Convergence analysis for iterative data-driven tight frame construction scheme

Chenglong Bao Department of Mathematics, National University of Singapore, Singapore 117543, Singapore Hui Ji Department of Mathematics, National University of Singapore, Singapore 117543, Singapore Zuowei Shen Department of Mathematics, National University of Singapore, Singapore 117543, Singapore

Numerical Analysis and Scientific Computing mathscidoc:2206.25005

Applied and Computational Harmonic Analysis, 38, (3), 510-523, 2015.5
Sparse modeling/approximation of images plays an important role in image restoration. Instead of using a fixed system to sparsely model any input image, a more promising approach is using a system that is adaptive to the input image. A non-convex variational model is proposed in [1] for constructing a tight frame that is optimized for the input image, and an alternating scheme is used to solve the resulting non-convex optimization problem. Although it showed good empirical performance in image denoising, the proposed alternating iteration lacks convergence analysis. This paper aims at providing the convergence analysis of the method proposed in [1]. We first established the sub-sequence convergence property of the iteration scheme proposed in [1], i.e., there exists at least one convergent sub-sequence and any convergent sub-sequence converges to a stationary point of the minimization problem. Moreover, we showed that the original method can be modified to have sequence convergence, i.e., the modified algorithm generates a sequence that converges to a stationary point of the minimization problem.
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@inproceedings{chenglong2015convergence,
  title={Convergence analysis for iterative data-driven tight frame construction scheme},
  author={Chenglong Bao, Hui Ji, and Zuowei Shen},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220614162348462411366},
  booktitle={Applied and Computational Harmonic Analysis},
  volume={38},
  number={3},
  pages={510-523},
  year={2015},
}
Chenglong Bao, Hui Ji, and Zuowei Shen. Convergence analysis for iterative data-driven tight frame construction scheme. 2015. Vol. 38. In Applied and Computational Harmonic Analysis. pp.510-523. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220614162348462411366.
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