用循环与对角矩阵构建高效可逆层，提升生成模型速度与表达能力。
CDFlow: Building Invertible Layers with Circulant and Diagonal Matrices
- 通过循环矩阵与对角矩阵的乘积设计可逆线性层，降低参数量
- 计算逆矩阵与行列式的时间复杂度分别降至O(mn log n)和O(mn)
- 适合需要高速采样与高维数据建模的生成模型应用
归一化流是通过可逆变换实现高效似然估计和采样的深度生成模型。核心挑战在于设计既具强表达力又计算高效的线性层。本文提出一种基于循环矩阵与对角矩阵乘积的新型可逆线性层，将参数复杂度从O(n²)降至O(mn)，其中n为输入维度，使用m个对角矩阵和m-1个循环矩阵。借助快速傅里叶变换，该方法将矩阵求逆时间复杂度从O(n³)降至O(mn log n)，行列式对数计算复杂度从O(n³)降至O(mn)。基于此层构建的循环-对角流（CDFlow）在自然图像数据集上实现强密度估计，并有效建模具有周期结构的数据。此外，CDFlow显著加速归一化流中的关键操作，为可扩展生成建模提供实际优势。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。