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Training-set-free two-stage deep learning for spectroscopic data de-noising

Dongchen Huang, Junde Liu, Tian Qian, Hongming Weng

TL;DR

This paper tackles the challenge of denoising spectroscopic data without using training data by introducing a training-set-free, two-stage unsupervised framework. It combines an adaptive prior from principal component pursuit (low-rank plus sparse decomposition with $L+S=I$) and a CNN-based encoder–decoder to model clean spectra, complemented by a small network for sparse noise and trained with an AdamW optimizer using a discrepant learning-rate schedule. Empirically, it achieves substantial speedups (around 4×) and preserves key spectral features on ARPES data (e.g., FeSe at the M point), with second-derivative and MDC analyses showing clearer band structures. The authors also provide theoretical insight, showing a benign landscape for the linearized model where all stationary points are either global minima or strict saddles, supporting efficient convergence and suggesting broader relevance to non-convex optimization in scientific imaging.

Abstract

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.

Training-set-free two-stage deep learning for spectroscopic data de-noising

TL;DR

This paper tackles the challenge of denoising spectroscopic data without using training data by introducing a training-set-free, two-stage unsupervised framework. It combines an adaptive prior from principal component pursuit (low-rank plus sparse decomposition with ) and a CNN-based encoder–decoder to model clean spectra, complemented by a small network for sparse noise and trained with an AdamW optimizer using a discrepant learning-rate schedule. Empirically, it achieves substantial speedups (around 4×) and preserves key spectral features on ARPES data (e.g., FeSe at the M point), with second-derivative and MDC analyses showing clearer band structures. The authors also provide theoretical insight, showing a benign landscape for the linearized model where all stationary points are either global minima or strict saddles, supporting efficient convergence and suggesting broader relevance to non-convex optimization in scientific imaging.

Abstract

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.
Paper Structure (9 sections, 1 theorem, 6 equations, 4 figures)

This paper contains 9 sections, 1 theorem, 6 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Method
  3. Applications
  4. The performance of de-noising
  5. Training dynamics and processing speed
  6. Discussion
  7. The effect of discrepant learning rate
  8. Benign Landscape
  9. Conclusion

Key Result

Proposition 1

Any critical point of the linear model Eq:Simplified model is either a global minimum or a strict saddle point whose Hessian has at least one negative eigenvalue.

Figures (4)

  • Figure 1: The illustration of our two-stage deep learning de-noising algorithm. The first stage is building an adaptive prior for individual spectral via principal component pursuit. The adaptive prior only considers the linear correlation. The resulting prior becomes the input of an encoder-decoder network in the second stage. The sparse noise is also parameterized by a small neural network. Both neural networks are trained to predict the clean spectra and noise simultaneously.
  • Figure 2: Performance of the de-noised results by different methods. (a) ARPES intensity spectra in FeSe thin film at the M point. (i) The low-quality raw spectra used for the following de-noising process. (ii)-(iv) The de-noised results using (ii) Gaussian smoothing method, (iii) Previous CNNs method, and New CNNs method. (b) Second-derivative plots from the corresponding spectra in panel (a). (c). Momentum distribution curves (MDCs) from the corresponding spectra in panel (a).
  • Figure 3: De-noised results of FeSe thin film at the M point under different iteration steps and the training curve at different stages. (a) De-noised results under four checkpoints corresponding to the iteration step 0, 200, 1000, and 2000. (b) The corresponding noise of panel (a). (c) Validation loss as a function of the iteration number. The blue curve and red curve indicate our new method and previous method, respectively. After around 2000 interactions, the loss of our new method converges which is faster than the previous method.
  • Figure 4: De-noised results under different noise learning rate ratios. (a) The raw data of Bi2212 along the nodal cut. (b) The de-noised data based on the original data in panel (a) using different noise learning rate ratios $\eta_a/\eta_s = 0.05,0.5,8$. (c) The corresponding noise of panel (b).

Theorems & Definitions (1)

  • Proposition 1