🤖 AI Summary
This paper addresses the problem of efficient and theoretically sound density estimation on injective manifolds. We propose Random Projection Flows (RPFs), a class of invertible generative models that project high-dimensional data onto lower-dimensional manifolds via random semi-orthogonal matrices drawn from the Haar distribution, enabling exact, differentiable, and reversible dimensionality reduction. Leveraging Riemannian geometry, we derive a closed-form volume correction term for the change of variables, yielding a tractable, training-free, plug-and-play probabilistic model. RPFs constitute the first unification of random projection theory with normalizing flows, preserving invertibility and theoretical rigor while drastically reducing computational overhead. Experiments demonstrate competitive performance in generative modeling tasks across diverse benchmarks. By bridging theoretical foundations with practical efficiency, RPFs establish a strong, principled baseline for unsupervised learning.
📝 Abstract
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 PCA-based 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.