Flow-based generative models as iterative algorithms in probability space

TL;DR

This paper presents a comprehensive mathematical framework for flow-based generative models, emphasizing their formulation as neural network-driven ODEs for precise density estimation and convergence analysis.

Contribution

It introduces a rigorous, accessible theoretical foundation for flow-based models, connecting empirical methods with mathematical principles like Wasserstein metrics and gradient flows.

Findings

Provides convergence guarantees for flow-based models

Links empirical practices with theoretical insights

Offers a unified mathematical framework for density evolution

Abstract

Generative AI (GenAI) has revolutionized data-driven modeling by enabling the synthesis of high-dimensional data across various applications, including image generation, language modeling, biomedical signal processing, and anomaly detection. Flow-based generative models provide a powerful framework for capturing complex probability distributions, offering exact likelihood estimation, efficient sampling, and deterministic transformations between distributions. These models leverage invertible mappings governed by Ordinary Differential Equations (ODEs), enabling precise density estimation and likelihood evaluation. This tutorial presents an intuitive mathematical framework for flow-based generative models, formulating them as neural network-based representations of continuous probability densities. We explore key theoretical principles, including the Wasserstein metric, gradient flows, and density evolution governed by ODEs, to establish convergence guarantees and bridge empirical advancements with theoretical insights. By providing a rigorous yet accessible treatment, we aim to equip researchers and practitioners with the necessary tools to effectively apply flow-based generative models in signal processing and machine learning.