Fourier Token Merging: Understanding and Capitalizing Frequency Domain for Efficient Image Generation
Authors: Jiesong Liu
Xipeng Shen
Venue: NeurIPS 2025 poster ()
Keywords: Efficient transformers
Primary area: deep_learning

Abstract: Image generation requires intensive computations and faces challenges due to long latency.
    Exploiting redundancy in the input images and intermediate representations throughout the neural network pipeline is an effective way to accelerate image generation.
    Token merging (ToMe) exploits similarities among input tokens by clustering them and merges similar tokens into one, thus significantly reducing the number of tokens that are fed into the transformer block.
    This work introduces Fourier Token Merging, a new method for understanding and capitalizing frequency domain for efficient image generation.
    By introducing frequency token merging, we find that transforming the token into the frequency domain representation for clustering can better exert the ability of clustering based on the underlying redundancy after de-correlation.
    Through analytical and empirical studies, we demonstrate the benefits of using Fourier clustering over the original time domain clustering.
    We experimented fourier token merging on the stable diffusion model, and the results show up to 25\% reduction in latency without impairing image quality. 
The code is available at https://github.com/Fred1031/Fourier-Token-Merging.

Fourier Token Merging: Understanding and Capitalizing Frequency Domain for Efficient Image Generation
Jiesong Liu Department of Computer Science North Carolina State University Raleigh, NC 27606 j1iu930ncsu. edu
Xipeng (Gracen) Shen Department of Computer Science North Carolina State University Raleigh, NC 27606 xshen5@ncsu, edu
Abstract
Image generation requires intensive computations and faces challenges due long latency. Exploiting redundancy in the input images and intermediate representa- tions throughout the neural network pipeline an effective way to accelerate image generation. Token merging (ToMe) exploits similarities among input tokens by clustering them and merges similar tokens into one, thus significantly reducing the number of tokens that are fed into the transformer block. This work introduces Fourier Token Merging, a new method for understanding and capitalizing frequency domain for efficient image generation. By introducing frequency token merging, we find that transforming the token into the frequency domain representation for clus- tering can better exert the ability of clustering based on the underlying redundancy after de-correlation. Through analytical and empirical studies, we demonstrate the benefits of using Fourier clustering over the original time domain clustering. We experimented Fourier Token Merging on the stable diffusion model, and the results show up to 25% reduction in latency without impairing image quality. The code is available https: //gi thub. com/ comm/Fred103//oouiiaae Token -Merging
Introduction
Image generation has achieved impressive results through advanced generative models such as stable diffusion (21), but these models face significant computational challenges and long latency, especially when deployed on essurce-constrainee devices (5) I 26). A primary avenue for addressing these issues involves exploiting redundancies within input images and intermediate neural network representations to streamline computation. Token merging (ToMe) represents one such approach, utilizing clustering methods to merge similar tokens, thus substantially reducing computational overhead within transformer blocks (2).
In this work, we introduce Fourier Token Merging, novel technique that leverages frequency-domal insights for efficient clustering and merging of tokens during image generation. Unlike traditional methods, Fourier Token Merging transforms tokens into frequency-domate representations, capitaliz- ing on the inherent redundancy revealed after decorrelation. By clustering tokens in this frequency space, our method enhances clustering effectiveness and significantly reduces computational latency.
We present both theoretical and empirical analyses demonstrating that frequency- clustering surpasses conventional time-domain clustering approaches in capturing token similarities. Specifically, we show analytically that Fourier-based clustering reduces token error within clusters, especially as the diffusion process progresses and tokens propagate into deeper transformer layers. Empirical results from extensive experiments on the stable diffusion v1.5 model validate our approach, achieving up 25% latency reduction without sacrificing image quality.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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To the best of our knowledge, this is the first work to explore frequency domain for speeding up image generation. Our results underline the potential of frequency-done techniques enhance the efficiency and practicality of advanced generative models, opening new possibilities for deploying high-quality image generation on edge devices and low-resource environments.
