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Categorical Flow Matching on Statistical Manifolds (2405.16441v2)

Published 26 May 2024 in cs.LG and stat.ML

Abstract: We introduce Statistical Flow Matching (SFM), a novel and mathematically rigorous flow-matching framework on the manifold of parameterized probability measures inspired by the results from information geometry. We demonstrate the effectiveness of our method on the discrete generation problem by instantiating SFM on the manifold of categorical distributions whose geometric properties remain unexplored in previous discrete generative models. Utilizing the Fisher information metric, we equip the manifold with a Riemannian structure whose intrinsic geometries are effectively leveraged by following the shortest paths of geodesics. We develop an efficient training and sampling algorithm that overcomes numerical stability issues with a diffeomorphism between manifolds. Our distinctive geometric perspective of statistical manifolds allows us to apply optimal transport during training and interpret SFM as following the steepest direction of the natural gradient. Unlike previous models that rely on variational bounds for likelihood estimation, SFM enjoys the exact likelihood calculation for arbitrary probability measures. We manifest that SFM can learn more complex patterns on the statistical manifold where existing models often fail due to strong prior assumptions. Comprehensive experiments on real-world generative tasks ranging from image, text to biological domains further demonstrate that SFM achieves higher sampling quality and likelihood than other discrete diffusion or flow-based models.

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Citations (5)
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Summary

  • The paper introduces a flow-matching framework that leverages the Fisher information metric to equip statistical manifolds with a Riemannian structure for discrete generation.
  • It presents an efficient training and sampling algorithm that uses diffeomorphism to address numerical stability, outperforming models on metrics like NLL, FID, and BPC.
  • The study demonstrates SFM's effectiveness on tasks such as Swiss Roll, binarized MNIST, Text8, and promoter design, highlighting its practical impact on generative modeling.

Statistical Flow Matching: A Geometric Approach to Discrete Generative Modeling

The paper presents a novel generative framework, Statistical Flow Matching (SFM), which leverages the intrinsic geometric properties of parameterized probability measures. The authors introduce SFM as a mathematically rigorous flow-matching framework inspired by results from information geometry. This method is particularly applied to discrete generation tasks by instantiating SFM on the manifold of categorical distributions, an area that has not been thoroughly explored in prior models.

Methodological Contributions

The primary contribution of this work is the integration of the Fisher information metric to equip the statistical manifold with a Riemannian structure. This approach allows the model to leverage the manifold's intrinsic geometries by following geodesics, or shortest paths, on this space. The authors develop an efficient training and sampling algorithm that addresses numerical stability issues via a diffeomorphism between manifolds. The framework enables the exact computation of likelihoods for arbitrary probability measures, which contrasts with previous models that often rely on variational bounds.

Numerical Results and Experimental Validation

  1. Swiss Roll on Simplex:
    • The authors demonstrate the effectiveness of SFM on a toy example by projecting the Swiss roll dataset onto a 2-simplex. This example illustrates SFM's ability to capture complex geometric shapes that other models, such as Dirichlet-based methods, fail to represent adequately.
    • The achieved NLL for SFM is notably better than for other models, showcasing its robustness in modeling intricate patterns.
  2. Binarized MNIST:
    • On the binarized MNIST dataset, the authors report both NLL and Fréchet inception distance (FID) to evaluate the model's performance. SFM significantly outperforms other discrete generative models like D3PM and DDSM, both in sample quality and likelihood.
  3. Text8 Dataset:
    • For the Text8 dataset, SFM shows competitive performance in terms of bits-per-character (BPC) against state-of-the-art methods. It even achieves better results than some existing flow-based and diffusion models, demonstrating its applicability in natural language processing tasks.
  4. Promoter Design:
    • SFM is applied to promoter DNA sequence design, a bioinformatics task with practical implications. Measured by the mean squared error (SP-MSE) between the predicted promoter activity of generated sequences and human genome sequences, SFM achieves the lowest SP-MSE compared to other baselines, underscoring its utility in practical computational biology tasks.

Theoretical Implications and Future Directions

The introduction of SFM has several theoretical implications:

  • Riemannian Geometry in Generative Modeling: By considering the Riemannian structure of statistical manifolds, this framework opens new avenues for exploring the geometric properties of probability distributions in generative tasks.
  • Exact Likelihood Calculation: The capability to compute exact likelihoods provides a significant advantage, particularly for models where precise probabilistic interpretation is crucial.
  • Optimal Transport: The application of optimal transport within this framework suggests potential improvements in training efficiency and performance by aligning noise distributions with target samples more effectively.

The paper speculates that future research could extend SFM to other statistical manifolds where closed-form geodesics are available, thus broadening its applicability. Additionally, investigating the application of this geometric approach in more complex generative tasks and higher-dimensional data could be a promising direction.

Conclusion

Statistical Flow Matching presents a novel approach to discrete generative modeling by integrating information geometry and Riemannian structures into the framework. By focusing on the intrinsic geometric properties of statistical manifolds, SFM achieves a higher quality of sampling and likelihood estimation than existing discrete diffusion or flow-based models. The results indicate that considering the true geometry of probability distributions can lead to more accurate and stable generative models, providing a new perspective for future developments in generative AI.

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