Date of Award

Summer 8-2026

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics & Statistics

Program/Concentration

Computational and Applied Mathematics

Committee Director

Yuesheng Xu

Committee Member

Frank Liu

Committee Member

Michael Pokojovy

Committee Member

Guohui Song

Committee Member

Ruhai Zhou

Abstract

This dissertation studies two complementary notions of convergence arising in modern neural network models: the convergence of recursively constructed neural network architectures and the convergence of optimization algorithms used for training neural-network-based image restoration models. The first part develops a convergence theory for deep neural networks (DNNs) viewed as recursively generated sequences of functions. Within an Lp framework motivated by statistical learning, sufficient conditions are established under which increasing-depth neural network sequences converge to well-defined limiting functions. The analysis covers both bounded-width and unbounded-width architectures, establishes explicit convergence rates, and motivates a network initialization strategy derived from the convergence conditions. Numerical experiments validate the theoretical predictions and demonstrate the effectiveness of the proposed initialization. The second part extends the analysis to multi-grade deep neural networks (MGDNNs), whose hierarchical construction naturally generates sequences of neural network approximations. Separate convergence theories are developed for free-weight and shared-weight formulations, and sufficient conditions are established for pointwise, uniform, and Lp convergence under general Lipschitz continuous activation functions. These results provide a rigorous characterization of the asymptotic behavior of multi-grade neural architectures as the number of grades increases. The final part considers a multi-grade sparse-guided implicit representation model for training-data-free image restoration. Unlike the preceding chapters, the convergence studied here concerns the optimization algorithm used to compute the network parameters. A fixed-point formulation of the associated proximal alternating algorithm is derived, and convergence to stationary points is established under suitable regularity assumptions. Together, these results develop new theoretical foundations for recursively constructed DNNs, hierarchical MGDNNs, and optimization methods for training-data-free image restoration.

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DOI

10.25777/mjfb-ay06

ISBN

9798193214403

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