Title: CDFlow: Building Invertible Layers with Circulant and Diagonal Matrices
Venue: NeurIPS 2025
Accepted: yes
Keywords: Normalizing Flows; Fast Fourier Transform (FFT); Matrix Inversion; Jacobian Determinant
Abstract: Normalizing flows are deep generative models that achieve efficient likelihood estimation and sampling through invertible transformations. A key challenge is designing linear layers that enhance expressiveness while enabling efficient computation of the Jacobian determinant and inverse. In this work, we introduce a novel invertible linear layer based on the product of circulant and diagonal matrices. This decomposition provides a parameter- and computation-efficient formulation, reducing the parameter complexity from $\mathcal{O}(n^2)$ to $\mathcal{O}(mn)$ by using $m$ diagonal matrices together with $m-1$ circulant matrices, while approximating arbitrary linear transformations.Furthermore, leveraging the Fast Fourier Transform (FFT), our method reduces the time complexity of matrix inversion from $\mathcal{O}(n^{3})$ to $\mathcal{O}(mn \log n)$ and matrix log-determinant from $\mathcal{O}(n^{3})$ to $\mathcal{O}(mn)$, where $n$ is the input dimension. Building upon this, we introduce a novel normalizing flow model called Circulant-Diagonal Flow (CDFlow). Empirical results demonstrate that CDFlow excels in density estimation for natural image datasets and effectively models data with inherent periodicity. In terms of computational efficiency, our method speeds up the matrix inverse and log-determinant computations by $1.17\times$ and $4.31\times$, respectively, compared to the general dense matrix, when the number of channels is set to 96.

CDFlow: Building Invertible Layers with Circulant and Diagonal Matrices

Abstract

Normalizing flows are deep generative models that achieve efficient likelihood estimation and sampling through invertible transformations. A key challenge is designing linear layers that enhance expressiveness while enabling efficient computation of the Jacobian determinant and inverse. In this work, we introduce a novel invertible linear layer based on the product of circulant and diagonal matrices. This decomposition provides a parameter- and computation-efficient formulation, reducing the parameter complexity from O ( n 2 ) to O ( mn ) by using m diagonal matrices together with m -1 circulant matrices, while approximating arbitrary linear transformations. Furthermore, leveraging the Fast Fourier Transform (FFT), our method reduces the time complexity of matrix inversion from O ( n 3 ) to O ( mn log n ) and matrix log-determinant from O ( n 3 ) to O ( mn ) , where n is the input dimension. Building upon this, we introduce a novel normalizing flow model called CirculantDiagonal Flow (CDFlow). Empirical results demonstrate that CDFlow excels in density estimation for natural image datasets and effectively models data with inherent periodicity. In terms of computational efficiency, our method speeds up the matrix inverse and log-determinant computations by 1 . 17 × and 4 . 31 × , respectively, compared to the general dense matrix, when the number of channels is set to 96.

1 Introduction

Deep generative models have gained significant attention due to their potential applications across various domains, including natural language processing [Dong et al., 2024], image restoration and generation [Yao et al., 2023], robot learning [Chao et al., 2024], and autonomous driving [Guan et al., 2024]. Among these models, normalizing flows hold a distinctive position because of their ability to provide accurate likelihood estimation and efficient sampling.

Flow-based models enable the probabilistic modeling of complex distributions by transforming them into simpler ones through a series of invertible mappings. Their uniqueness lies in the strict requirements for the construction of these functions: on one hand, the functions must be invertible; on the other hand, the determinant of the corresponding Jacobian matrix should be easy to compute. However, these constraints may limit the expressive power of the models.

To address this challenge, NICE [Dinh et al., 2014] introduced additive coupling layers, which were later extended by RealNVP [Dinh et al., 2016] to affine coupling layers. These coupling layers have since become a fundamental component of most flow-based models. However, the design of these non-linear layers, which only transform a subset of variables, requires coordination with linear layers to effectively capture dependencies across arbitrary dimensions. As a result, recent

efforts have focused on developing specialized linear layer designs, ranging from fixed channeldisrupting operations [Dinh et al., 2014, 2016] to 1 × 1 convolutions [Kingma and Dhariwal, 2018] and even d × d convolutions [Hoogeboom et al., 2019] with learnable parameters. However, dense convolutions can result in significant memory overhead as a result of the large weight matrices required for high-dimensional inputs. One approach to address this is the construction of reversible butterfly layers [Meng et al., 2022] using butterfly matrices [Dao et al., 2019], which help maintain the model's expressiveness while reducing the number of parameters.

Structured matrices and Fourier transform acceleration have been demonstrated in numerous studies [Cheng et al., 2015, Ding et al., 2017, Karami et al., 2019, Jensen et al., 2021] to be highly effective in reducing the number of parameters and computational complexity of neural networks. Research by [Huhtanen and Perämäki, 2015] demonstrates that any matrix M ∈ R n × n can be represented as the alternating product of a cyclic matrix and a diagonal matrix, requiring no more than 2 n -1 factors. This factorization structure has demonstrated versatility and effectiveness in domains such as deep neural networks [Moczulski et al., 2015, Araujo et al., 2019] and parameter-efficient fine-tuning [Ding et al., 2025]. However, these studies have not focused on model reversibility.

