Temporal Source Covariance Matching

Which source covariance should a few-step flow-matching model use? We formulate source selection as a finite-step numerical design problem. For Gaussian population fields under independent coupling and linear interpolation, we prove that covariance matching uniquely minimizes endpoint Wasserstein error for every $N\ge 2$ uniform Euler steps, including non-commuting source covariances. The resulting covariance-only oracle has second-order excess risk near the optimum. Temporal Source Covariance Matching (TSCM) implements this principle with an offline, length-indexed Gaussian source estimated from within-channel temporal second moments and used in both training and sampling, without changing the architecture or objective. On HumanML3D, a controlled comparison of 16 sources with three training seeds each gives mean 24-step FIDs of $0.108$ for TSCM, $0.135$ for uncentered Toeplitz, and $0.497$ for white noise, with source-specific constant guidance selected on validation. Oracle risk is associated with these guided test FIDs (log–log Pearson $r=0.93$). Frozen-field controls also reveal learned-flow differences that persist under reference integration. Further analyses characterize solver-dependent calibration, conditional shared-source regret, and local covariance-deficit minimaxity beyond Gaussian targets. A separate guidance analysis identifies an early covariance bias; matched source replacements in other sequence generators assess empirical transfer.