Gaussian Flow-Matching Schedules: Implications for Sampling and Training

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, and a factorization, which leaves this flow unchanged while controlling irreducible regression variance. On the sampling side, we characterize 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.