Geometric and functional representations of stochastic neural dynamical systems: from realization theory to controlled approximation
Veeravalli, Tanya
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https://hdl.handle.net/2142/129887
Description
Title
Geometric and functional representations of stochastic neural dynamical systems: from realization theory to controlled approximation
Author(s)
Veeravalli, Tanya
Issue Date
2025-07-18
Director of Research (if dissertation) or Advisor (if thesis)
Raginsky, Maxim
Doctoral Committee Chair(s)
Raginsky, Maxim
Committee Member(s)
Srikant, Rayadurgam
Belabbas, Mohamed Ali
Zhao, Zhizhen
Department of Study
Electrical & Computer Eng
Discipline
Electrical & Computer Engr
Degree Granting Institution
University of Illinois Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Optimal Control
Neural Dynamical Systems
Stochastic Realization Theory
Nonlinear Controllability
Function Approximation
Language
eng
Abstract
There has been a great deal of interest in understanding continuous-time processes in deep learning, in particular methods related to (stochastic) control for improving diffusion models. In this work, we explore various facets of function approximation and realization problems through the lens of dynamical systems theory, neural stochastic differential equations (neural SDEs), and differential geometry. A neural SDE is an Itô diffusion process whose drift and diffusion matrices are elements of some parametric families. We cover topics from estimating the transition density of both uniformly elliptic and possibly degenerate diffusion processes by leveraging tools from sub-Riemannian geometry and stochastic control. There are many nuanced insights we can get into the behavior of deep neural networks and diffusion models by studying properties of associated problems in optimal control theory and drawing on other tools from the rich mathematical physics literature. The geometric insights explain the underlying noise structure and controllability properties of a stochastic dynamical system while also explaining the expressive power of the stochastic system.
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