  We propose a kernel-spectral embedding algorithm for learning low-dimensional
nonlinear structures from high-dimensional and noisy observations, where the
datasets are assumed to be sampled from an intrinsically low-dimensional
manifold and corrupted by high-dimensional noise. The algorithm employs an
adaptive bandwidth selection procedure which does not rely on prior knowledge
of the underlying manifold. The obtained low-dimensional embeddings can be
further utilized for downstream purposes such as data visualization, clustering
and prediction. Our method is theoretically justified and practically
interpretable. Specifically, we establish the convergence of the final
embeddings to their noiseless counterparts when the dimension and size of the
samples are comparably large, and characterize the effect of the
signal-to-noise ratio on the rate of convergence and phase transition. We also
prove convergence of the embeddings to the eigenfunctions of an integral
operator defined by the kernel map of some reproducing kernel Hilbert space
capturing the underlying nonlinear structures. Numerical simulations and
analysis of three real datasets show the superior empirical performance of the
proposed method, compared to many existing methods, on learning various
manifolds in diverse applications.
