# CInC Flow: Characterizable Invertible 3x3 Convolution

arXiv: https://arxiv.org/abs/2107.01358 · HTML: https://mlsanity.com/paper/2107.01358/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 16 references that are in the map.
- First posted on arXiv 3 Jul 2021.

## Builds on (most cited first, 12 of 12)

- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [Excessive Invariance Causes Adversarial Vulnerability](https://mlsanity.com/paper/1811.00401/) (2018)
- [Emerging Convolutions for Generative Normalizing Flows](https://mlsanity.com/paper/1901.11137/) (2019)
- [Using transfer learning to detect galaxy mergers](https://mlsanity.com/paper/1805.10289/) (2018)

## Built on by (most cited first, 4 of 4)

- [StyleFlow: Disentangle Latent Representations via Normalizing Flow for Unsupervised Text Style Transfer](https://mlsanity.com/paper/2212.09670/) (2022)
- [MANGO: Multimodal Attention-based Normalizing Flow Approach to Fusion Learning](https://mlsanity.com/paper/2508.10133/) (2025)
- [Adaptation of the super resolution SOTA for Art Restoration in camera capture images](https://mlsanity.com/paper/2309.13655/) (2023)
- [Parallel Backpropagation for Inverse of a Convolution with Application to Normalizing Flows](https://mlsanity.com/paper/2410.14634/) (2024)

## Abstract (arXiv)

Normalizing flows are an essential alternative to GANs for generative modelling, which can be optimized directly on the maximum likelihood of the dataset. They also allow computation of the exact latent vector corresponding to an image since they are composed of invertible transformations. However, the requirement of invertibility of the transformation prevents standard and expressive neural network models such as CNNs from being directly used. Emergent convolutions were proposed to construct an invertible 3×3 CNN layer using a pair of masked CNN layers, making them inefficient. We study conditions such that 3×3 CNNs are invertible, allowing them to construct expressive normalizing flows. We derive necessary and sufficient conditions on a padded CNN for it to be invertible. Our conditions for invertibility are simple, can easily be maintained during the training process. Since we require only a single CNN layer for every effective invertible CNN layer, our approach is more efficient than emerging convolutions. We also proposed a coupling method, Quad-coupling. We benchmark our approach and show similar performance results to emergent convolutions while improving the model's efficiency.