Velocity Ambiguity Profiles: Time-Resolved Bayes-Risk Diagnostics for Flow Matching

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Original abstract

Flow matching trains a continuum of velocity regression problems but usually reports a loss averaged over interpolation time, obscuring which regions are limited by intrinsic ambiguity and which remain data-limited. We introduce the Velocity Ambiguity Profile (VAP), $\mathcal{A}_v(t)$, the time-local Bayes-risk floor of velocity prediction, and decompose time-resolved loss as $L_N(t)=\mathcal{A}_v(t)+R(t;N)$, where only the excess term $R(t;N)$ is reducible by more data or a different estimator. For $N$ training samples and a $K$-component mixture in the structured example, we prove two statistical benchmarks. In an intrinsic local-regression benchmark, $\mathcal{A}_v(t)$ enters the time-dependent noise factor of the standard nonparametric rate. In separated isotropic Gaussian mixtures, the Bayes velocity becomes locally affine, and a label-oracle residual-regression benchmark has parametric excess-risk scale $K\mathcal{A}_v(t)/N$, matched within the same separated residual Gaussian-location subproblem. Controlled Gaussian-mixture experiments validate this diagnostic through a dense transition sweep, calibrated oracle-to-blind and empirical-proxy audits, and a local controlled time-sampler stress test in which a frozen reducible-error proxy reduces reducible loss while cross-cell sensitivity delimits aggressive sampling. On CIFAR-100 DINOv2 feature flow, guided time sampling reduces uniformly evaluated velocity loss by $6.78\%$ and global feature Fréchet distance (FD) by $23.6\%$, while class-conditional kernel inception distance (KID) shows no improvement. VAP turns an averaged flow-matching objective into a time-resolved diagnostic for locating ambiguity-limited and data-limited regions.

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