Manifold-Aligned Generative Transport

Abstract: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.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2602.19600 [stat.ML]

(or arXiv:2602.19600v2 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2602.19600 Focus to learn more arXiv-issued DOI via DataCite

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