Inspired by previous work, we propose a novel normalizing flow model called Circulant-Diagonal Flow (CDFlow), which is built on invertible linear layers constructed using circulant and diagonal matrices. This construction preserves the expressive power of the model while significantly reducing storage demands, as only the diagonal and circulant elements need to be stored instead of the full n × n weight matrix. In addition, the distinct properties of circulant and diagonal matrices enable the use of the Fast Fourier Transform (FFT) to enhance computation efficiency [Ding et al., 2025]. This facilitates faster calculations for both the determinant of Jacobian matrices and the inversion of weight matrices. Overall, the main contributions of our work can be summarized as following:

We propose CDFlow, a new class of flow-based generative models. The linear layer in CDFlow is designed to have a storage complexity that scales linearly with the input dimension, as it only requires storing a collection of circulant and diagonal vectors - namely, the diagonal elements of diagonal matrices and the eigenvalue vectors of circulant matrices. In practice, the linear layer weights can be constructed using a small number of circulant and diagonal matrices, enabling efficient transformations across both spatial and channel dimensions.

We leverage the unique properties of circulant and diagonal matrices to significantly reduce computational complexity. A key property is that the determinant of any diagonalizable matrix equals the product of its eigenvalues. Accordingly, by performing an O ( n log n ) FFT once during initialization to precompute the eigenvalues of circulant matrices, we enable O ( n ) log-determinant evaluation at runtime. Additionally, the complexity of matrix inversion is reduced to O ( n log n ) .

Wedemonstrate through image generation experiments that our CDFlow performs effectively while also offering superior computational efficiency in terms of both log-determinant computation and matrix inverse.

2 Background

2.1 Flow-based Models

The normalizing flow model maps the complex data distribution p X ( x ) to a known simple probability distribution p Z ( z ) via an invertible transformation f θ : x → z for data generation with θ being the model parameter. The log-likelihood log p θ ( x ) can be obtained according to the change of variables formula:

where J f θ ( · ) calculates the Jacobian matrix of the transformation f θ ( x ) , and det( · ) computes the matrix determinant.

It should be noted that f θ can be composed of multiple layers in the case of using deep learning models, and each layer needs to be an invertible function to guarantee the invertibility of f θ :

where ◦ is for function composition and there are L layers in the model. According to the chain rule in differentiability, the determinant of f θ is now transformed into layer-wise computation:

Therefore, it is important to have an efficient log-determinant computation when training normalizing flow models. When f i is a linear layer with weight matrix W , its Jabobian matrix is equivalent to the weight matrix:

where the determinant computation complexity is O ( n 3 ) .

Normalizing flow model can perform data generation using f -1 θ by feeding with samples z from p Z ( z ) . Therefore, it is important to have an efficient f -1 θ to achieve fast data generation. In particular, when f i is a linear layer, its inverse is equivalent to the linear transformation built on top of the inverse of weight matrix W :

In general, the matrix inverse can take O ( n 3 ) computation complexity.

Overall, log-determinant computation needs to be fast for training normalizing flow models, while data generation requires fast inverse computation.

Nonlinear Coupling Layer. The coupling layer [Dinh et al., 2014, 2016] is a simple and effective invertible transformation, where the Jacobi matrix is a triangular matrix. It divides the input into two parts x a and x b , then transforms x b using a scaling parameter s θ and a translation parameter b θ that can be a nonlinear function of x a , while z a is directly equal to x a , and the final output concatenates the two parts together:

Coupling layers remain a crucial component in most flow-based models [Kingma and Dhariwal, 2018, Hoogeboom et al., 2019, Meng et al., 2022]. However, since coupling layers only transform a subset of the variables, they must be combined with linear layers to facilitate information fusion across channels. As a result, recent research has increasingly focused on the design of linear layers to enhance model performance.

Invertible Linear Layer. A well-designed linear layer can significantly enhance both the coupling layer's ability to learn the target distribution and the overall expressive power of the model. Furthermore, the computational cost associated with matrix inversion and determinant computation serves as an important indicator of the efficiency of linear layer design.

Glow [Kingma and Dhariwal, 2018] pioneered the use of 1 × 1 convolution with learnable parameters as a linear layer in flow-based models. This method is more effective at capturing correlations between variables compared to simpler channel shuffling operations [Dinh et al., 2014, 2016]. Additionally, Glow introduces a weight design based on LU decomposition, reducing the Jacobian determinant computation from O ( n 3 ) to O ( n ) .

Similarly, [Hoogeboom et al., 2019] extended convolution to d × d and proposed periodic convolution, leveraging the Fast Fourier Transform (FFT) to accelerate the convolution process. However, the periodic convolution method involves generalized determinant computation, leading to a complexity of O ( n 3 ) . In contrast, our proposed method not only accelerates the convolution process using FFT, but also reduces the determinant computation complexity to O ( n ) .

