用随机投影构建高效流模型，提升低维流形密度估计
Random Projection Flows for Efficient Manifold Density Estimation
- 用随机半正交矩阵投影数据至低维空间，保持几何结构
- 可闭式计算黎曼体积修正项，无需训练额外参数
- 适合高维流形数据建模，作为生成模型基准方法
准确的密度估计对理解复杂高维数据至关重要，但当数据位于或靠近低维流形时，传统方法面临挑战。随机投影提供了一种自然的降维方式，能在近似保持几何结构的同时实现有效密度估计。本文提出随机投影流（RPFs），一种基于随机矩阵理论和随机投影几何的可注入归一化流框架。RPFs使用从高斯矩阵经QR分解得到的海勒分布正交阵，将数据投影至低维潜在空间以定义基分布。与主成分分析流或学习型可注入映射不同，RPFs无需训练、即插即用，并能获得黎曼体积修正项的闭式表达。实验表明，RPFs兼具理论严谨性与实际有效性，为生成建模提供了强大基线，并架起了随机投影理论与归一化流之间的桥梁。
原文摘要 · Abstract (English)
Accurate density estimation is crucial for understanding complex high-dimensional data, but it becomes challenging when the data lies on or near low-dimensional manifolds. Random projections provide a natural way to reduce dimensionality while approximately preserving geometric structure, enabling effective density estimation in these settings. We introduce \emph{Random Projection Flows} (RPFs), a principled framework for injective normalizing flows that leverages tools from random matrix theory and the geometry of random projections. RPFs employ random semi-orthogonal matrices, drawn from Haar-distributed orthogonal ensembles via QR decomposition of Gaussian matrices, to project data into lower-dimensional latent spaces for the base distribution. Unlike principal component analysis flows or learned injective maps, RPFs are plug-and-play, efficient, and yield closed-form expressions for the Riemannian volume correction term. We demonstrate that RPFs are both theoretically grounded and practically effective, providing a strong baseline for generative modeling and a bridge between random projection theory and normalizing flows.
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