Flow-based generative models as iterative algorithms in probability space (2502.13394v1)

Abstract: Generative AI (GenAI) has revolutionized data-driven modeling by enabling the synthesis of high-dimensional data across various applications, including image generation, LLMing, 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.

• The paper introduces a mathematical framework for flow-based generative models by leveraging invertible ODE mappings and optimal transport theory.
• It contrasts discrete-time and continuous-time normalizing flows, emphasizing exact likelihood computation and scalable sampling in high-dimensional spaces.
• An iterative training approach inspired by the JKO scheme facilitates structured convergence, enhancing applications in anomaly detection and probabilistic inference.

A Mathematical Framework for Flow-Based Generative Models in Probability Space

The paper "Flow-based generative models as iterative algorithms in probability space" by Yao Xie and Xiuyuan Cheng presents an in-depth exploration of flow-based generative models within a probabilistic framework. The traditional applications of Generative AI (GenAI) span across various fields, including image synthesis, LLMing, and biomedical image processing. The paper explores flow-based generative models, which are gaining attention due to their unique ability to provide exact likelihood estimation, efficient sampling, and deterministic transformations, contrary to the stochastic approaches used in diffusion models.

Mathematical Foundations

The authors establish a solid mathematical foundation for understanding flow-based generative models through the use of invertible mappings governed by Ordinary Differential Equations (ODEs). These models handle the flow of a data distribution through a sequence of transformations, ensuring invertibility and allowing exact likelihood computations via the continuous change of variables formula. The paper leverages the Wasserstein metric to gauge convergence and optimal transport to maintain tractability across these transformations. By grounding the models in these rigorous mathematical principles, the paper provides a framework for designing and analyzing flow-based generative models with guarantees on convergence and generative capabilities.

Algorithmic Approach

Flow-based models perform transformations on data through a series of invertible mapping steps that are reminiscent of neural network layers, specifically Residual Networks (ResNet). The authors describe two primary types: discrete-time normalizing flows, which align closer to traditional neural models consisting of a series of layers, and continuous-time normalizing flows that utilize the neural ODE approach. This continuous-time aspect allows for scalable computation of transformations in high-dimensional spaces with fewer constraints on network architecture, unlike some prescribed forms in discrete-time normalizing flows which ease the computation of Jacobian determinants.

Through leveraging neural ODEs in continuous normalizing flows, the paper delineates a path towards generating complex data distributions from simple ones, such as Gaussian distributions. The model's flexibility in parameterizing velocity fields across time allows for powerful transformations in high-dimensional and complex datasets.

Iterative Training and Implementation

Implementing flow models as iterative algorithms offers a compelling approach towards achieving convergence to desired distributions. Inspired by the Jordan-Kinderlehrer-Otto (JKO) scheme, the paper presents an iterative framework where each step optimizes a localized version of the model objective, progressively transforming a distribution from data to noise and vice-versa. This facilitates a structured, piecewise training that may be more efficient and theoretically tractable than holistic, end-to-end training methods.

Practical Implications and Future Directions

The implications of this work are substantial in areas requiring accurate modeling of complex, high-dimensional data distributions, including anomaly detection and probabilistic inference. By providing insights into the theoretical underpinnings and computational strategies for flow-based models, this paper supports their expansion into wider applications.

Further research could expand on improving the computational efficiency of sampling and optimization through this framework. Advances leveraging parallel processing or GPU computing could be explored to handle real-world, large-scale datasets with higher dimensionality more effectively.

In conclusion, the paper by Xie and Cheng offers a robust and theoretically grounded perspective on flow-based generative models. By developing a cohesive framework that synergizes mathematical theory with practical algorithm design, the research opens avenues for both fundamental advancements and real-world applications in AI.