Despite numerous works exploiting special matrix properties to enhance model expressiveness, most focus on improving the efficiency of the Jacobian determinant computation, with limited optimization of the matrix inversion process during the sampling phase of the flow model. In these methods, including the aforementioned works [Kingma and Dhariwal, 2018, Hoogeboom et al., 2019, Meng et al., 2022], the computational complexity for the inverse sampling operation is typically O ( n 2 ) . The Woodbury Transformation [Lu and Huang, 2020] achieves a theoretical complexity of O ( dn ) , where d is the dimensionality of smaller matrices (e.g., 8 or 16). Although this is better in theory than the O ( n log n ) complexity of our method, in practice Woodbury requires multiple input transformations, which makes its actual efficiency inferior to ours.

2.2 Weight Matrix Construction and Fast Multiplication

A general matrix M ∈ R n × n can be factorized into the alternating product of circulant and diagonal matrices, with the total number of matrices not exceeding 2 n -1 [Huhtanen and Perämäki, 2015]. Specifically, it can be written as:

where D 2 j -1 and C 2 j are diagonal and circulant matrix for j = 1 , 2 , . . . , n . This decomposition can theoretically approximate any dense matrix, and the approximation error can be controlled by adjusting the number of factors. [Ding et al., 2025] have successfully applied such representation to fine-tuning large language models.

Denote the input vector as x ∈ R n . The weight matrix W ∈ R n × n can be represented by m diagonal matrices and m -1 circulant matrices, thereby m ≤ n . For each j ∈ { 1 , 2 , . . . , m } , the j -th diagonal matrix is defined by the vector d 2 j -1 ∈ R n × 1 , and the j -th circulant matrix is defined by the vector c 2 j ∈ R n × 1 . Thus, the model weight matrix can be represented as:

where diag ( · ) and circ ( · ) represent the construction of diagonal and circulant matrices, respectively. It should also be noted that circulant matrix can be factorized into:

where F is the order n discrete Fourier transform matrix. In practice, it is better to implement with the frequency domain parameter ˆ c 2 j as weight parameter rather than real valued c 2 j to save computation costs in matrix vector product, logarithm determinant, and matrix inversion.

3 Invertible Linear Layer Design

In this section, we introduce our invertible linear layer that consist of circulant and diagonal matrices. We discuss the fast algorithms for the matrix vector product, logarithm of matrix determinant and matrix inversion. Additionally, we also analyze the theoretical computation cost associated with these operations.

3.1 Matrix Log-determinant

According to Eq. 10, The weight matrix is constructed as an alternating product of diagonal and circulant matrices. The determinant of this matrix can be efficiently computed by leveraging the special properties of these matrices. Recall that determinant of matrix product is equal to product of matrix determinant:

To compute the logarithm of the matrix determinant of W , we can express into the sum of each matrix determinant:

Therefore, the overall log-determinant computation can be finished by simple summation with a complexity of O ( mn ) .

3.2 Matrix Inverse

The inversion of the weight matrix for the data sampling process can be efficiently implemented by leveraging the decomposition of the weight matrix into alternating diagonal and circulant matrices. This section introduces the inverse calculation process, including its objective, step-by-step procedure, and computational complexity. The objective of the matrix inversion is to recover the input vector x from the output y , given the weight matrix W . This is achieved by solving the equation:

where the inverse of W is represented as:

Diagonal Matrices. For each diagonal matrix D 2 j -1 , its inverse is obtained by taking the reciprocal of its diagonal elements.

which has a computation complexity of O ( n ) .

Circulant Matrices. For each circulant matrix C j , its inverse is similar to a diagonal matrix according to Eq. (11):

It can be seen that matrix inversion W -1 can be immediately determined given d 2 j -1 and c 2 j . Therefore, the complexity of Eq. (16) is equivalent to matrix vector product complexity, i.e., O ( mn log n ) .

3.3 Generative Modeling with CDFlow

We present our CDFlow model that consists of L blocks, where each block is formed by stacking K layers of flow modules. Each flow module consists of the ActNorm layer, the invertible convolutional layer, and the coupling layer, working together to transform the input data. The multi-scale structure helps the model learn complex relationships within the data by processing it at different levels of abstraction. Figure 1 illustrates how the flow module is constructed for each layer and how the entire model is repeated and integrated to learn and generate the target data.

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Actnorm Layers. The ActNorm layer [Kingma and Dhariwal, 2018] is a type of normalization technique that uses affine transformations to perform class normalization. Unlike traditional methods like BatchNorm [Ioffe and Szegedy, 2015], which rely on the statistics of the entire batch, the ActNorm layer initializes its parameters based on the statistics of the first batch. Once initialized, these parameters are treated as learnable variables. This initialization method is advantageous for small batch sizes, as it avoids the performance degradation that BatchNorm often encounters when dealing with small batches. ActNorm helps in stabilizing the training of normalizing flows and is especially useful for flow-based models that require flexible scaling and translation invariance.

Invertible Convolutional Layer. In CDFlow, the CD-Convolution layer leverages special weight matrices, specifically circulant and diagonal matrices, to enable efficient computation. This design significantly reduces the computational cost of matrix inversion and the computation of the Jacobi determinant, which are essential operations for flow-based models. A current limitation of our approach is its reliance on 1 × 1 convolution, and our future work will extend to d × d convolutions. A detailed explanation of this design can be found in Section 3.