2
Background
Token Merging for Image Generation. Token merging 2 3 has emerged as an effective technique to optimize computational efficiency in generative image models such stable diffusion. By grouping similar tokens and merging them, these methods substantially reduce computational overhead and latency during image generation processes. While most token merging methods focus standard Vision Transformers, Token Merging with Attention (ToMA) [14] is a key prior work that first adapted token merging for DīT-like models (e.g., SDXL and Flux) by introducing strategy to handle their specific architectural challenges. More related work to reduce transformer computaion is in Appendix C
Discrete Fourier Transform. Central to our approach the Fast Fourier Transform (FFT), a widely used computational algorithm that efficiently computes the Discrete Fourier Transform (DFT). The DFT decomposes a spatiotemporal signal into frequency-domann representations, revealing correlations at different scales and orientations. For given imput x (zo.-.. NNN-1 1,, the DFT is defined as:
N-1
F(x)+ =
Σ ==0
nk 0
S k N 1. 1.
(1)
The inverse transform, which reconstructs the original signal from its frequency-domain represen- tation, is simply the conjugate transpose of the forward transform, emphasizing the linearity and unitarity of the DFT. The FFT algorithm [6 8 significantly reduces computational complexity from O(N2) to O(N log N), facilitating rapid signal processing and making it practical for real-world applications. Its computational efficiency and versatility have made it a standard tool across various domains, notably signal and image processing. This background underpins our novel Fourier Token Merging technique, leveraging frequency-domate insights to enhance image generation efficiency.
Fourier Token Merging for Image Generation
To accelerate inference in diffusion models on the fly without compromising visual fidelity, we introduce a novel merging framework that leverages frequency-domain representations. Our approach, termed Fourier Token Merging, augments existing token reduction techniques by incorporating global structural priors via the Discrete Fourier Transform (DFT). Figure Tillustrates the architecture the proposed system.
3.1 Modular Integration in the Diffusion Pipeline
Our framework is integrated into the denoising backbone of diffusion model, such as the U-Net used in Stable Diffusion. At each sampling timestep t, the model receives a noisy latent 2 and computes its denoised counterpart through modified 6g network that supports frequency-domat token merging.
The Original Merging Module represents baseline spatial-domain merging. The data are first clustered in a token-level granularity and then each cluster uses the centroid to represent the whole cluster and uses only that one token for further transformer block computation. The bipartite softmatch clustering component partitions the tokens into two sets (src and dst) and computes the similarity score between each of the and dst pair. For each token in the src, the module clusters it with the corresponding token the dst with the highest similarity score. The module then computes attention operations with the reduced tokens. Since the subsequent ResNet requires the full number of tokens, tokens are unmerged to their original length, optimizing the computation speed in the attention module. ResNet blocks and MIP layers remain unchanged.
The Fourier Merging Module transforms each token x E Rd into the frequency domain via the Discrete Fourier Transform (DFT), denoted by F(x) as introduced in Section To reduce noises

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Fourier Merging
DFT
Trunc
Softmatch
tolamn 3ssigned
Keep the
Merge contacted
Diffusion Process
Aftratton Rediinal
Softmatch
Cluster Centroid
Atteatian Residual
U-Nett
T-NNEE
Merging
X
Mergo
Output
Original Merging
Figure The proposed Fourier Token Merging Framework and the original Token Merging for Diffusion Models. Fourier Token Merging integrates frequency-aware merging mechanism into the diffusion generation process. The Original Merging Module from the traditional method performs token merging based on similarity in the spatial domain. In contrast, the Fourier Merging Module processes intermediate visual representations using the Discrete Fourier Transform before truncating the higher frequencies, enabling structured token merging in the frequency domain using clustering method called seftmatch. The tokens are merged into the Attention Residual U-Ner (Eş), which computes the denoised prediction at each timestep. The figure shows the Diffusion Pipeline, where the model iteratively denoises latent variables from 2n down to the final output. This modular design allows Fourier-basse merging in place of existing merging techniques, improving both generation efficiency and preservation of global structure.
while preserving structural semantics, we retain only the dominant wwwfrreuency components and discard the rest. This is done by truncating the real part of the DFT output according to a threshold (0, 1]:
x = Trunc, (Re (F(x)))
Here, F(x) E Cd is the full complex-valued spectrum, Re(-) extracts the real-valued coefficients, and Trunc- (.) retains only the lowest 7d frequency components. The resulting representation then used for similarity computation and token clustering, effectively emphasizing coarse-grained structure while suppressing frequeryy noise.
