# 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/en/arxiv-id/2006.13070v1/

## Key points

- Authors address the challenge of leveraging the underlying structure of real-world data to improve generative algorithms
- Focus on facial images as an example of data hypothesized to lie on a low-dimensional manifold
- Normalizing flows have limitations in learning lower-dimensional representations due to successive variable transformations
- Noisy injective flows (NIF) introduced as a generalization of normalizing flows to effectively operate across dimensions
- NIF explicitly maps latent space to a learnable manifold within high-dimensional data space using injective transformations
- Additive noise model incorporated into NIF to accommodate deviations from the manifold and establish a stochastic inverse of the generative process
- Implementation of NIF within existing flow architectures leads to significant improvements in sample quality and facilitates generation of separable data embeddings

## Summary

In their paper titled "Normalizing Flows Across Dimensions," authors Edmond Cunningham, Renos Zabounidis, Abhinav Agrawal, Madalina Fiterau, and Daniel Sheldon address the challenge of leveraging the underlying structure of real-world data to improve generative algorithms. The authors focus on facial images as an example of data that is hypothesized to lie on a low-dimensional manifold. While state-of-the-art generative algorithms have been developed to learn low-dimensional representations based on this hypothesis, the popular model known as normalizing flows has limitations in taking advantage of this structure. Normalizing flows rely on successive variable transformations that are inherently unable to learn lower-dimensional representations. To overcome this limitation, the authors introduce noisy injective flows (NIF) as a generalization of normalizing flows that can effectively operate across dimensions. NIF works by explicitly mapping the latent space to a learnable manifold within a high-dimensional data space using injective transformations. Additionally, an additive noise model is incorporated into NIF to accommodate deviations from the manifold and establish a stochastic inverse of the generative process. Through empirical demonstrations, the authors show that implementing their method within existing flow architectures leads to significant improvements in sample quality and facilitates the generation of separable data embeddings. By introducing NIF as an innovative approach to extend normalizing flows across dimensions while addressing deviations from underlying manifolds, this research contributes valuable insights to the field of generative algorithms and data representation techniques.