Coupling Layers. The coupling layer [Dinh et al., 2014, 2016] is a core component of normalizing flow models. The coupling layer splits the input into two parts: one part is passed through a neural network to compute a transformation, while the other part is left unchanged. This operation enables the model to learn complex transformations while maintaining a simple and computationally efficient structure. The coupling layer is combined with the reversible convolutional layer to form the flow module, allowing the CDFlow model to capture complex distributions and generate high-quality samples.

Multi-scale architecture. CDFlow incorporates a multi-scale architecture [Dinh et al., 2016], which improves the model's ability to capture complex and high-dimensional data distributions. Within each block, the split and squeeze operations play an important role:

The split operation divides the input tensor along the channel dimension, which allows the model to independently process different features. This enhances the model's expressive power by enabling it to focus on diverse aspects of the data.

The squeeze operation reduces the spatial resolution of the data by downsampling the spatial dimensions. This operation allows the model to focus on higher-level features while also reducing computational cost by focusing on the most important data representations.

4 Experiments

In this section, we present the experimental results to evaluate the performance of the proposed method. The experiments are conducted in three main parts:

We first evaluate the method on commonly used natural image datasets to demonstrate its general effectiveness under standard conditions.

Next, we apply the method to datasets with specialized structures to investigate its performance on data exhibiting periodic patterns.

Finally, we conduct a runtime analysis to assess the computational efficiency of the method.

The following subsections present the detailed results and discussions for each of these experiments.

We further investigated the effectiveness of the proposed linear layer within the Flow Matching model, and the results can be found in Appendix C.

4.1 Experimental Settings and Model Parameters

Datasets and Baselines. To compare our method with previous approaches, we evaluate its performance on the CIFAR-10 [Krizhevsky, 2009] and ImageNet [Deng et al., 2009] datasets, using bits per dimension (BPD) as the evaluation metric. BPD is a commonly used measure for assessing the quality of generative models, indicating the number of bits required to encode each dimension of the data. Metrics such as IS and FID are not used in this work, as they do not generalize well across different datasets and fail to account for overfitting. For the baseline, we select several models related to CDFlow, including MAF [Papamakarios et al., 2017], Real NVP [Dinh et al., 2016], Glow [Kingma and Dhariwal, 2018], Emerging [Hoogeboom et al., 2019], i-ResNet [Behrmann et al., 2019], Woodbury [Lu and Huang, 2020] and ButterflyFlow [Meng et al., 2022].

Implementation Details. To facilitate a fair comparison with prior architectures, we adopt consistent backbone settings for flow-based baselines that share a similar multi-scale design, including Glow, Woodbury, Emerging, and ButterflyFlow. On standard images datasets, all of these models are configured with three blocks and 32 flow steps. On the structured dataset, we instead adopt a lighter configuration with two blocks and eight flow steps. For Woodbury, we adhere to the original configuration with d s = d c = 16 , while for ButterflyFlow we use the default setting with butterfly level = 1 . For our proposed CDFlow, we set m = 2 , meaning we use two diagonal vectors and a circulant vector.

During training, we apply spectral normalization [Miyato et al., 2018] to the convolutional layers to ensure stability. Thanks to the structure of circulant and diagonal matrices, the maximum singular value of each weight matrix can be efficiently computed and used to rescale the weights. In addition, we employ a channel-aware learning rate scaling for the CD-Convolution parameters, which prevents abrupt updates to the structured matrices. Empirically, this combination further stabilizes the training of CDFlow.

Hyperparameter Selection Each linear transform in our method is decomposed into 2 m -1 matrices (i.e., m diagonal and m -1 circulant). While larger m values increase the approximation power, they also introduce additional parameter and computational costs. The key objective is to balance expressiveness, efficiency, and runtime performance. To this end, we systematically evaluate the effect of varying m on model behavior.

As shown in Figure 2(a), increasing m on CIFAR-10 leads to only marginal improvements in expressiveness, as reflected by BPD values, while significantly increasing the parameter count of the

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CD-Convolution layers. Figures 2(b) and (c) further report the runtime measurements for forward and inverse operations. Although larger m values introduce additional latency, our method remains more efficient than the other methods reported in Section 4.4. In addition, we separately evaluated the log-determinant computation time across channel sizes ranging from 16 to 1024, where the runtime remains consistent across different m , with values between 0.095 ms and 0.097 ms.

Based on these results, we adopt m = 2 as the default setting for all experiments, including both the main and supplementary studies, unless otherwise specified.

4.2 Density Estimation on Standard Images

Standard Image Datasets. To enable direct comparison with prior work [Meng et al., 2022], we evaluate our method on the CIFAR-10 [Krizhevsky, 2009] and ImageNet-32×32 [Deng et al., 2009] datasets. For both datasets, training is conducted with unified dequantization and standard data augmentation techniques.

