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Michael Thompson McCann

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TMLR Journal 2025 Journal Article

Are We Really Learning the Score Function? Reinterpreting Diffusion Models Through Wasserstein Gradient Flow Matching

An Vuong
Michael Thompson McCann
Javier E. Santos
Yen Ting Lin

Diffusion models are commonly interpreted as learning the score function, i.e., the gradient of the log-density of noisy data. However, this learning target is a conservative vector field (i.e., a vector field that is the gradient of some function), a property not enforced by neural network architectures used in practice. We show numerically that trained diffusion networks violate both the integral and differential constraints that conservative vector fields must satisfy, indicating that the learned vector fields are not score functions of any density. Despite this, the models perform remarkably well as generative mechanisms. To explain this paradox, we propose a new theoretical perspective: diffusion training is better understood as \emph{flow matching} to the velocity field of a Wasserstein Gradient Flow (WGF), rather than as score learning for a reverse-time stochastic differential equation. Under this view, the "probability flow" arises naturally from the WGF framework, eliminating the need to invoke reverse-time SDE theory and clarifying why generative sampling remains successful, even when the neural vector field is not a true score. We further show that non-conservative errors from neural approximation do not necessarily harm density transport. Our results advocate adopting the WGF perspective as a principled, elegant, and theoretically grounded framework for understanding diffusion generative models.

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ICLR Conference 2023 Conference Paper

Diffusion Posterior Sampling for General Noisy Inverse Problems

Hyungjin Chung
Jeongsol Kim
Michael Thompson McCann
Marc Louis Klasky
Jong Chul Ye

Diffusion models have been recently studied as powerful generative inverse problem solvers, owing to their high quality reconstructions and the ease of combining existing iterative solvers. However, most works focus on solving simple linear inverse problems in noiseless settings, which significantly under-represents the complexity of real-world problems. In this work, we extend diffusion solvers to efficiently handle general noisy (non)linear inverse problems via the Laplace approximation of the posterior sampling. Interestingly, the resulting posterior sampling scheme is a blended version of diffusion sampling with the manifold constrained gradient without a strict measurement consistency projection step, yielding a more desirable generative path in noisy settings compared to the previous studies. Our method demonstrates that diffusion models can incorporate various measurement noise statistics such as Gaussian and Poisson, and also efficiently handle noisy nonlinear inverse problems such as Fourier phase retrieval and non-uniform deblurring.

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