Manyfold learning: a geometric framework for the analysis, optimization, and convergence of nonlinearly parametrised models
File(s)
Author(s)
Dogra, Akshunna Shaurya
Type
Thesis
Abstract
Nonlinearly parametrised models, in conjunction with iterative optimization methods, have profitably tackled a rapidly growing set of problems in the recent years, especially in Machine and Deep Learning contexts. Empirical observations hint that a small toolbox of such models and methods leads to sufficient performance across a remarkably large landscape of tasks.
However, the parametric dynamics in these regimes are generically non-trivial, lacking the broad characterizations and convergence guarantees that abound for linearly parametrised models. These gaps hold even if the original problem of interest is nominally \textit{well-behaved} (possesses properties that make it \textit{solvable} in some precise mathematical sense, at least in principle).
We begin by positing a versatile hypothesis on when a problem may be considered nominally well-behaved. We then focus on three critical aspects of using nonlinearly parametrised models to solve such problems: \textbf{(i)} which sets of models inherit sufficient structure from well-behaved problems to ensure tractable optimization dynamics, \textbf{(ii)} how such model sets may be extended in a computationally feasible manner to obtain a sequence of approximations limiting to the true solution, and \textbf{(iii)} what kinds of problems can be shown to fit our hypothesis and when can our framework improve upon the existing models and optimisation methods for them.
However, the parametric dynamics in these regimes are generically non-trivial, lacking the broad characterizations and convergence guarantees that abound for linearly parametrised models. These gaps hold even if the original problem of interest is nominally \textit{well-behaved} (possesses properties that make it \textit{solvable} in some precise mathematical sense, at least in principle).
We begin by positing a versatile hypothesis on when a problem may be considered nominally well-behaved. We then focus on three critical aspects of using nonlinearly parametrised models to solve such problems: \textbf{(i)} which sets of models inherit sufficient structure from well-behaved problems to ensure tractable optimization dynamics, \textbf{(ii)} how such model sets may be extended in a computationally feasible manner to obtain a sequence of approximations limiting to the true solution, and \textbf{(iii)} what kinds of problems can be shown to fit our hypothesis and when can our framework improve upon the existing models and optimisation methods for them.
Version
Open Access
Date Issued
2025-10-02
Date Awarded
2026-03-01
Copyright Statement
Attribution-NonCommercial 4.0 International Licence (CC BY-NC)
License URL
Advisor
Webster, Kevin N.
Lamb, Jeroen S. W.
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)