Impact of signal-to-noise ratio and bandwidth on graph Laplacian spectrum from high-dimensional noisy point cloud

We systematically {study the spectrum} of kernel-based graph Laplacian (GL) constructed from high-dimensional and noisy random point cloud in the nonnull setup, where the point cloud is sampled from a low-dimensional geometric object, like a manifold, and corrupted by high-dimensional noise. We quantify how the signal and noise interact over different regimes of signal-to-noise ratio (SNR), and report {the resulting peculiar spectral behavior} of GL. In addition, we explore the choice of kernel bandwidth on the spectrum of GL over different regimes of SNR, which leads to an adaptive choice of bandwidth that coincides with the common practice in real data. This result provides a theoretical support for what practitioner do when the dataset is noisy.