Teacher–student decomposition for the Eulerian two-time denoiser on the simplex

Determine whether there exists a teacher–student decomposition of the Eulerian characterization of the two-time denoiser δ_{s,t}—the simplex-valued clean-data predictor defined by δ_{s,t}(x) = x + (1 − s) v_{s,t}(x), where v_{s,t} is the average velocity of the flow map—that remains entirely on the probability simplex at each token position, thereby enabling a cross-entropy-based training objective analogous to the semigroup formulation.

Background

The paper introduces the two-time denoiser δ{s,t} to reparameterize the flow map into a simplex-valued predictor suitable for cross-entropy training in the language modeling setting, where each token’s output lies on the probability simplex. They translate flow map characterizations (Lagrangian, Eulerian, and semigroup) into conditions on δ{s,t}.

For the semigroup condition, both δ{s,u} and δ{u,t} are simplex-valued and combine convexly, yielding a natural teacher–student formulation. The Lagrangian condition also admits a simplex-valued teacher via the composite denoiser D_t(X_{s,t}(x)). However, for the Eulerian condition, the authors state they were unable to find a teacher–student decomposition that clearly stays on the simplex, leaving an explicit unresolved question about its existence and formulation.