Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs
This provides incremental theoretical improvements for researchers in machine learning and data analysis working with graph-based methods on manifolds.
The paper tackles the problem of improving spectral convergence rates for graph-based approximations of Laplace-Beltrami operators from random data, showing that eigenvalues and eigenvectors converge at a rate of O(n^{-1/(m+4)}) up to log factors for optimal graph connectivity, where m is the manifold dimension and n is the number of vertices.
In this paper we improve the spectral convergence rates for graph-based approximations of Laplace-Beltrami operators constructed from random data. We utilize regularity of the continuum eigenfunctions and strong pointwise consistency results to prove that spectral convergence rates are the same as the pointwise consistency rates for graph Laplacians. In particular, for an optimal choice of the graph connectivity $\varepsilon$, our results show that the eigenvalues and eigenvectors of the graph Laplacian converge to those of the Laplace-Beltrami operator at a rate of $O(n^{-1/(m+4)})$, up to log factors, where $m$ is the manifold dimension and $n$ is the number of vertices in the graph. Our approach is general and allows us to analyze a large variety of graph constructions that include $\varepsilon$-graphs and $k$-NN graphs.