From Distributions to Stochastic Processes: Neural Approximation of Measure-Valued Maps
Learning mappings between probability distributions arises naturally when inputs and outputs are represented by populations of samples rather than individual observations. We develop an approximation-theoretic framework for distribution-to-distribution learning and extend it to mappings between stochastic processes. For continuous operators on $W_2$-compact families of finite-dimensional probability laws, we establish uniform neural approximation in the 2-Wasserstein metric using finite law statistics, a simplex-valued neural map, and a shared atomic output support that guarantees valid probability measures. We further extend this principle to probability laws on separable Hilbert spaces ...
| Authors | Yichen Wang, Ziyi Wang, Wenlian Lu et al. |
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| Primary category | cs.LG |
| Published | 27 September 2026 |
| Last updated at source | 27 September 2026 |
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