aiqbench analysis
This research introduces CDFlow, a novel invertible linear layer utilizing circulant and diagonal matrices to significantly reduce computational complexity in normalizing flows. By leveraging the Fast Fourier Transform, the method accelerates matrix inversion and log-determinant calculations while maintaining high expressiveness for density estimation on natural images. The approach offers practical efficiency gains for scalable generative modeling without sacrificing model performance.
CDFlow: Building Invertible Layers with Circulant and Diagonal Matrices
arXiv:2510.25323v4 Announce Type: replace Abstract: 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 $\mathcal{O}(n^2)$ to $\mathcal{O}(mn)$ 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 $\mathcal{O}(n^3)$ to $\mathcal{O}(mn\log n)$ and that of computing the log-determinant from $\mathcal{O}(n^3)$ to $\mathcal{O}(mn)$, 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.
Abstract: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 $\mathcal{O}(n^2)$ to $\mathcal{O}(mn)$ 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 $\mathcal{O}(n^3)$ to $\mathcal{O}(mn\log n)$ and that of computing the log-determinant from $\mathcal{O}(n^3)$ to $\mathcal{O}(mn)$, 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.
Accepted at NeurIPS 2025. 10 pages, 12 figures, 2 tables
Machine Learning (cs.LG)
arXiv:2510.25323 [cs.LG]
(or arXiv:2510.25323v4 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2510.25323 Focus to learn more arXiv-issued DOI via DataCite
https://doi.org/10.52202/085713-0124 Focus to learn more DOI(s) linking to related resources
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Full text under the publisher's open licence · 267 words
