# Normalizing Flows Across Dimensions

**Authors:** Edmond Cunningham, Renos Zabounidis, Abhinav Agrawal, Madalina Fiterau, Daniel Sheldon  
**arXiv:** [2006.13070](https://arxiv.org/abs/2006.13070)  
**Date:** 2020-06-23  
**Summary page:** https://www.summarizepaper.com/fr/arxiv-id/2006.13070v1/

## Abstract

Real-world data with underlying structure, such as pictures of faces, are hypothesized to lie on a low-dimensional manifold. This manifold hypothesis has motivated state-of-the-art generative algorithms that learn low-dimensional data representations. Unfortunately, a popular generative model, normalizing flows, cannot take advantage of this. Normalizing flows are based on successive variable transformations that are, by design, incapable of learning lower-dimensional representations. In this paper we introduce noisy injective flows (NIF), a generalization of normalizing flows that can go across dimensions. NIF explicitly map the latent space to a learnable manifold in a high-dimensional data space using injective transformations. We further employ an additive noise model to account for deviations from the manifold and identify a stochastic inverse of the generative process. Empirically, we demonstrate that a simple application of our method to existing flow architectures can significantly improve sample quality and yield separable data embeddings.