Many modern generative modeling methods, including diffusion models, normalizing flows, and flow matching, estimate transport maps or plans between distributions without explicitly targeting an optimal transport (OT) map. In applications like generative modeling, the transport cost itself is irrelevant, and this makes it natural to target maps which are more tractable from either a statistical or computational standpoint. In this short note, we formalize the task of estimating any valid transport map in a rigorous minimax framework. One consequence of this framing is that it yields sample complexity lower bounds for any method whose learned object is evaluated as a transport map or plan, including flow matching and diffusion-based generative models, in settings where direct analysis would be challenging due to the analytic complexity of the methods and their target maps. We observe that, under standard, though strong, stability assumptions from the OT literature, estimating any valid transport map is statistically as hard as estimating the OT map. We complement these results with some examples showing that when these stability assumptions fail, alternative transport maps can be learned substantially more accurately than the OT map. Our minimax framing provides a rigorous foundation for understanding the statistical limits of modern transport-based generative methods and clarifies when targeting sub-optimal maps can provide real statistical advantages.
Many high-dimensional datasets concentrate near a low-dimensional structure embedded in the ambient space. Generative models for such data must control off-support mass while remaining computationally practical. Diffusion models use iterative denoising at inference, whereas standard normalizing flows require invertible, dimension-preserving maps. We propose MAGT (Manifold-Aligned Generative Transport), a direct transport from a low-dimensional base distribution to the data space. Its core objective compares the data and generator-induced scores at a selected Gaussian smoothing level. A posterior identity expresses this score through a latent conditional mean, which is approximated by self-normalized importance sampling over a finite anchor set. After training, generation requires one evaluation of the transport, whose image also carries an intrinsic density with respect to manifold volume. We establish a minimax-optimal Wasserstein convergence rate for an explicitly constructed localized spline-RePU coordinate transport estimator, and treat finite-anchor approximation separately. Experiments on synthetic, image, and tabular benchmarks compare fidelity, support alignment, and sampling cost with diffusion, flow-matching, and adversarial baselines.
These notes recapitulate the high level mathematical principles behind different techniques for generative modeling. I show the connections between optimal transport and standard techniques such as Schr{ö}dinger bridge and flow matching.
Flow matching models learn to transport samples from a simple prior distribution to a complex data distribution. When prior-data pairs are coupled via optimal transport (OT), the learned trajectories are straight and non-crossing, enabling fast, even single-step, generation. However, computing the OT coupling in high dimensions is intractable, and existing methods attempt to solve the OT problem, at the cost of persistent bias or significant overhead. Rather than solving for the OT coupling, we reformulate the problem. Once the prior is treated as a design choice rather than a fixed input, the OT coupling between prior and data is no longer unique. Many priors admit an OT-optimal identity coupling to the data, leaving us free to choose one that is also tractable to sample. We identify low-frequency projection of natural images as such a choice. The identity coupling between data and its low-frequency representation is empirically OT-optimal, the prior is structured enough to be sampled by a lightweight model at inference, and the remaining flow-matching task reduces to synthesizing high-frequency detail. Interpolating the prior with Gaussian noise further improves generation quality while preserving the OT coupling. The approach requires no modifications to the flow model itself, and integrates naturally with latent-space models, classifier-free guidance, and one-step generation frameworks. Across all benchmarks, our method reduces trajectory curvature by more than 2× compared to existing flow matching methods, yielding better generation quality in the few-step regime.