Preconditioned Flow Matching

arXiv:2603.02337v5 Announce Type: replace-cross Abstract: Flow matching (FM) learns vector fields by regressing stochastic velocity targets along intermediate distributions $p_t$. We identify a geometric optimization bottleneck in this regression problem: when the covariance $\Sigma_t$ of $p_t$ is ill-conditioned, gradient-based training rapidly fits high-variance directions while making slow progress along low-variance ones. In an exactly solvable Gaussian setting, we prove that the excess risk is weighted by $\Sigma_t$, and that both gradient descent and stochastic gradient descent inherit c

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