilon,N}$ a normalized graph Laplacian defined on a set of $N$ points $x_1, \ldots, x_N$ sampled $\textit{i.i.d.}$ from a probability distribution with smooth density, we establish Bernstein-type lower bounds on the probability that $||U_{\lambda,\epsilon,N}^t[u] - U_{\lambda,\epsilon}^t[u]||_{L^{\infty}} \leq \delta$ with $U_{\lambda,\epsilon,N}^t := e^{-i t \sqrt{\Delta}_{\lambda,\epsilon,N}}$. We apply this to coherent states to show that the geodesic flow on $\mathcal{M}$ can be approximated by matrix dynamics on the discrete sample set, namely that $\textit{with high probability}$, $c_{t,N}^{-1} \sum_{j=1}^N |U_{\lambda,\epsilon,N}^t[\psi_h](x_j)|^2 u(x_j) = u(x_t) + O(h)$ for $c_{t,N} := \sum_{j=1}^N |U_{\lambda,\epsilon,N}^t[\psi_h](x_j)|^2$ and $x_t$ the projection of $\Gamma^t(x_0, \xi_0)$ onto $\mathcal{M}$.