Manifold-Aligned Generative Transport

TL;DR

MAGT introduces a flow-like generative model that learns a manifold-aligned transport in a single step, improving sampling speed and fidelity by focusing on data manifold structure.

Contribution

It proposes MAGT, a novel one-shot, manifold-aligned transport method that combines the advantages of flows and diffusion models, with stable training and likelihood evaluation.

Findings

Samples faster than diffusion models

Achieves higher fidelity on benchmark datasets

Concentrates probability near data manifolds

Abstract

High-dimensional generative modeling is fundamentally a manifold-learning problem: real data concentrate near a low-dimensional structure embedded in the ambient space. Effective generators must therefore balance support fidelity -- placing probability mass near the data manifold -- with sampling efficiency. Diffusion models often capture near-manifold structure but require many iterative denoising steps and can leak off-support; normalizing flows sample in one pass but are limited by invertibility and dimension preservation. We propose MAGT (Manifold-Aligned Generative Transport), a flow-like generator that learns a one-shot, manifold-aligned transport from a low-dimensional base distribution to the data space. Training is performed at a fixed Gaussian smoothing level, where the score is well-defined and numerically stable. We approximate this fixed-level score using a finite set of latent anchor points with self-normalized importance sampling, yielding a tractable objective. MAGT samples in a single forward pass, concentrates probability near the learned support, and induces an intrinsic density with respect to the manifold volume measure, enabling principled likelihood evaluation for generated samples. We establish finite-sample Wasserstein bounds linking smoothing level and score-approximation accuracy to generative fidelity, and empirically improve fidelity and manifold concentration across synthetic and benchmark datasets while sampling substantially faster than diffusion models.