Results. Table 2 presents the quantization results of the model on the CIFAR-10 and ImageNet 32 × 32 datasets. CDFlow outperforms Glow, Emerging, Woodbury, and ButterflyFlow (all using the same modeling framework) and performs comparably to models like Residual Flows and i-DenseNet. The generated sample results are visualized in Figure 3a and Figure 3b.

4.3 Density Estimation on Structured Images

To demonstrate the capability of CDFlow in modeling periodic structures, we apply it to galaxy images [Ackermann et al., 2018], conducting an experiment on this real-world dataset.

Galaxy Datasets. When the boundary pixels of an image are relatively uniform or connected, the data can be considered periodic. A typical example of periodic data is galaxy images, which often

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feature scattered light sources with predominantly dark boundaries. The Galaxy datasets [Ackermann et al., 2018] is a small classification dataset that includes 3,000 merged galaxies and 5,000 non-merged galaxies in the training and test sets. We train CDFlow on non-merged galaxy images and compare its performance to other models under the same experimental setup.

Results. Table 2 also presents the BPD results on the galaxy dataset. It is clear that CDFlow outperforms all other models, achieving a 4.95% improvement in BPD compared to the Glow model. In addition, Figure 3c presents a visual comparison between our method and two strong baselines: Emerging and Butterfly. Experimental results demonstrate that our proposed CDFlow model can well model data with periodic qualities.

4.4 Runtime

To demonstrate the effectiveness of our method in reducing computational complexity, we separately evaluated the runtime of the forward operation, log-determinant (logdet) computation, and inverse operation. We compared our method with standard 1 × 1 convolution, emerging and periodic convolutions [Hoogeboom et al., 2019], the Woodbury transformation and ME-Woodbury with a fixed latent dimension of d = 16 [Lu and Huang, 2020], and the butterfly layer with level 10 [Meng et al., 2022]. For a fair comparison, all methods were implemented in PyTorch, and all experiments were conducted on an NVIDIA A800 GPU.

Forward Time. The forward operation refers to the runtime required to apply the tested layer function and compute the log-determinant, i.e., y = f ( x ) and log | det( W ) | , under gradient computation mode. As shown in Figure 4, although the theoretical complexity of the 1 × 1 convolution is O ( n 3 ) , its actual runtime remains relatively stable across varying channel sizes, likely due to GPU parallelism. All baseline methods are slower than the 1 × 1 convolution, while our method achieves even faster performance. Notably, although the periodic convolution leverages Fourier transforms, its reliance on two-dimensional FFT introduces substantial memory overhead, leading to sharp runtime increases with more channels. In contrast, our method combines circulant and diagonal matrix constructions to reduce log-determinant complexity, and employs an optimized convolutional implementation to accelerate forward computation, as detailed in Sections 2.2 and 3.1.

In addition, we specifically measured the runtime of the log-determinant computation, which depends solely on the size of the weight matrix, i.e., the number of input channels. Figure 4 illustrates how the runtime of different methods varies with increasing channel dimensions. Our method not only outperforms all baselines but also maintains stable performance without significant growth as the number of channels increases.

Inverse Time. The inverse operation refers to the time required to apply the inverse function of the tested layer, i.e., x = f -1 ( y ) , under no-gradient mode. We first analyze the impact of channel number on runtime, as shown in Figure 4. The Emerging convolution suffers from poor parallelism, while the

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periodic convolution relies on two-dimensional Fourier transforms, leading to a significant increase in runtime as the number of channels grows. Other baseline methods perform slightly worse than the 1 × 1 convolution, possibly due to the lack of optimized inverse implementations and additional transformation overhead. In contrast, our method reduces the inversion complexity to O ( n log n ) , and despite requiring data reshaping, consistently outperforms other baselines, particularly at large channel sizes.

We further evaluate how spatial resolution affects runtime, with the number of channels and batch size fixed. The results show that all methods experience increased runtime with larger spatial sizes. Our method outperforms the 1 × 1 convolution at small spatial sizes, and becomes slightly less efficient at larger sizes, since it mainly reduces channel-wise rather than spatial complexity. Nevertheless, our method still clearly outperforms all other baselines except the 1 × 1 convolution.

5 Conclusion

In this paper, we introduce the CDFlow model, a novel flow-based generative model. Instead of relying on traditional weight matrices, we employ the interleaved multiplication of circulant and diagonal matrices, leveraging the properties of these matrices along with the Fast Fourier Transform (FFT) to efficiently compute the Jacobian determinant and perform the inverse transformation. Our experiments demonstrate that CDFlow excels in density estimation for standard image datasets and performs well on real-world data with periodic properties, outperforming existing baselines.

Acknowledgments and Disclosure of Funding

This work was financially supported by the National Key R&D Program of China (Grant No. 2024YFA1211400).

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A Impact of Linear Transformation Constraints on Flow-based Model Performance

To further validate the effectiveness of our proposed method, we reproduced the experiment described in Appendix A.5 of Chao et al. [2023] and included an additional variant based on our Diagonal-Circulant Decomposition (DCD) linear layers. The compared linear layer types include F -type (full matrix), L -type (lower-triangular), U -type (upper-triangular), LU -type (the product of lower and upper triangular matrices), and our proposed DCD -type.

