Random Projection Flows for Efficient Manifold Density Estimation

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 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.