Metadata
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). Specifically, the graph Laplacian’s kernel bandwidth (bias term) $\sqrt{\epsilon }$ must decay faster than the semiclassical parameter $h$, 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 $\langle U_{\lambda ,\epsilon }^{-t} \operatorname{Op}_h(a) U_{\lambda ,\epsilon }^t \psi _h, \psi _h \rangle = a \circ \Gamma ^t(x_0, \xi _0) + O(h)$.
In the discrete setting, for a normalized graph Laplacian $\Delta _{\lambda ,\epsilon ,N}$ defined on a set of $N$ points $x_1, \dots , x_N$ sampled i.i.d. from a probability distribution with smooth density, we establish Bernstein–type lower bounds on the probability that $\Vert U_{\lambda ,\epsilon ,N}^t u - U_{\lambda ,\epsilon }^t u \Vert _{L^{\infty }} \le \delta $ with $U_{\lambda ,\epsilon ,N}^t := e^{-i t \sqrt{\Delta }_{\lambda ,\epsilon ,N}}$. We apply these bounds to coherent states to show that the geodesic flow on $\mathcal{M}$ can be approximated by matrix dynamics on the discrete sample set. Specifically, with high probability, the evolved state concentrates at the projected geodesic $x_t$ of $\Gamma ^t(x_0, \xi _0)$, satisfying $c_{t,N}^{-1} \sum _{j=1}^N |U_{\lambda ,\epsilon ,N}^t \psi _h(x_j)|^2 u(x_j) = u(x_t) + O(h)$ for the normalization $c_{t,N} := \sum _{j=1}^N |U_{\lambda ,\epsilon ,N}^t \psi _h(x_j)|^2$.
References
[1] Fractional diffusion maps, Appl. Comput. Harmon. Anal., Volume 54 (2021), pp. 145-175 | DOI | MR | Zbl
[2] The dynamics of the Schrödinger flow from the point of view of semiclassical measures, Spectral geometry (Proceedings of Symposia in Pure Mathematics), Volume 84, American Mathematical Society, 2012, pp. 93-116 | DOI | MR | Zbl
[3] Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Ann. Inst. Fourier, Volume 57 (2007) no. 7, pp. 2465-2523 | DOI | MR | Zbl | Numdam
[4] On square roots of ${M}$-matrices, Linear Algebra Appl., Volume 42 (1982), pp. 119-132 | DOI | MR | Zbl
[5] Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Commun. Math. Phys., Volume 178 (1996) no. 1, pp. 83-105 | MR | Zbl | DOI
[6] Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, 194, Springer, 1971 | DOI | MR | Zbl
[7] A graph discretization of the Laplace–Beltrami operator, J. Spectr. Theory, Volume 4 (2014) no. 4, pp. 675-714 | DOI | MR | Zbl
[8] Convergence of Laplacian eigenmaps (Advances in Neural Information Processing Systems), Volume 19, MIT Press (2006), pp. 129-136
[9] Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J., Volume 111 (2002) no. 2, pp. 223-252 | DOI | MR | Zbl
[10] Applications of Semiclassical Analysis on the Quantized Torus, University of California, Berkeley, USA (2023) | MR
[11] Improved spectral convergence rates for graph Laplacians on $\varepsilon $-graphs and $k$-NN graphs, Appl. Comput. Harmon. Anal., Volume 60 (2022), pp. 123-175 | DOI | MR | Zbl
[12] Spectral embedding norm: looking deep into the spectrum of the graph Laplacian, SIAM J. Imaging Sci., Volume 13 (2020) no. 2, pp. 1015-1048 | DOI | MR | Zbl
[13] Probabilistic Weyl laws for quantized tori, Commun. Math. Phys., Volume 299 (2010) no. 2, pp. 305-334 | DOI | MR | Zbl
[14] Control of eigenfunctions on surfaces of variable curvature, J. Am. Math. Soc., Volume 35 (2022) no. 2, pp. 361-465 | DOI | MR | Zbl
[15] Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, 1999 | DOI | MR | Zbl
[16] Order structure and topological methods in nonlinear partial differential equations. Vol. 1: Maximum principles and applications, Series on Partial Differential Equations and Applications, 2, World Scientific, 2006 | DOI | Zbl | MR
[17] How well does the finite Fourier transform approximate the Fourier transform?, Commun. Pure Appl. Math., Volume 58 (2005) no. 10, pp. 1421-1435 | DOI | MR | Zbl
[18] Distribution-free pointwise consistency of kernel regression estimate, Ann. Stat., Volume 12 (1984) no. 4, pp. 1570-1575 | DOI | MR | Zbl