The clustering component employing the same bipartite soft matching then identifies and clusters tokens based on the frequency structures. This improves clustering according to the underlying structures of the tokens after decorrelation of the time domain information. The clustering results are then used guide the merging process, according to which the original tokens are merged. The reduced token set is then forwarded to the self-attention and cross coossatttnntinn layers within the U-Net.
Once merged, the modified U-Net computes the denoised residual co(2,, (), which is used in standard reverse diffusson step to obtain 2t-1- This process is repeated iteratively until the final clean latent 2g is reached. By performing Fourier Token Merging every timestep. the model enjoys consistent computational savings throughout the generation trajectory.
In contrast to purely spatial methods, Fourier Token Merging enables:
underlying redundancy detection: Frequency representations capture long- long-range dependencies compactly, aiding the retention of layout and structure. The module exerts the ability to cluster based on the underlying redundancy after decorrelation.
(2) compact representation: Many high-resolution regions exhibit sparsity in the frequency domain, allowing more aggressive merging without degradation.
compatibility: The module is lightweight, plug-and-play. and can be applied to pretrained diffusion models without retraining.
3.2 Theoretical Approximation Error Analysis
We now analyze the approximation error introduced by Token Merging as a preparation for analyzing
the benefits of Fourier Token Merging. For given input sequence X = X1, x2,"..,,",,
N D= of N feature vectors with bounded norm || |x,|| S R. 1, ...., N, self-attention transforms
X into the output sequence H in the following two steps:
3

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1. The input sequence X is projected into the query matrix Q, the key matrix K, and the value matrix V via three linear transformations
Q WQ K= W V WrX,
where We. WK E DDDD and Wy E DDxD2 are the weight matrices. Let Q := 91,... ] K == k.
kN), = M vNl where qi k; V, are query, key, and
value vectors respectively.
2. The output sequence H
h hx) is computed
QQK V
A
H
VT
VD
VD
where a denotes the softmax function applied row-wise to A = Q Let ai denote the attention scores.
Assuming linear attention, where o(x = xT, the output for query qi is:
N Σx
Σ q kj Vj h Wr
W Wox.
Suppose that we cluster the token set into C' disjoint clusters {Cx and approximate each token X, by its corresponding cluster centroid Xk())- Define the approximated attention output by
C Σ ECIII-XxX
h
Wv
W k W (Qx.())
(4)
where c(k) and c(i) denote the indices of the clusters containing tokens X₁ and x,, respectively. Now, analyze the effect of token merging by introducing the following theorem.
Theorem 1 (Token Merging Error Bound). Let {C*) be a clustering of the token set {x, 1N and let R 1c. Ejec, Xj be the centroid of cluster Ck- Suppose x; and all xj E C& replaced by their respective centroids. Then the approximation error between original and merged attention satisfies:
h,
14-hh h merged
Ih . |x;
Xc(9) + L2 Σ
Σ ||x - -xoll
k=1jECa
where c(i) is the index of the cluster containing x;, and constants L₁, L2 depend network parameters and operator norms.
Furthermore, following Cluster Distortion Decay Assumption 7 the total error satisfies the asymptotic bound
h merged
N
CLL.D.
indicating that merging accuracy improves as the number of clusters C increases.
The proof is provided in Appendix
3.3 Empirical Structural Analysis
To validate the theoretical insight presented in Theorem I we begin by analyzing the structural properties of token clusters formed during the merging process. Specifically, we examine whether Fourier-based token merging results in tighter and more coherent token groupings-two key factors that directly affect the magnitude of the approximation error bound. The results shown below are conducted with fixed truncation ratio = 0.7.
