Flow Matching with General Discrete Paths: A Kinetic-Optimal Perspective (2412.03487v1)

Abstract: The design space of discrete-space diffusion or flow generative models are significantly less well-understood than their continuous-space counterparts, with many works focusing only on a simple masked construction. In this work, we aim to take a holistic approach to the construction of discrete generative models based on continuous-time Markov chains, and for the first time, allow the use of arbitrary discrete probability paths, or colloquially, corruption processes. Through the lens of optimizing the symmetric kinetic energy, we propose velocity formulas that can be applied to any given probability path, completely decoupling the probability and velocity, and giving the user the freedom to specify any desirable probability path based on expert knowledge specific to the data domain. Furthermore, we find that a special construction of mixture probability paths optimizes the symmetric kinetic energy for the discrete case. We empirically validate the usefulness of this new design space across multiple modalities: text generation, inorganic material generation, and image generation. We find that we can outperform the mask construction even in text with kinetic-optimal mixture paths, while we can make use of domain-specific constructions of the probability path over the visual domain.

• The paper proposes a comprehensive framework for discrete generative models leveraging arbitrary discrete probability paths beyond masked approaches.
• The paper introduces kinetic-optimal velocity formulations by decoupling velocity and probability path optimization in discrete spaces.
• Empirical validation shows improved performance across text, materials, and images, suggesting future flexibility and efficiency in discrete domains.

Summary of "Flow Matching with General Discrete Paths: A Kinetic-Optimal Perspective"

This paper presents a novel approach to designing and optimizing discrete generative models, focusing on generative flows in discrete spaces. The work mainly addresses the challenge of creating flexible discrete-space diffusion and flow models by integrating continuous-time Markov chains (CTMC) to manage discrete probability paths. It emphasizes utilizing kinetic-optimal principles to optimize the velocity fields that drive these discrete generative processes. The significant contributions and findings of this research are elaborated as follows:

1. General Discrete Flow Matching Framework: The authors propose a comprehensive framework that allows for the construction and optimization of discrete generative models, leveraging arbitrary discrete probability paths—referred to as 'corruption processes'. This expands beyond the traditional masked approaches predominately seen in discrete spaces.
2. Kinetic Optimality for Velocity: By optimizing symmetric kinetic energy, the paper introduces specific velocity formulations applicable to any given probability path. This decoupling of probability and velocity is a pivotal advancement, permitting the specification of probability paths based on domain expertise. The research identifies special constructions, labeled as kinetic optimal (KO) mixture probability paths, that optimize kinetic energy in discrete contexts.
3. Velocity and Probability Path Optimization: The proposed methodology distinctly separates the optimization of velocity fields and probability paths. Through a closed-form solution for kinetic energy in discrete spaces, the authors present a new design space where conditional paths can be tailored to specific domains, while maintaining optimal generation kinetics. The kinetic optimization extends to finding optimal probability paths, leading to state-dependent schedulers.
4. Empirical Validation: The effectiveness of this expanded design space is empirically validated across multiple modalities including text, inorganic material, and image generation. For instance, in text generation, kinetic-optimal mixture paths demonstrated superior performance over masked constructions. Additionally, domain-specific probability path configurations showed high versatility and capability to outperform autoregressive models.
5. Implications for Future Developments: The paper's framework suggests broader implications for the generative model landscape, particularly emphasizing flexibility and efficiency in discrete domains. Future developments might revolve around exploring domain-specific expert systems enhanced through tailored probability paths, harnessing the full potential of discrete generative flows. The insights from this paper may spur further research into more complex discrete space applications, expanding computational efficiency and model accuracy in generative AI tasks.

In conclusion, this paper significantly contributes to the ongoing advancement in discrete generative modeling by introducing a structured approach to using kinetic energy optimizations. The potential to decouple probabilities and velocities and to craft specialized paths could lead to substantial improvements in efficiency and applicability of discrete generative flows across diverse data modalities. As AI models continue to grapple with discrete data spaces, this research provides foundational insights and tools for generating discrete data with enhanced precision and flexibility.