[19] Error estimates for spectral convergence of the graph Laplacian on random geometric graphs toward the Laplace–Beltrami operator, Found. Comput. Math., Volume 20 (2020) no. 4, pp. 827-887 | DOI | Zbl | MR
[20] A variational approach to the consistency of spectral clustering, Appl. Comput. Harmon. Anal., Volume 45 (2018) no. 2, pp. 239-281 | DOI | Zbl | MR
[21] From graphs to manifolds–weak and strong pointwise consistency of graph Laplacians, Learning Theory (Lecture Notes in Computer Science), Volume 3559, Springer (2005), pp. 470-485 | DOI | Zbl
[22] Geometrical aspects of statistical learning theory, Technische Universität Darmstadt (2005)
[23] Opérateurs de Schrödinger avec champs magnétiques faibles et constants, Sémin. Équ. Dériv. Partielles, Volume 1988-1989 (1989), 12, 11 pages | MR | Zbl | Numdam
[24] The analysis of linear partial differential operators. I: Distribution theory and Fourier analysis, Classics in Mathematics, Springer, 2003 | DOI | MR | Zbl
[25] The analysis of linear partial differential operators. III: Pseudo-differential operators, Classics in Mathematics, Springer, 2007 | DOI | MR | Zbl
[26] Does a quantum particle know the time?, Emerging applications of number theory (The IMA Volumes in Mathematics and its Applications), Volume 109, Springer, 1999, pp. 355-371 | DOI | MR | Zbl
[27] Shining light on data: Geometric data analysis through quantum dynamics (2022) | arXiv
[28] On the semiclassical functional calculus for $h$-dependent functions, Ann. Global Anal. Geom., Volume 52 (2017) no. 1, pp. 57-97 | DOI | MR | Zbl
[29] Diffusion maps and geometric harmonics, Yale University, USA (2004) | MR
[30] Consistency of spectral clustering, Ann. Stat., Volume 36 (2008) no. 2, pp. 555-586 | DOI | MR | Zbl
[31] The Schrödinger flow in a compact manifold: high-frequency dynamics and dispersion, Modern aspects of the theory of partial differential equations (Operator Theory: Advances and Applications), Volume 216, Birkhäuser, 2011, pp. 275-289 | DOI | MR
[32] On square roots of ${M}$-operators, Linear Algebra Appl., Volume 223-224 (1995), pp. 501-520 | DOI | MR | Zbl
[33] Diffusion maps, spectral clustering and reaction coordinates of dynamical systems, Appl. Comput. Harmon. Anal., Volume 21 (2006) no. 1, pp. 113-127 | DOI | Zbl | MR
[34] On the pointwise behavior of semi-classical measures, Commun. Math. Phys., Volume 175 (1996) no. 2, pp. 229-258 | DOI | MR | Zbl
[35] Coherent states and applications in mathematical physics, Theoretical and Mathematical Physics, Springer, 2021 | DOI | Zbl | MR
[36] A remark on Stirling’s formula, Am. Math. Mon., Volume 62 (1955), pp. 26-29 | DOI | MR | Zbl
[37] From graph to manifold Laplacian: the convergence rate, Appl. Comput. Harmon. Anal., Volume 21 (2006) no. 1, pp. 128-134 | DOI | MR | Zbl
[38] How do wave packets spread? Time evolution on Ehrenfest time scales, J. Phys. A. Math. Gen., Volume 45 (2012) no. 21, 215307, 28 pages | DOI | MR | Zbl
[39] Vector diffusion maps and the connection Laplacian, Commun. Pure Appl. Math., Volume 65 (2012) no. 8, pp. 1067-1144 | DOI | MR | Zbl
[40] Pseudodifferential operators, Princeton Mathematical Series, 34, Princeton University Press, 1981 | MR | Zbl | DOI
[41] Tangent space estimation for smooth embeddings of Riemannian manifolds, Inf. Inference, Volume 2 (2013) no. 1, pp. 69-114 | DOI | MR | Zbl
[42] Almost sure Weyl law for quantized tori, Commun. Math. Phys., Volume 378 (2020) no. 2, pp. 1539-1585 | DOI | MR | Zbl
[43] Spectral convergence of diffusion maps: improved error bounds and an alternative normalization, SIAM J. Numer. Anal., Volume 59 (2021) no. 3, pp. 1687-1734 | DOI | MR | Zbl
[44] The FBI transform on compact $\mathscr{C}^\infty $ manifolds, Trans. Am. Math. Soc., Volume 353 (2001) no. 3, pp. 1151-1167 | DOI | MR | Zbl
[45] Semiclassical Analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, 2012 | DOI | MR | Zbl