Figure presents comparison of structural integrity between Fourier and non-Fourier merging. Each point in the figure refers to specific merging layer. There are five merging layers for each diffusion step. The left plot reports the cluster similarity, defined as the average cosine similarity among token vectors within each cluster. Higher values indicate stronger internal coherence. Fourier merging

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clusters even consistently achieves greater intra-cluster similarity, suggesting that it identifies more semantically aligned groups. The right plot shows the token error within cluster, measured as the mean &g distance from each token to its cluster centroid. Again, Fourier merging results in significantly lower intra- cluster variance, supporting the claim that frequency-done Looking into the individual layers in each diffusion step, for both the Fourier merging and non-Fourier merging, the cluster similarity increases as the layer becomes deeper, meaning the similarity of tokens in deeper layers is better exploited by abstracting the features of the image.
groupings are geometrically compact.
Similarit
0.9
0.8
Non-Fourier
0.7
Fourier
Token Liror
1.5
1.0
0.5
Non-Fourrer
Fourier
10
20
30
40
50
Diffusion Step
Figure 2: Comparison of Cluster Similarity and Token Error across Diffusion Steps. This figure presents two metries averaged over diffusion steps to evaluate the structural behavior of token merging strategies. Up: The average cosine similarity among token vectors within the same Cluster.Fourier-base based merging maintains consistently higher similarity, indicating more coherent token grouping across the generation process. Down: The average token error within each cluster, defined as the mean distance between tokens and their cluster centroid. Fourier merging yields lower intra-cluster error, suggesting more compact and semantically aligned clusters. These trends highlight the advantage of Frequency-based merging in preserving structural consistency over time.
To further support this observation, Table D reports aggregated statistics over different merging ratios for both traditional and Fourier-based methods. As the merging ratio-i.e the degree of token count reduction per transformer block-increases, Fourier-based merging consistently maintains higher cluster similarity and lower token error. Notably, at low merging ratios (e.g.. 0.1), Fourier merging achieves a O. 16 reduction token error compared traditional merging, while involving a similar number of active cluster centers (Here, active cluster refers to the number of unique dst tokens receiving merged src tokens at each diffusion step. For example, there are 231 unique dst tokens receiving merged src tokens). As the ratio increases, Fourier merging slightly expands the number of such active clusters, yet still demonstrates superior semantic compactness, as reflected by the lower metric. These results suggest that Fourier- merging distributes merges more evenly while selectively preserving semantically important structures, leading to under higher compression.
Traditional Token Merging
Fourier Token Merging
Ratio Similarity (1)
Active Clusters
Token
Error
Similarity (1)
Active Clusters
Token
Error
Error
Metric
Error
Metric
(t)
())
(L)
(f)
(L)
())
0.1
0.90
231
0.73
13.85
0.91
231
0.57
11.94
0.2
0.88
396
0.74
19.23
0.89
399
0.76
19.12
0.3
0.87
507
0.98
33.61
0.89
523
0.86
27.11
0.4
0.87
624
1.23
46.96
0.88
605
0.98
40.56
0.5
0.86
706
1.32
59.29
0.86
716
1.38
58.82
Table I: Comparison between non-Fourier and Fourier token merging across different merge ratios.
Bold numbers indicate better performance (f: higher is better; 1: lower is better).
In Figure we further investigate cluster statisties across diffusion steps. The left subplot reports the number of active cluster centers (i.c., dst tokens) that the src tokens merged into at each step.
5

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This effectively reflects the distribution spread of token merges and indicates the level of compression granularity. The shaded region denotes the standard deviation across five runs. Interestingly, despite its structure-preservinng nature, Fourier-based merging results in comparable or even higher number of utilized cluster centers than its non-Fourier counterpart, suggesting broader merge distribution and reduced concentration bias.
The right subplot depicts an aggregate error metric within cluster, designed reflect intra-cluster dispersion as it appears in the second term of the inequality in Theorem | Although traditional merging performs clustering directly the time domain, Fourier-based merging surprisingly achieves lower within-cluster error throughout the generation trajectory.
