Speaker
Description
Normalizing Flows (NFs) enable tractable density evaluation by modelling data through invertible neural transformations. However, this reliance on global bijectivity severely restricts their expressiveness when the target distribution lies on a low-dimensional manifold or exhibits complex topology. To overcome this limitation, we introduce Principal Surjective Flows (PSFs), a framework that replaces invertibility with carefully constructed surjective mappings. Leveraging the Smooth Co-area Formula, we derive a principled likelihood expression in which the density transformation is governed by the normal Jacobian, which is equivalently the square root of a Gram determinant formed from the Jacobian’s nonzero singular values. This reformulation enables stable and geometrically meaningful density computation for dimension-reducing generative maps. Through Gaussian examples, we show that integrating densities over fibres reduces to evaluating Gaussians on affine subspaces determined by the row space and nullspace of the map, revealing a direct connection between surjective flows and core numerical linear algebra concepts including low-rank operators, projection matrices, and conditioning. Overall, PSFs provide a mathematically grounded and computationally efficient extension of normalizing flows, broadening their applicability while preserving exact and tractable density estimation.