Flow-Based Generative Models as Iterative Algorithms in Probability Space: An intuitive mathematical framework [Special Issue on the Mathematics of Deep Learning]

Flow-based generative models have emerged as a powerful class of deep generative models, offering exact likelihood estimation, invertible sample transformations, and reliable and efficient sampling, making them particularly well-suited for applications in signal processing, anomaly detection, and structured data synthesis. Unlike diffusion models, which rely on stochastic differential equations (SDEs) for progressive denoising, flow-based models define deterministic transformations governed by ordinary differential equations (ODEs), allowing for faster inference and interpretable probabilistic modeling. This tutorial presents a rigorous mathematical framework for flow-based generative models, positioning them as iterative algorithms in probability space and exploring their connections to optimal transport and Wasserstein gradient flows. We discuss key algorithmic insights, including continuous normalizing flows (CNFs), flow-matching (FM), and distributionally robust optimization (DRO), which enable efficient, high-dimensional generative modeling with theoretical guarantees. We also examine the role of flow-based models in progressive training schemes, demonstrating their convergence properties and generative guarantees under structured optimization frameworks. By bridging mathematical theory with practical implementation, this tutorial aims to provide researchers and practitioners with a comprehensive foundation in flow-based generative modeling and its applications in signal processing and beyond.