This apparent contradiction arises from the fact that Fourier clustering operates denoised, low- frequency representation of the tokens, suppressing high-frequency fluctuations that often introduce spurious local variation. As result, the clusters it produces to transient noise. Thus, even though Fourier merging makes clustering decisions compressed space, the resulting assignments tend to yield lower dispersion in the original domain.
semantically coherent and robust
Number of Active Clusters
Error Metric Cluster
Non-Fourier
80
Non-Fourier
Fourier
Fourier
675
70
60
650-
Active Clusters
Erno Metric
50
625
40
600
30
575
20
550
10
50
30
50
Diffussio Step
Diffusion Step
Figure 3: Evolution of Error Metrics and Number of Active Cluster Centers across Diffusion Steps. This figure analyzes the dynamics of token merging through two global metrics. Left: Number unique dst tokens receiving merged src tokens at each diffusion step. This reflects the effective number of cluster centers actively used during token merging. Fourier-based merging exhibits a more stable and often broader distribution merges, indicating smoother and semantically consistent clustering behavior. Right: A global error metric measuring total intra-cluster variation. The Fourier merging consistently achieves lower eтооr across the generation timeline, indicating better information retention. Together, these plots validate the efficiency and accuracy benefits of incorporating Fourier-domain priors in token merging strategies.
Together, these results confirm that frequency-domain merging yields clusters with higher semantic reduce the approximation emor in attention computations.
coherence and reduced variance-two properties that, according to our theoretical analysis, directly
4
Adaptive Fourier Token Merging Scheme
4.1 Cluster Visualization Analysis
To better understand the behavior of frequency-basee token merging, we begin with a qualitative visualization of clustering results. Figure presents examples from cat images in the CIFAR-10 dataset, where clustering is applied to the red channel the original 32 x 32 images. For each pixel, extract 5 5 local patch centered that pixel and treat it as a token. The figure shows the largest 10 clusters obtained using both Fourier and non- non-Fourier token merging, visualized as 2D scatter plots.
We observe that Fourier-based clustering produces more structured and semantically coherent clusters. By suppressing high-frequency components, it is able to group together patches that are visually similar in texture or pattern, even when their raw pixel values differ significantly due to noise or local variance. This indicates that frequency-domain clustering can effectively abstract away irrelevant variations, yielding more meaningful token groupings.

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Cluster
Cluster
Claster 3
Claster
Claster
Cluster 6
Clusser ?
Cluster 8
Claster Cluster 10
The original (Red Channel)
Clusters
alt
Non-Fouriir Chasters
Fourier
Non-Fourier Clasters
Figure 4: Visualization of the top 10 clusters for Fourier and non-Fourier merging on two CIFAR- 10 cat images (red channel only). Each point corresponds token belonging to particular cluster. Fourier-based clustering yields denser and more structured groupings.
local patch treated
To assess whether these benefits generalize to deeper representations, we extend our analysis in Appendix B (Figure to intermediate activations from a CifarNet-like model (e.g., outputs from the second convolutional layer). Unlike raw pixel inputs, these activations resemble multi-channel heatmaps with more diffuse spatial organization.
In this higher-level feature space, find that clustering results remain structured, but now prioritize semantic similarity over local texture. Clusters appear less periodic and more abstract, indicating a shift in the type of information being preserved.
These findings suggest that the optimal degree of frequency truncation should be input-dependent: For low-level features (e.g., raw RGB patches), retaining more frequency information helps capture spatial regularities. For high-level activations, more aggressive truncation can be beneficial, suppressing noise and redundancy highlight semantically relevant patterns.
4.2 Adaptive Scheme for Truncation Ratio Scheduling
Motivated by the above analysis, we introduce an adaptive truncation scheme that dynamically adjusts the truncation ratio based diffusion timestep and layer depth-two factors that affect token complexity and noise level.
First, we adapt truncation along the diffusion step axis. In early diffusion steps, the input represen- tations are highly stochastic and noisy. This favors stronger truncation to eliminate gigh-frequency noise and improve clustering robustness. As generation proceeds and signals become more refined, we gradually reduce the truncation ratio to preserve finer details.
Second, we adapt along the layer depth within the network. Early layers tend to encode low-level details such as texture and edges, which are sensitive to frequency suppression. retain these features, we apply a higher truncation ratio (i.e., less truncation). In deeper layers, the tokens become more abstract and semantically structured, allowing for more aggressive frequency pruning without degrading cluster coherence.
