Manifold-Aligned Generative Transport

2026-02-23

AI Summary

High-dimensional generative modeling often struggles to simultaneously achieve manifold fidelity and sampling efficiency. This work proposes the Manifold-Aligned Generative Transport (MAGT) model, which learns a single-step transport map from a low-dimensional base distribution to the data manifold. Trained under a fixed Gaussian smoothing level, MAGT leverages latent-space anchors and self-normalized importance sampling to efficiently approximate the score function. It is the first method to enable single-step generation while preserving both manifold concentration and computable intrinsic likelihoods, and it establishes a finite-sample Wasserstein error bound that quantifies the relationship between smoothing levels and generation fidelity. Experiments demonstrate that MAGT significantly improves generation quality and manifold alignment on both synthetic and real-world datasets, with sampling speeds substantially exceeding those of diffusion models.

Technology Category

• Machine Learning: Learning with Manifolds
• Natural Language Processing: Generation
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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.

Problem

• generative modeling
• manifold learning
• sampling efficiency
• support fidelity
• high-dimensional data

Innovation

• manifold-aligned transport
• generative modeling
• score approximation
• one-shot sampling
• intrinsic density