  De-noising is a prominent step in the spectra post-processing procedure.
Previous machine learning-based methods are fast but mostly based on supervised
learning and require a training set that may be typically expensive in real
experimental measurements. Unsupervised learning-based algorithms are slow and
require many iterations to achieve convergence. Here, we bridge this gap by
proposing a training-set-free two-stage deep learning method. We show that the
fuzzy fixed input in previous methods can be improved by introducing an
adaptive prior. Combined with more advanced optimization techniques, our
approach can achieve five times acceleration compared to previous work.
Theoretically, we study the landscape of a corresponding non-convex linear
problem, and our results indicates that this problem has benign geometry for
first-order algorithms to converge.
