  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 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 $W^{(\alpha)}_{ij} = k_0( \frac{ \|
x_i - x_j \|^2}{ \epsilon \hat{\rho}(x_i)
\hat{\rho}(x_j)})/\hat{\rho}(x_i)^\alpha \hat{\rho}(x_j)^\alpha$, 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 $C^0$
consistency for $\hat{\rho}$ which bounds the relative estimation error
$|\hat{\rho} - \bar{\rho}|/\bar{\rho}$ uniformly with high probability, where
$\bar{\rho} = p^{-1/d}$, 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.
