Convergence of Graph Laplacian with kNN Self-tuned Kernels

Abstract: Kernelized Gram matrix $W$ constructed from data points ${x_i}<em>{i=1}<sup>N$ as $$W</em>{ij}= k_0( \frac{ | x_i - x_j |^2} {\sigma^2} )$$ is widely used in graph-based geometric data analysis and unsupervised learning. An important question is how to choose the kernel bandwidth $\sigma$, and a common practice called self-tuned kernel adaptively sets a $\sigma_i$ at each point $x_i$ by the $k$-nearest neighbor (kNN) distance. When $x_i$'s are sampled from a $d$-dimensional manifold embedded in a possibly high-dimensional space, unlike with fixed-bandwidth kernels, theoretical results of graph Laplacian convergence with self-tuned kernels have been incomplete. This paper proves the convergence of graph Laplacian operator $L_N$ to manifold (weighted-)Laplacian for a new family of kNN self-tuned kernels $$W^{{(\alpha)}_{ij}} = k_0( \frac{ | x_i - x_j |^2}{ \epsilon \hat{\rho}(x_i) \hat{\rho}(x_j)})/\hat{\rho}(x_i)^\alpha \hat{\rho}(x_j)<sup>\alpha$$, where $\hat{\rho}$ is the estimated bandwidth function {by kNN}, and the limiting operator is also parametrized by $\alpha$. When $\alpha = 1$, the limiting operator is the weighted manifold Laplacian $\Delta_p$. Specifically, we prove the point-wise convergence of $L_N f$ and convergence of the graph Dirichlet form with rates. Our analysis is based on first establishing a $C<sup>0$ consistency for $\hat{\rho}$ which bounds the relative estimation error $|\hat{\rho} - \bar{\rho}|/\bar{\rho}$ uniformly with high probability, where $\bar{\rho} = p<sup>{-1/d}$, and $p$ is the data density function. Our theoretical results reveal the advantage of self-tuned kernel over fixed-bandwidth kernel via smaller variance error in low-density regions. In the algorithm, no prior knowledge of $d$ or data density is needed. The theoretical results are supported by numerical experiments on simulated data and hand-written digit image data.