# Normalizing Flows Across Dimensions

arXiv: https://arxiv.org/abs/2006.13070 · HTML: https://mlsanity.com/paper/2006.13070/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jun 2020.

## Built on by (most cited first, 19 of 19)

- [Rectangular Flows for Manifold Learning](https://mlsanity.com/paper/2106.01413/) (2021)
- [Trumpets: Injective Flows for Inference and Inverse Problems](https://mlsanity.com/paper/2102.10461/) (2021)
- [Tractable Density Estimation on Learned Manifolds with Conformal Embedding Flows](https://mlsanity.com/paper/2106.05275/) (2021)
- [Generalized Normalizing Flows via Markov Chains](https://mlsanity.com/paper/2111.12506/) (2021)
- [Lifting Architectural Constraints of Injective Flows](https://mlsanity.com/paper/2306.01843/) (2023)
- [Universal Joint Approximation of Manifolds and Densities by Simple Injective Flows](https://mlsanity.com/paper/2110.04227/) (2021)
- [Density estimation on low-dimensional manifolds: an inflation-deflation approach](https://mlsanity.com/paper/2105.12152/) (2021)
- [VQ-Flows: Vector Quantized Local Normalizing Flows](https://mlsanity.com/paper/2203.11556/) (2022)
- [COMET Flows: Towards Generative Modeling of Multivariate Extremes and Tail Dependence](https://mlsanity.com/paper/2205.01224/) (2022)
- [Simulation-based Inference for High-dimensional Data using Surjective Sequential Neural Likelihood Estimation](https://mlsanity.com/paper/2308.01054/) (2023)
- [Foundation Models and Transformers for Anomaly Detection: A Survey](https://mlsanity.com/paper/2507.15905/) (2025)
- [Nonlinear Isometric Manifold Learning for Injective Normalizing Flows](https://mlsanity.com/paper/2203.03934/) (2022)
- [Principal Manifold Flows](https://mlsanity.com/paper/2202.07037/) (2022)
- [Deterministic training of generative autoencoders using invertible layers](https://mlsanity.com/paper/2205.09546/) (2022)
- [ManiFlow: Implicitly Representing Manifolds with Normalizing Flows](https://mlsanity.com/paper/2208.08932/) (2022)
- [Joint Manifold Learning and Density Estimation Using Normalizing Flows](https://mlsanity.com/paper/2206.03293/) (2022)
- [Deep Invertible Approximation of Topologically Rich Maps between Manifolds](https://mlsanity.com/paper/2210.00577/) (2022)
- [Generative structured normalizing flow Gaussian processes applied to spectroscopic data](https://mlsanity.com/paper/2212.07554/) (2022)
- [Out-of-distribution detection using normalizing flows on the data manifold](https://mlsanity.com/paper/2308.13792/) (2023)

## Abstract (arXiv)

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.