  We systematically study the spectrum of kernel-based graph Laplacian (GL)
constructed from high-dimensional and noisy random point cloud in the nonnull
setup. The problem is motived by studying the model when the clean signal is
sampled from a manifold that is embedded in a low-dimensional Euclidean
subspace, and corrupted by high-dimensional noise. We quantify how the signal
and noise interact over different regions of signal-to-noise ratio (SNR), and
report the resulting peculiar spectral behavior of GL. In addition, we explore
the impact of chosen kernel bandwidth on the spectrum of GL over different
regions of SNR, which lead to an adaptive choice of kernel bandwidth that
coincides with the common practice in real data. This result paves the way to a
theoretical understanding of how practitioners apply GL when the dataset is
noisy.
