# Regularized Autoencoders via Relaxed Injective Probability Flow

arXiv: https://arxiv.org/abs/2002.08927 · HTML: https://mlsanity.com/paper/2002.08927/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Feb 2020.

## Built on by (most cited first, 23 of 23)

- [Rectangular Flows for Manifold Learning](https://mlsanity.com/paper/2106.01413/) (2021)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Recent Advances for Quantum Neural Networks in Generative Learning](https://mlsanity.com/paper/2206.03066/) (2022)
- [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)
- [Embrace the Gap: VAEs Perform Independent Mechanism Analysis](https://mlsanity.com/paper/2206.02416/) (2022)
- [A survey on Variational Autoencoders from a GreenAI perspective](https://mlsanity.com/paper/2103.01071/) (2021)
- [Lifting Architectural Constraints of Injective Flows](https://mlsanity.com/paper/2306.01843/) (2023)
- [Conditional Injective Flows for Bayesian Imaging](https://mlsanity.com/paper/2204.07664/) (2022)
- [A Review of Change of Variable Formulas for Generative Modeling](https://mlsanity.com/paper/2308.02652/) (2023)
- [Bounds all around: training energy-based models with bidirectional bounds](https://mlsanity.com/paper/2111.00929/) (2021)
- [VQ-Flows: Vector Quantized Local Normalizing Flows](https://mlsanity.com/paper/2203.11556/) (2022)
- [Neural Implicit Manifold Learning for Topology-Aware Density Estimation](https://mlsanity.com/paper/2206.11267/) (2022)
- [Rank Reduction Autoencoders](https://mlsanity.com/paper/2405.13980/) (2024)
- [Spread Flows for Manifold Modelling](https://mlsanity.com/paper/2109.14216/) (2021)
- [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)
- [Injective flows for star-like manifolds](https://mlsanity.com/paper/2406.09116/) (2024)
- [On Geometry Regularization in Autoencoder Reduced-Order Models with Latent Neural ODE Dynamics](https://mlsanity.com/paper/2603.03238/) (2026)
- [Variational Rejection Particle Filtering](https://mlsanity.com/paper/2103.15343/) (2021)
- [Weakly Supervised Label Learning Flows](https://mlsanity.com/paper/2302.09649/) (2023)
- [Exploring bidirectional bounds for minimax-training of Energy-based models](https://mlsanity.com/paper/2506.04609/) (2025)
- [On the flow matching interpretability](https://mlsanity.com/paper/2510.21210/) (2025)

## Abstract (arXiv)

Invertible flow-based generative models are an effective method for learning to generate samples, while allowing for tractable likelihood computation and inference. However, the invertibility requirement restricts models to have the same latent dimensionality as the inputs. This imposes significant architectural, memory, and computational costs, making them more challenging to scale than other classes of generative models such as Variational Autoencoders (VAEs). We propose a generative model based on probability flows that does away with the bijectivity requirement on the model and only assumes injectivity. This also provides another perspective on regularized autoencoders (RAEs), with our final objectives resembling RAEs with specific regularizers that are derived by lower bounding the probability flow objective. We empirically demonstrate the promise of the proposed model, improving over VAEs and AEs in terms of sample quality.