Variable Projected Augmented Lagrangian Methods in Modern Differentiable Frameworks Open Access

Abstract

This thesis investigates the Variable Projected Augmented Lagrangian (VPAL) framework for inverse problems and explores its use in modern differentiable settings beyond the classical matrix-based case. After reviewing the background on inverse problems, regularization, ADMM, variable projection, and VPAL, a PyTorch-based implementation is developed using automatic differentiation. This allows VPAL to be applied when the forward operator is represented implicitly, including latent-space reconstruction through a decoder and learned forward operators. The method is tested on latent-space reconstruction problems and a time-conditioned heat-diffusion inverse problem. The experiments show that VPAL can be successfully implemented in these settings. Latent-space reconstruction gives better results than direct image-space reconstruction, while experiments with learned forward operators show that reconstruction quality is affected by forward-model mismatch.

Table of Contents

Introduction

Background

Approach

Experiments

Conclusion

Appendix A: Code Availability

Bibliography

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Applied Mathematics
Keyword	Inverse Problems Latent-Space Reconstruction Generalized Lasso Differentiable Forward Models Variable Projected Augmented Lagrangian (VPAL)
Committee Chair / Thesis Advisor	Chung, Matthias, Emory University
Committee Members	Roth, Robert, Emory University Manetta, Manuela, Emory University

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