On the use of the Belopol'skaya-Daletskii representation of a diffusion on a Riemann manifold to construct path integrals

What the math gives to ML

The paper provides a geometrically equivariant construction of finite-dimensional diffusion path measures using the Riemannian exponential map and the Belopol'skaya-Daletskii formulation, avoiding coordinate-dependent descriptions of stochastic increments. Its transferable asset is a manifold-valued stochastic transition in which drift and noise are combined in the tangent space and mapped back to the state manifold by \(\operatorname{Exp}_x\), with the metric induced by the inverse diffusion tensor. A strong neural-network application is a coordinate-equivariant stochastic residual layer or diffusion sampler for latent representations constrained to a learned manifold, together with a path-likelihood correction based on the exponential-map volume Jacobian.