# CDFlow: Building Invertible Layers with Circulant and Diagonal Matrices

arXiv: https://arxiv.org/abs/2510.25323 · HTML: https://mlsanity.com/paper/2510.25323/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 32 references that are in the map.
- First posted on arXiv 29 Oct 2025.

## Builds on (most cited first, 24 of 24)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [DreamLLM: Synergistic Multimodal Comprehension and Creation](https://mlsanity.com/paper/2309.11499/) (2023)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [World Models for Autonomous Driving: An Initial Survey](https://mlsanity.com/paper/2403.02622/) (2024)
- [Learning Fast Algorithms for Linear Transforms Using Butterfly Factorizations](https://mlsanity.com/paper/1903.05895/) (2019)
- [CirCNN: Accelerating and Compressing Deep Neural Networks Using Block-CirculantWeight Matrices](https://mlsanity.com/paper/1708.08917/) (2017)
- [Local Implicit Normalizing Flow for Arbitrary-Scale Image Super-Resolution](https://mlsanity.com/paper/2303.05156/) (2023)
- [Emerging Convolutions for Generative Normalizing Flows](https://mlsanity.com/paper/1901.11137/) (2019)
- [Maximum Entropy Reinforcement Learning via Energy-Based Normalizing Flow](https://mlsanity.com/paper/2405.13629/) (2024)
- [ACDC: A Structured Efficient Linear Layer](https://mlsanity.com/paper/1511.05946/) (2015)
- [Invertible DenseNets with Concatenated LipSwish](https://mlsanity.com/paper/2102.02694/) (2021)
- [ButterflyFlow: Building Invertible Layers with Butterfly Matrices](https://mlsanity.com/paper/2209.13774/) (2022)
- [Using transfer learning to detect galaxy mergers](https://mlsanity.com/paper/1805.10289/) (2018)
- [Woodbury Transformations for Deep Generative Flows](https://mlsanity.com/paper/2002.12229/) (2020)
- [Training Energy-Based Normalizing Flow with Score-Matching Objectives](https://mlsanity.com/paper/2305.15267/) (2023)
- [Understanding and Training Deep Diagonal Circulant Neural Networks](https://mlsanity.com/paper/1901.10255/) (2019)
- [Parameter-Efficient Fine-Tuning with Circulant and Diagonal Vectors](https://mlsanity.com/paper/2505.00580/) (2025)

## Built on by (most cited first, 1 of 1)

- [Compact Circulant Layers with Spectral Priors](https://mlsanity.com/paper/2602.21965/) (2026)

## Abstract (arXiv)

Normalizing flows are deep generative models that enable efficient likelihood estimation and sampling through invertible transformations. A key challenge is to design linear layers that enhance expressiveness while maintaining efficient computation of the Jacobian determinant and inverse. We introduce a novel invertible linear layer based on the product of circulant and diagonal matrices. This decomposition reduces parameter complexity from O(n2) to O(mn) using m diagonal matrices and m-1 circulant matrices while still approximating general linear transformations. By leveraging the Fast Fourier Transform, our approach reduces the time complexity of matrix inversion from O(n3) to O(mnlog n) and that of computing the log-determinant from O(n3) to O(mn), where n is the input dimension. We build upon this layer to develop Circulant-Diagonal Flow (CDFlow), which achieves strong density estimation on natural image datasets and effectively models data with inherent periodic structure. Furthermore, CDFlow significantly accelerates key operations in normalizing flows, providing practical benefits for scalable generative modeling.