Improved convergence rate of kNN graph Laplacians

In graph-based data analysis, $k$-nearest neighbor ($k$NN) graphs are widely used due to their adaptivity to local data densities. Allowing weighted edges in the graph, the kernelized graph affinity provides a more general type of $k$NN graph where the $k$NN distance is used to set the kernel bandwidth adaptively. In this work, we consider a general class of $k$NN graph where the graph affinity is $W_{ij} = \epsilon^{-d/2} \; k_0 ( \| x_i - x_j \|^2 / \epsilon \phi( \widehat{\rho}(x_i), \widehat{\rho}(x_j) )^2 ) $, with $\widehat{\rho}(x)$ being the (rescaled) $k$NN distance at the point $x$, $\phi$ a symmetric bi-variate function, and $k_0$ a non-negative function on $[0,\infty)$. Under the manifold data setting, where $N$ i.i.d. samples $x_i$ are drawn from a density $p$ on a $d$-dimensional unknown manifold embedded in a high dimensional Euclidean space, we prove the point-wise convergence of the $k$NN graph Laplacian to the limiting manifold operator (depending on $p$) at the rate of $O(N^{-2/(d+6)}\,)$, up to a log factor, when $k_0$ and $\phi$ have $C^3$ regularity and satisfy other technical conditions. This fast rate is obtained when $\epsilon \sim N^{-2/(d+6)}\,$ and $k \sim N^{6/(d+6)}\,$, both at the optimal order to balance the theoretical bias and variance errors. When $k_0$ and $\phi$ have lower regularities, including when $k_0$ is a compactly supported function as in the standard $k$NN graph, the convergence rate degenerates to $O(N^{-1/(d+4)}\,)$. Our improved convergence rate is based on a refined analysis of the $k$NN estimator, which can be of independent interest. We validate our theory by numerical experiments on simulated data.