Only the gradient component of score errors affects marginal distributions in diffusion models, so L2 error can be arbitrarily large with perfect match; this yields an impossibility result, a gradient-only KL bound, and a Sobolev estimator that correlates better with quality.
A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem.Numerische Mathematik, 84:375–393, 2000
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Diffusion Models Observe Only Gradients: A Geometric Perspective on Score Matching Errors
Only the gradient component of score errors affects marginal distributions in diffusion models, so L2 error can be arbitrarily large with perfect match; this yields an impossibility result, a gradient-only KL bound, and a Sobolev estimator that correlates better with quality.