What Does a Stream Model Buy You in Flow Matching?
Jian Xu
Abstract
Stream-level flow matching replaces the linear interpolant of conditional flow matching (CFM) by a Gaussian-process (GP) stream connecting each source--target pair, and reports lower sample error than \icfm{} on 2-Gaussian, MNIST and CIFAR-10 benchmarks. We ask what such a stream model actually contributes. Three results answer the question. (i)~\emph{Reduction.} The stream-level CFM objective depends on the stream law only through the per-time joint law of $(s_t,\sdot_t)$, so the conditional paths a Gaussian stream can reach are exactly the Gaussian conditional paths CFM already parametrises; in the coordinate-wise, shared-scalar-kernel construction gpcfm actually uses, the entire design space collapses to two scalar curves $(m_t,v_t)$, and cross-time covariance affects only estimator variance. (ii)~\emph{The GP is a constrained chart of that space.} One kernel sets both $m_t$ and $v_t$, so the paper's own recipe for widening coverage-shrinking the SE length-scale---destroys the interpolant (the midpoint mean weight falls from $1.03$ to $0.00$). On the 2-Gaussian benchmark this makes the GP chart diverge on $15/200$ runs at high coverage against $0/200$ for a decoupled $(m_t,v_t)$ chart ($p=6.6\times10^{-5}$), and crossing the two curves shows the divergence tracks the mean, not the variance. On MNIST the same sweep does not diverge and the ordering reverses, so whether the coupling is harmful is benchmark-dependent; what holds on both is that the recipe buys nothing---no coverage level beats the paper's own, and past $\max_t\sqrt{v_t}\approx0.6$ both charts degrade. (iii)~\emph{Audit.} The released code does not implement the mechanism it describes: state and velocity are drawn independently ($\mathrm{corr}=0.00\pm0.01$ against an intended $\pm0.83$--$0.99$).