Convergence of Laplacian Spectra from Random Samples

Convergence of Laplacian spectra from random samples

Eigenvectors and eigenvalues of discrete graph Laplacians are often used for manifold learning and nonlinear dimensionality reduction. It was previously proved by Belkin and Niyogi [3] that the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled ind...

Spectral convergence of the connection Laplacian from random samples

Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as DiffusionMaps and Laplacian Eigenmaps, are often used for manifold learning and nonlinear dimensionality reduction. It was previously shown by Belkin&Niyogi (2007, Convergence of Laplacian eigenmaps, vol. 19. Proceedings of the 2006 Conference on Advances in Neural Information Processing System...

Loose laplacian spectra of random hypergraphs

Let H = (V , E) be an r-uniform hypergraph with the vertex set V and the edge set E. For 1 ≤ s ≤ r/2, we define a weighted graph G(s) on the vertex set (Vs) as follows. Every pair of s-sets I and J is associated with a weight w(I , J), which is the number of edges in H containing I and J if I ∩ J = ∅, and 0 if I ∩ J = ∅. The s-th Laplacian L(s) of H is defined to be the normalized Laplacian of ...

Laplacian Spectral Properties of Graphs from Random Local Samples

The Laplacian eigenvalues of a network play an important role in the analysis of many structural and dynamical network problems. In this paper, we study the relationship between the eigenvalue spectrum of the normalized Laplacian matrix and the structure of ‘local’ subgraphs of the network. We call a subgraph local when it is induced by the set of nodes obtained from a breath-first search (BFS)...

Mesoscopic structures and the Laplacian spectra of random geometric graphs