提出MAGT模型,单次前向生成并精准贴近数据流形。
Manifold-Aligned Generative Transport
- 基于流形对齐设计单步生成器,结合流形结构优化采样路径。
- 在合成与基准数据集上提升流形聚焦度,采样速度远超扩散模型。
- 支持基于流形体积测度的似然评估,适合需高效高质生成场景。
高维生成建模本质上是流形学习问题:真实数据集中在嵌入在高维空间中的低维结构附近。有效生成器必须平衡支持保真度(将概率质量置于数据流形附近)与采样效率。扩散模型常能捕捉近流形结构,但需多次去噪迭代且易产生离支持样本;归一化流可单步采样,却受限于可逆性与维度保持。我们提出MAGT(Manifold-Aligned Generative Transport),一种类似流的生成器,学习从低维基分布到数据空间的一次性、流形对齐传输。训练在固定高斯平滑水平下进行,此时得分函数定义良好且数值稳定。我们通过有限个潜在锚点结合自归一化重要性采样来近似该固定水平得分,获得可计算目标。MAGT在单次前向传播中采样,概率集中于学习到的支持集,并诱导出相对于流形体积测度的内在密度,实现生成样本的合理似然评估。我们建立了有限样本下关于平滑水平与得分近似精度的Wasserstein边界,连接其与生成保真度的关系;实证表明,在合成与基准数据集上均提升了生成保真度与流形聚焦度,同时采样速度显著快于扩散模型。
原文摘要 · Abstract (English)
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.
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