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About this paper
Takashi Furuya, Nicholas H. Nelsen, Frank Cole
A theoretical analysis establishes when continuous, measure-dependent pushforwards can uniformly approximate arbitrary continuous operators between probability-measure spaces. Universality holds on compact non-atomic input families satisfying a uniform level-set condition, while atomic inputs create unavoidable obstructions; the result is extended to measure-theoretic transformers and continuously varying source measures.
Key insights
- Deterministic pushforwards are not universal on atomic domains: a Dirac input remains a Dirac measure, and finite atomic inputs cannot generally produce arbitrary reweightings or diffuse outputs.
- The uniform level-set condition provides a continuous uniform latent coordinate inside every input measure; compactness then enables uniform approximation across the entire family.
- Universality concerns approximation, not exact transport representation: the target operator need not itself preserve particles or support, even though the approximating model is a deterministic pushforward.
Summary by alphaXiv.
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