As shown in Fig. 5, the model employing the F-type layers achieved the lowest negative loglikelihood (NLL), consistent with Chao et al. [2023]. Although our DCD-type layer does not surpass the fully learnable F-type, it achieves the second-best performance, outperforming the LUtype design. This result indicates that DCD retains much of the expressiveness of the unconstrained full matrix while maintaining a more efficient and structured parameterization.

B Runtime

Specifically, our approach requires transforming the input tensor from a four-dimensional structure [ B,C,H,W ] to a two-dimensional matrix [ B × H × W,C ] . While the runtime reported in Section 4.4 omits this transformation, we further measured the total runtime including it to ensure a fair comparison. As illustrated in Figure 6, our method continues to demonstrate strong computational efficiency even after accounting for the transformation overhead. It is worth noting that the computation of the log-determinant is independent of the input tensor, and therefore the reshaping operation has no impact on its runtime.

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C Additional Experiments on Flow Matching Models

To further validate the effectiveness of the proposed linear layer, we conducted experiments within the Flow Matching framework Lipman et al. [2022], which employs a 5-layer MLP on 2D toy datasets. To systematically evaluate the method, we performed experiments on three datasets: Chessboard, Moons, and Circles. Within the original framework, we replaced the linear layers of the standard MLP with four structured variants for comparison: EMLP (Emerging Layer) Hoogeboom et al. [2019], WMLP (Woodbury Layer) Lu and Huang [2020], BFMLP (Butterfly Layer) Meng et al. [2022], and CDMLP (Ours).

The experimental results are summarized in Table 3. CDMLP achieves the best or near-best performance across most cases while requiring significantly fewer parameters. For instance, it attains the lowest NLL on the Chessboard and Circles datasets and the best MSE and FID on Moons, outperforming or matching larger models such as MLP and EMLP. In addition, Figures 7-8 further visualize the learned distributions, showing that the model successfully evolves from Gaussian initialization toward the target distributions. These findings not only confirm the modeling capability of the proposed method within the Flow Matching framework but also highlight its applicability and generality across different data structures.

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D Visualization of Generation Results on Image Datasets

