Lipschitz regularity of graph Laplacians on random data clouds

Abstract

In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study arises in data analysis in the context of graph-based learning and contains, as important examples, the equations satisfied by graph Laplacian eigenvectors. In particular, we prove high probability interior and global Lipschitz estimates for solutions of graph Poisson equations. Our results can be used to show that graph Laplacian eigenvectors are, with high probability, essentially Lipschitz regular with constants depending explicitly on their corresponding eigenvalues. Our analysis relies on a probabilistic coupling argument of suitable random walks at the continuum level, and an interpolation method for extending functions on random point clouds to the continuum manifold. As a byproduct of our general regularity results, we obtain high probability L ∞ and approximate C 0,1 convergence rates for the convergence of graph Laplacian eigenvectors towards eigenfunctions of the corresponding weighted Laplace-Beltrami operators. The convergence rates we obtain scale like the L 2 -convergence rates established in [13].

Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation

“…The approach was extended to kNN graphs in [7], where the rate of eigenvalue and 2-norm eigenvector convergence was also improved to match the point-wise rate of the ε-graph or kNN graph Laplacians, leading to a rate of Õ(N −1/(d+4) ) when d/2+2 = Ω( log N N ). The same rate was shown for ∞-norm consistency of eigenvectors in [8], combined with Lipschitz regularity analysis of empirical eigenvectors using advanced PDE tools.…”

“…The eigenvalue LB, however, is more difficult, as has been pointed out in [6]. In [6] and following works taking the variational principle approach, the LB analysis is by "interpolating" the empirical eigenvectors to be functions on M. Unlike with the population eigenfunctions which are known to be smooth, there is less property of the empirical eigenvectors that one can use, and any regularity property of these discrete objects is usually non-trivial to obtain [8].…”

“…The operator point-wise convergence has been intensively studied and established in a series of works [19,18,4,10,27], and extended to variant settings, such as different kernel normalizations [23,36] and general class of kernels [31,5,9]. The eigen-convergence, namely how the empirical eigenvalues and eigenvectors converge to the population eigenvalues and eigenfunctions of the manifold Laplacian, is a more subtle issue and has been studied in [4,34,6,35,28,14] (among others) and recently in [32,7,11,8].…”

Boundary Estimation from Point Clouds: Algorithms, Guarantees and Applications

“…Our convergence proofs for the solutions of PDEs on point clouds in Section 5 utilize the maximum principle, building upon previous related works in the field [13,15,42,48,77]. We also expect that recent advances in the studies of PDEs on point clouds [20,21,47] can also be applied in this setting, to obtain, for example, spectral convergence for the Dirichlet graph Laplacian. There are many methods in the numerical analysis literature for solving PDEs on unstructured meshes or point clouds.…”

“…The proof of this is expected to be more involved than Theorem 5.8, since we cannot use the maximum principle to obtain strong discrete stability results. We expect discrete to continuum convergence results to hold for the eigenvector problem (5.24) using the combined variational and PDE methods from [20,21,47]. We show in Figure 10 the first 7 Dirichlet eigenfunctions on the disk computed by solving (5.24) over a graph constructed with n = 10 5 random variables independent and uniformly distributed on the disk.…”