Convergence of graph Laplacian with kNN self-tuned kernels

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