An Explicit Surrogate for Gaussian Mixture Flow Matching with Wasserstein Gap Bounds

Flow matching models are typically trained to transport a Gaussian source to the data distribution, with source and target samples paired independently. We replace the source with a fitted Gaussian mixture and couple each data point to a mixture component through an entropic optimal transport problem. We show that the resulting transport reduces to the standard Gaussian-source problem up to a guidance-like correction, which enables the reuse of a pretrained velocity field. This correction is a closed-form function of the fitted mixture plus a term that we learn with a small correction network. We test our model on published checkpoints for CIFAR-10 and CelebA-HQ. It improves FID at every step count, and on CelebA-HQ it reaches the quality of the baseline with about half the steps. The closed-form term alone, with no training, already delivers most of this improvement; the small correction network accounts for the rest.

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)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)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)~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 $\pm0.83$--$0.99$).

In flow matching (FM), a velocity model $v_θ$ is trained using a predefined path $g_t$ that connects data and noise samples (e.g., $g_t(x_0, x_1) = (1 - t) x_0 + t x_1$). In this work, we study the choice of this path from an optimization perspective by analyzing the variance of stochastic gradients. We consider the class $G(p_t,v^\star_t)$ of paths that induce the same marginal distributions $p_t$ and marginal velocity field $v^\star_t$, and therefore the same FM objective. Our main finding is that the choice of path $g_t$ can fundamentally change the convergence rate of SGD, even when the FM objective remains exactly the same. (i) For a linear velocity model and one-dimensional Gaussian data, we derive a tight bound on the SGD iteration complexity up to logarithmic factors and find an analytically optimal path that minimizes this bound among linear paths inducing the same FM problem. (ii) We then extend the variance analysis to general FM problems and formulate path selection at a fixed $θ$ as the variance-minimization problem PathOpt$_θ$, constrained to $g_t\in G(p_t,v^\star_t)$. We show that this constraint is essential: reducing variance without it can lead to slower convergence. (iii) Since the constraint $g_t \in G(p_t,v^\star_t)$ cannot generally be verified directly, we derive an equivalent formulation with constraints that can be estimated from samples, allowing paths to be found numerically. Our theoretical results are supported by experiments with Gaussian data, Gaussian mixture models, and real datasets.