Formally, model the truncation ratio function of timestep and layer index L, normalized within their respective ranges:
T(t, E) interpolate (Tmin: Tmaxi A ntrm(t) + X norm(f))
where d and As control the relative importance of timestep and layer depth, and Tmins Tmax define the bounds of the truncation ratio. These are considered as hyperparameters.
This adaptive scheduling mechanism ensures that Fourier Token Merging remains sensitive to the structural demands of each layer and each stage of the generation process, leading improved latency tradeoffs across the board.
5 Experiments
We conduct a series of experiments on the Fourier Token Merging to validate its efficacy of for efficient image generation.
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5.1 Experimental Setup
Methodology. To evaluate the effectiveness of Fourier Token Merging, integrate the proposed frequency-ban based method into Stable Diffusion (SD) (CC-BY 4.0) models [21]. We experiment with various merging ratios-ie.., different degrees of token count reduction per transformer block-to assess performance under different merging configurations. Results are compared against those the original Token Merging baseline. The Fourier Token Merging is applied to transformer blocks with a downsampling level of at most I, covering a total of five layers. We select Tmin in {0.3, 0.4, 0.5) and Tmax in {0.6, 0.8, 1.0], respectively, and traverse X X in {0, 0.1, 1.00 and report the best result.
Platform. Experiments are conducted on a system equipped with 1 12-core (8 Performance-cores and Efficient-cores) Intel Core i7-12700K CPU operating at a base frequency of 3.60GHz, 128GB of RAM, and NVIDIA GeForce RTX 4090 GPU with 24GB of GDDR6X memory. The GPU features 16,384 CUDA cores based on the Ada Lovelace architecture, offering a memory bandwidth of up to 1,008GB/s.
Workloads. The goal of Fourier Token Merging is to achieve better performance SD models while maintaining generation quality close to that of the full-token baseline. To evaluate this, we sample two images per class from all 1,000 ImageNet (CC-BY 4.0) categories, using prompts of the form "a photo of a CLASS". Comparisons are made with fixed random seed to ensure consistency across runs.
We use the Fréchet Inception Distance (FID) 9 to assess the distributional similarity between images generated by the diffusion model and 5,000 classbbalanced ImageNet- val examples. In addition, LPIPS [32] is used to measure pairwise perceptual similarity between images generated by the full model and the ones that use token merging. while MS-SSIM [27] 28 quantifies pairwise structural consistency. Both LPIPS and MS-SSIM are computed at the image level to capture deviations from the full-token baseline under a fixed random seed.
5.2 Experimental Results
We evaluate Fourier Token Merging by measuring its impact both image quality and generation efficiency. Figure reports latency against three metrics: FID, LPIPS, and MS-SSIM.
Latency FID
I Latency LPIPS
Latency MS-SSIM
2.0
Baseline
1.7
Baseline
1.340
Baseline
Lateney (sfm)
1.8
Ours
Lateney (sim)
1.6
Ours
Latency (sim)
1.335
Ours
1.6
1.5
1.330
1.4
1.325
1.4
1.3
gg
1.320
42
44
0.70
0.72
0.0775
0.0800 0.0825
FID (1)
LPIPS (1)
MS-SSIM (1)
Figure 5: Comparison of latency versus image quality metrics between the baseline and method.
The different points on a curve correspond to different token merging ratios. Left: FID vs. latency, showing that our method achieves lower latency for comparable better FID. Middle: LPIPS vs. latency, demonstrating improved perceptual similarity over baseline. Right: MS-SSIM vs. latency, indicating stronger structural consistency with minimal latency overhead. Note that Pareto frontiers are selected from a set of hyperparameter settings, and therefore differ across subplot. Overall, our tradeoff across multiple evaluation metrics.
approach consistently improves the quality-
FID. Our method achieves comparable or improved FID relative to the baseline across all configura- tions, while consistently reducing generation latency. This indicates that frequency-domain merging preserves the global semantics of generated images.
Despite aggressive compression, Fourier Token Merging maintains perceptual similarity on par with the baseline. In several cases, it achieves lower LPIPS at lower latency, suggesting improved retention of local visual features.
LPIPS.