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a new efficient parameterization for linear mixing layers in normalising flow architectures. It relies on a circular-diagonal decomposition of general matrices and leveraging the FFT for efficient matrix inverses and Jacobion computation required in flow models. The resulting speedup is significantly more scalable, where the matrix inversion in the inverse flow direction goes from O(n^3) to O(m n log n) as it is not bottlenecked by the FFT, and the log determinant computation becomes O(mn) due to the simplicity of Jacobians of diagonal and circulant matrices. Empirical results on natural image data with and without periodic features show that it achieve competitive or even superior performance to alternative mixing layer models, while being computationally efficient allowing significant speedup of the training process.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Simple but practical idea that should be easily included in many flow implementation packages and pipelines\n- Proven idea of circulant-diagonal decomposition used in other fields like variational Gaussian processes, e.g. [Jensen, Kristopher, et al. \"Scalable Bayesian GPFA with automatic relevance determination and discrete noise models.\" Advances in Neural Information Processing Systems 34 (2021): 10613-10626.]\n- Competitive performance and evidence of advantages in structured datasets with minimal complexity overhead\n\nWeaknesses:\n- Experiments only consider image data\n- Hyperparameter choice of the method (number of matrices in the decomposition) seems a bit obscure\n\nQUESTIONS:\n- It is not clear how to choose the m value in a systematic way, is this just done heuristically? If there are any heuristic/mathematical guidelines please do include this in the method/discussion\n- Could you clarify the choices of the baselines hyperparameters, e.g. the Woodbury and butterfly settings described in 4.3?\nAre the same baseline hyperparameters for the linear transform layers used in 4.1 and 4.2? This should be stated clearly at the start of 4 if all experiments use similar baseline setups including hyperparameters.\n- A note: In the abstract, I realise nowhere the meaning of “m” is mentioned despite being in the complexity expression (related to number of diagonal and circulant matrices in the decomposition). This should be added for clarity and standalone role of the abstract.\n\nClarification on these points will help me assess whether the framework proposed actually has considered these design and thought points and can help me adjust the score ratings.\n\nLIMITATIONS:\nIt would be good to mention avenues for further work and limitations such as the scaling of m required to achieve sufficient performance in general compared to dense transformations, or some discussion of the trade off between performance and the choice of m. I will be able to boost my review scores if a good discussion aspect is added.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors present a new flow architecture that uses a product of diagonal and circulant matrices for the weight matrices. This enables parsimonious representation of weight matrices, and efficient forward, backward, and determinant calculation. The authors also show that this computation allows for good density estimation, and is often computationally more efficient than existing competing architectures.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n - The paper is well-written and easy to follow.\n - The core idea is simple, elegant and well-executed.\n - The presentation of the mathematics is clear and intuitive.\n - The numerical experiments, especially those presented in Section 4.3, clearly valorizes the work.\n\nWeaknesses:\n - The work has many small typos scattered throughout. Please see the questions section.\n - The experiments show that CDFlow performs well in practice, but the gains in terms of density estimation are modest, at best.\n\nQUESTIONS:\nQuestions/Comments\n\n - In the abstract, what is meant by \"This decomposition provides a compact representation?\" I don't understand what the phrase \"compact representation\" means here. Could you please explain?\n - In line 59, \"...storing a collection of circulant and diagonal vectors.\" The vectors are not circulant or diagonal, the matrices are, correct?\n - In line 63, \"By utilizing the Fast Fourier Transform (FFT), the determinant computation is simplified to $O(n)$.\" This sentence is not right. The runtime of the determinant computation doesn't follow from the FFT. It follows because you store the eigenvalues of the circulant layers directly.\n - Related to the point above, line 63 \"the determinant of a diagonal matrix equals the product of its diagonal elements, and the determinant of a circulant matrix is the product of its eigenvalues.\" This sentence is very odd. The determinant of every diagonalizable matrix is the product of its eigenvalues. It is not a special property of circulant matrices, and is the reason a diagonal matrix's determinant is the product of its diagonal entries.\n - In the paragraph in lines 189 - 194, I don't understand what is being said. What is \"the invertible convolutional layer?\" Are you referring to the circulant layers? Are you building a specific convolution kernel via the product of diagonal and circulant matrices? How is this similar to the 1x1 convolutions from Dinh et al.? I think this point needs to be clarified considerably, especially given the discussion in section 4.3\n\nTypos:\n - In line 134, there is a missing space between the period and reference.\n - In line 151, the t in The should not be capitalized.\n - In line 166 \"...its inverse is to compute the reciprocal...\" There is a typo here. \n - In line 169 \"is similar to diagonal matrix\" should be \"is similar to a diagonal matrix.\"\n - In line 171, the word \"with\" is unnecessary. \n - Line 203, \"handle complex and complex distributions.\" I don't understand what is meant here. \n - Line 252, there is a space missing after the word model.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents and explores a flow architecture, the distinctive feature of which is the usage of the product of interleaving circulant and diagonal matrices as the weights for 1x1 convolutional layers, constituting the non-linear transformation of the coupling layer. The main advantages of this approach are the reduced time complexity for logdet and inverse computation, as well as decreased space consumption.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**:\n1. The paper is written very clearly, with all the necessary prerequisites explained in Section 2 (“Background”).\n2. In order to prove the superiority of their method, authors do not only compute the theoretical complexity, but provide experimental time measurements in Section 4.3 as well.\n3. The proposed method, indeed, has a competitive time complexity, both in theory and in practice.\n\n**Weaknesses**:\n\nHere are some conceptual weaknesses of the paper:\n\n1. As the diagonal-circulant neural networks have already been proposed in [1], the novelty of the paper lies exclusively in applying such networks to the construction of a flow model. While the efficiency of this approach is confirmed experimentally, the trade-off between the number of parameters of the flow layers and the quality of the resulting network is not explored. According to [2], this might be a non-negligible trade-off. **Possible action items**: 1) Please check if it would be helpful to rearrange the results from Table 2 into a graph (or two graphs) with number of parameters (or runtime measurements from Figure 3) on one axis and quality on the other. 2) Could you reproduce the experimental setting of Appendix A.5 in [2] with the proposed method?\n2. The experimental section of the paper is scarce. While it is clearly shown that the computation of the Jacobian logdet is much faster than its alternatives, especially for large numbers of channels, which might indeed speed the training up, the sampling quality from Table 2 is average. **Possible action item**: The paper would have benefited much from applying the proposed layers to a real experimental setting, e.g. in a flow matching model or comparing to Woodbury in a setup similar to [3]. Speed up and quality measurements obtained this way would be much more valuable and would help other researchers to choose the appropriate method of flow layer compression.