Flow-matching schedules affect both sampling dynamics and the variance of the regression target.
For centered commuting Gaussians, we show that a direction-dependent schedule decomposes into two independent design choices:
- a variance path, which fully determines the intermediate laws and probability flow,
- a factorization, which leaves this flow unchanged while controlling irreducible regression variance.
On the sampling side, we analyze finite-step Euler accuracy and derive a necessary drift bound for exact N -step sampling, connecting the geodesic and the logarithmic path.
On the training side, for any fixed path, we derive closed-form factorizations that either minimize time-averaged regression variance or make it constant along the path.