MS-SSIM. We observe consistently higher MS-SSIM scores with minimal latency overhead, indi- cating stronger structural coherence the generated outputs. This aligns with the hypothesis that frequency-based grouping enhances token alignment.
8

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Across all metrics, Fourier Token Merging delivers favorable the theoretical benefits of frequency-aware token reduction in diffusion models.
tradeoffs, validating
5.3 Ablation Study
To better understand the contribution of each component in our design, we perform an ablation study across two axes: (i) the effect of the truncation ratio under fixed token merging ratios, and (ii) the impact of using Fourier-based clustering versus alternative diqquency-domaln such as wavelet transforms.
Effect of Truncation Ratio. To investigate the influence of the truncation ratio conduct controlled ablation by varying the fixed E 0.2, ..3..... 1.0 throughout the model across five fixed merging ratios (0.1 0.5). The resulting FID scores and latencies are reported in Figure 6 While the latency remains roughly the same across the truncation ratio, the FID score can change with different levels of truncation. We note that the optimal truncation level depends on the merging aggressiveness. When the merging ratio is low (i.e., fewer tokens are merged), preserving high-frequent information helps maintain generation fidelity. However, merging becomes more aggressive, discarding more components improves output quality. This suggests that low- low-frequency clustering becomes increasingly beneficial under stronger compression, possibly due its robustness to local noise and redundancy.
Merging Ratio 0.1
Merging Ratio 0.2
Merving Ratio 0.3
Mering Ratio
Merging Ratio
44
FID Score
42
0.5
1.0
0.5
1.0
0.5
1.0
0.5
1.0
0.5
1.0
Truncation
Truncation Ratio
Truncation
Truncation
Ratio
Figure 6: FID vs. Truncation Ratio under varying token merging ratios.
Wavelet vs. Fourier. We further compare the proposed DFT- based approach against a wavelet-based clustering scheme (Haar wavelet). Results in Figure 7 indicate that wavelet-based merging exhibits higher latency and inferior FID compared our method. This gap is mainly attributed to the extra over- head from multi-resolution decomposition and the lack of effi- cient GPU methods to implement wavelet transformation. Our Fourier approach, in contrast, allows for efficient transforma- tion and preserves salient global structure, leading to more favorable accuracy-efficieng, tradeoffs.
2.0
Wavelet
Baseline
Latency (s/im)
1.8
Ours
1.4
Score (1)
Figure 7: Wavelet vs. Fourier To- ken Merging: FID vs. Latency.
Real-coefffcients of Fourier tokens vs. full complex tokens. For efficiency, we adopt only the real coefficients of the Fourier tokens. Empirically, using the real part consistently achieves comparable or superior performance. For example, it attains 84.33% accuracy, slightly outperforming both the absolute-valu representation (84.29%) and the full complex representation (84%).
Fourier Token Merging
Baseline
Token Merging
5.4 Results for Generalizability
70%
50%
20%
0%
20%
50%
70%
Generalization to Diffusion Trans- formers (DiTs), Our Fourier Token Merging method aims for broader ap- plicability. Existing token merging methods often struggle with DiT's spe- cialized architecture and ROPE em- beddings. Specifically, our implemen- tation of Fourier Token Merging for the DIT architecture, which handles the complex structure of Joint/Single transformers, follows the integration
Figure 8: Visualization of Token Merging and Fourier Token Merging on the FLUX model.

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strategy introduced by Token Merging with Attention (ToMA) [14]. We have done experiments on DiT models, specifically FLux.I-dew tailoring FTM their specific transformer blocks. Results show benefits. Even with large merging ratio (0.75), we maintain high image quality (1.83 FID increase) while achieving over 20% latency reduction. This suggests FTM's strong generalization capabilities to diverse diffusion models, including DiT-like architectures, distinguishing our approach and highlighting its potential for current foundation models.
Generalization to Image Classification. We experimented our Fourier Token Merging in ViT models. Results show that our method generalizes well on image classification tasks, For ImageNet val- Ik dataset on the 4090 machine, with the same level of throughput (e.g. around 2000 im/s), our method increases the accuracy by 0.7 point (from 83.09% 83. (3.79% With the same level of accuracy, our method achieves 1.2 x speedups.