\n3. A couple of words about the presented results: while I understand that the proposed method does not have to be SOTA, as its main goal is efficiency, the results from Table 2 and Figure 2 are not very convincing. In Table 2 the results are presented without confidence intervals. Also, it is unclear why MAF and i-ResNet are not evaluated on ImageNet and even more methods are not evaluated on Galaxy. In Figure 2 the images are too small to see any difference, but to me it seems that images from the proposed method have some strange artifacts which other methods do not have. **Action items**: Please report the results for all methods, add confidence intervals and show larger pictures.\n4. The authors did not mention some very relevant related articles, which is alarming, as their work could have benefited much from using tips for training from [1] (this paper answers the questions of expressivity and efficient training methods of networks with circulant-diagonal layers) and comparing their method to a close in time complexity method from [4] (with both Jacobian determinant and inverse computation complexity being O(nlogn)). Also, similar parameterizations of linear layers as the product of circulant and diagonal matrices were proposed in [5], [6]. **Action item**: Please compare to [4] (please correct me if they are not comparable).\n\nSome small inaccuracies in the manuscript:\n\n1. In equation (3) there are L - 1 terms, while there should be L, with the first one being df_1 / dx.\n2. Lines 125-127: it is written that the proposed complexity O(n log n) is more efficient than O(dn) from Woodbury, where d is a constant, which is not true, especially considering the fact that the proposed complexity is actually O(mn log n), where m is a constant.\n3. In Figure 2, there is a typo in the name of “Woodbury”.\n\n[1] Araujo, Alexandre, et al. \"Understanding and training deep diagonal circulant neural networks.\" ECAI 2020. IOS Press, 2020. 945-952.\n\n[2] Chao, Chen-Hao, et al. \"Training energy-based normalizing flow with score-matching objectives.\" Advances in Neural Information Processing Systems 36 (2023): 43826-43851.\n\n[3] Lee, Jaewoo, et al. \"Differentially private normalizing flows for synthetic tabular data generation.\" Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 36. No. 7. 2022.\n\n[4] Karami, Mahdi, et al. \"Invertible convolutional flow.\" Advances in Neural Information Processing Systems 32 (2019).\n\n[5] Cheng, Yu, et al. \"An exploration of parameter redundancy in deep networks with circulant projections.\" Proceedings of the IEEE international conference on computer vision. 2015.\n\n[6] Moczulski, Marcin, et al. \"Acdc: A structured efficient linear layer.\" arXiv preprint arXiv:1511.05946 (2015).\n\nQUESTIONS:\nPlease see **action items** and **possible action items** in “Weaknesses”. I will be ready to accept the paper if the first three weaknesses are thoroughly addressed. Moreover, I have some smaller questions:\n\n1. Could you please provide experimental results (both regarding sampling quality and runtime efficiency) for different choices of hyperparameter m?\n2. What can be done to represent the non-square weight matrix?\n3. Could you please explain why there is such a large margin between the proposed method and other methods, especially in logdet computation? Some of the methods have the same theoretical complexity, why are they so much slower?\n4. Did you try using the initialization scheme and the placement of non-linearities proposed in [1]? I think it might increase the quality of the final model.\n\nLIMITATIONS:\nThe authors adequately addressed the limitations and potential negative societal impact of their work.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 1, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper is concerned with the efficient implementation of normalizing flows for sampling from complex probability distributions. These models require the construction of invertible deep networks, with the extra condition that the determinant of the Jacobian matrix should be easy to compute. The inverse is important primarily for the sampling process of the model, while the log-det is important primarily for the training process. Based on this, the authors focus on the efficient implementation of linear transformations (that is, linear layers) whose determinants and inverses are easy to compute. They leverage a result on the approximation of dense matrices by the product of structured matrices: namely the alternating product of diagonal and circulant matrices. The diagonal matrices have easily computable inverses and determinants, and the circulant matrices can be similarly handled using FFTs.\n\nSTRENGTHS AND WEAKNESSES:\nStrenghths:\n\nThe results and methods put forth in this work have the strength of being quite simple and easy to understand. The work leverages established results on structured matrix approximation to good effect, yielding a method that I can see being easily adapted by practitioners. The methods based on these results attain good asymptotic complexity, only comparable to the Woodbury method of [Lu and Huang, 2020], which is made up for with superior performance on test datasets and in runtime.\n\nWeaknesses:\n\nThere are some missing references on prior works that use circulant-diagonal linear layers in machine learning applications. For instance, see “ACDC: A structured efficient linear layer” by Moczulski et. al. Papers that use such methods generally weren’t using circulant-diagonal linear layers to implement normalizing flows, so it is not necessary to “compare” as far as experiments go, but it would be best to acknowledge them. Additionally, you set $m=2$ in your experiments, which is presumably quite a bit less than the $m=n$ necessary for exact representation of any dense matrix. Although it is acknowledged that “the approximation error can be controlled by adjusting the number of factors,” not much else is said on the expressivity of approximation as a function of $m$.\n\nQUESTIONS:\nMy questions are primarily around the theoretical properties of this approach as they pertain to training normalizing flow models.\n\nAs noted in the weaknesses, there is not much discussion on how the choice of $m$ affects the expressivity of the model. Of course, as $m$ increases the linear layer can parameterize a larger family of matrices. When $m$ is small, what kind of matrices are expected to be well-approximated? For instance, the composition of convolutions and pointwise multiplications might do a good job approximating “spatially-varying convolutions,” pseudodifferential operators, or something similar.\n\nThis method is put forth with the idea of enabling efficient matrix inversion and log-determinant calculation. However, these fail when the component (diagonal/circulant) matrices are not well-conditioned. Did you employ regularization in training to ensure that this does not happen? Is it even worthwhile to add penalties to enforce well-conditioned linear layers?\n\nYou have elected to focus on products of diagonal and circulant matrices, particularly since the latter can be easily parameterized using FFTs. However, there are many other types of structured matrices that admit similar efficient $n\\log n$ computations; see the paper “Butterfly factorization” by Li et. al. Do the algorithms and approximations in that paper have any bearing on potential generalizations of your proposed method? A *thorough* discussion on this would make this paper much more significant, even if not much is done experimentally.\n\nThere is a claim on limitations that I do not understand. On pg. 6, you say that “A current limitation of [your] approach is its reliance on 1x1 convolutions.” But your convolutions are implemented directly in the Fourier domain, so it does not seem that you have any control over the spatial extent of the convolutions implemented by the circulant matrices. What did you mean by this statement?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}