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 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(mnlogn) 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.
Explore similar work
Jun 29, 2026cs.LG
Normalizing flows are powerful generative models that learn an invertible mapping between complex data distributions and simple latent distributions, typically a standard normal density. However, this choice of latent density can impose unnecessary complexity on the learned flow transformation due to the topological mismatch between the latent and data densities, leading to slower training and suboptimal performance. In this work, we propose using mixtures of probabilistic principal component analyzers (MPPCA) as the latent density for normalizing flows. We simplify the learned flow transformation by learning a latent distribution that more closely aligns with the data distribution in terms of KL divergence, thus enabling faster convergence and improved generative performance. Critically, MPPCA models can be fit quickly and cheaply using the expectation-maximization algorithm, making them a practical choice for initializing latent distributions even in high-dimensional generative tasks. We validate our method on both tabular and image datasets, demonstrating consistent gains in training efficiency and generation quality compared to baselines.
Liam A. Kruse, Houjun Liu, Alexandros E. Tzikas +2
Apr 21, 2026cs.CV
Normalizing Flows (NFs) are a classical family of likelihood-based methods that have received revived attention. Recent efforts such as TARFlow have shown that NFs are capable of achieving promising performance on image modeling tasks, making them viable alternatives to other methods such as diffusion models. In this work, we further advance the state of Normalizing Flow generative models by introducing iterative TARFlow (iTARFlow). Unlike diffusion models, iTARFlow maintains a fully end-to-end, likelihood-based objective during training. During sampling, it performs autoregressive generation followed by an iterative denoising procedure inspired by diffusion-style methods. Through extensive experiments, we show that iTARFlow achieves competitive performance across ImageNet resolutions of 64, 128, and 256 pixels, demonstrating its potential as a strong generative model and advancing the frontier of Normalizing Flows. In addition, we analyze the characteristic artifacts produced by iTARFlow, offering insights that may shed light on future improvements. Code is available at https://github.com/apple/ml-itarflow.
Tianrong Chen, Jiatao Gu, David Berthelot +2
Apr 27, 2026cs.LG
Learning matrix-valued distributions from high-dimensional and possibly incomplete training data is challenging: ambient-space generative modeling is computationally expensive and statistically fragile when the matrix dimension is large but the sample size is limited. We propose CoreFlow, a geometry-preserving low-rank flow model that learns shared row/column subspaces across the matrix distribution, and then trains a continuous normalizing flow only on the induced low-dimensional core. CoreFlow is designed for settings where shared low-rank matrix geometry is present, especially in high-dimensional limited-sample regimes. This separates shared matrix geometry from sample-specific variation, preserves matrix structure, and substantially improves training efficiency. The same framework also handles incomplete training matrices through masked Riemannian updates and iterative completion. Across real and synthetic benchmarks, CoreFlow substantially improves spectral and moment-level generation quality in few-sample regimes while remaining competitive in data-rich settings, even under compression to 9% of the ambient dimension and with up to 40% missing training entries.
Dongze Wu, Linglingzhi Zhu, Yao Xie