2021/12/20 by Kumar, Akshat
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2112.10748
We establish conditions for which graph Laplacians Δλ,ε on compact, boundaryless, smooth submanifolds M of Euclidean space are semiclassical pseudodifferential operators (ΨDOs): essentially, that the graph Laplacian's kernel bandwidth (bias term) √ε decays faster than the semiclassical parameter h, i.e., h ≫ √ε and we compute the symbol. Coupling this with Egorov's theorem and coherent states ψh localized at (x0, ξ0) ∈ T^*M, we show that with Uλ,εt := e^-i t √Δλ,ε spectrally defined, the (co-)geodesic flow Γt on T^*M is approximated by ⟨ Uλ,ε-t Oph(a) Uλ,εt ψh, ψh ⟩ = a ∘ Γt(x0, ξ0) + O(h). Then, we turn to the discrete setting: for Δλ,ε,N a normalized graph Laplacian defined on a set of N points x1, …, xN sampled i.i.d. from a probability distribution with smooth density, we establish Bernstein-type lower bounds on the probability that ||Uλ,ε,Nt[u] - Uλ,εt[u]||L∞ ≤ δ with Uλ,ε,Nt := e^-i t √Δλ,ε,N. We apply this to coherent states to show that the geodesic flow on M can be approximated by matrix dynamics on the discrete sample set, namely that with high probability, ct,N-1 ∑j=1N |Uλ,ε,Nt[ψh](xj)|2 u(xj) = u(xt) + O(h) for ct,N := ∑j=1N |Uλ,ε,Nt[ψh](xj)|2 and xt the projection of Γt(x0, ξ0) onto M.