Regarding deployment efficiency, our empirical measurements confirm that the overhead is minimal. The wall-cloek time for the FFT is under 0.1 ms on average, which accounts for only 0.25% to 0.4% of the total inference time.
Overall, this study highlights the value of frequency-domain structural cues and validates our design choices in achieving compression with minimal quality degradation.
6 Conclusion
This work introduces Fourier Token Merging, a novel method for accelerating image generation by exploiting token redundancy in the frequency domain. By transforming tokens via the Discrete Fourier Transform and performing clustering in the low-frequency subspace, our method improves the semantic alignment and compactness of token groups, leading lower approximation error in self-attention computations. Theoretical analysis formalizes the error bounds introduced by token merging, and extensive empirical studies across structural metrics, attention distortion, and image quality evaluations (FID, LPIPS, MS- confirm the effectiveness of our approach.
Compared to baseline token merging methods, Fourier Token Merging achieves up 25% latency reduction while maintaining or even improving generation quality. Ablation studies further demon- strate the importance of frequency-aware clustering and validate the impact of truncation ratios under varying merging aggressiveness. We also explore adaptive schemes that adjust frequency truncation based on layer depth and diffusion timestep, offering flexible and robust extension the proposed framework. One limitation is that Fourier transform may be computationally intensive for resource-constrained devices. Its societal impacts are neutral.
Overall, our results highlight the potential of leveraging frequency-domain structure for efficient generative modeling, and provide a modular, plug-and-play solution applicable to a wide range of diffusion-based image generation systems.
Acknowledgement
We thank the anonymous reviewers for the constructive comments. We would also like to thank Wenbo Lu for sharing their implementation code for Token Merging with Attention (ToMA), which was instrumental in enabling our Flux. fllx.1--de 1-dev experiments. This material is based upon work supported by the National Institutes of Health (NIH) under Grant No. 1R01H2108473-01 and National Science Foundation (NSF) under Grant No. DAC-2417850 and P24-001771. Any opinions, findings, and conclusions recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NIH or NSF.
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A Proof of Theorem 1
Assumption Although the clustering method used in our token merging strategy not necessarily k-means, assume that it produces clusters with distortion comparable those obtained by k- means. That is, we assume the mean-squared intra-cluster error satisfies the standard quantization rate:
N
xx - -x())) (cr)
MSEcluster
Σ
N
where dy is the input feature dimension, and C' the number of clusters. This assumption is commonly satisfied under mild regularity conditions when the data lies on a compact manifold or is sampled from distribution with bounded support.
Proof. We express the attention output as:
h; Σ f(x,,))), where x(x,, (Wox,) (Wxx)) ((xxx))(((x)))) Wyx,.
Under token merging, all x, E Cg are replaced by Xk, and the query x; E Co(s) replaced by xx()- The merged attention output is:
h merged
f(xc()) Xx))
jEC.
The total error is:
8h, h merged
h;
Σ
(S(X)(9) Rx) [(x,, (xx,)x))
k=1jEC.
For each pair (1,j), we apply first-order Taylor expansion around (x,;
[(Xc()) &n) [xx x,) s Vx, (8e(0) x:) Vx, S (8x Xz)
Taking the norm and summing over all j, we obtain:
K
Joh
xll [|x.00 x; Vx, -|xx; xx
k= jECk
Bounding gradients by constants L1, L2. we arrive at:
|bb,|| S L1 ||xx(0) x; L2
|xj - X
A=IjECt
We then look into the second half of the theorem. Following and the total approximation error can be decomposed as
2h
hJ Wy [EE, Ex W W Qx, E W W.WWolxx xx(0))
where >i
Σラ x and 2 = De [
We now bound each term individually. Let ej Xj 8c() denote the clustering error. Then
N 11
Σ - E'IFF 2RN
2RN
MSE
N
-=1
13

---

If the input distribution satisfies mild regularity conditions (e.g., bounded support or sub- Lub-Gaussian), classical results from quantization theory imply
N
MSEcluster -0 C2/D=
|2 Eall-o
C1/Da
Meanwhile, the second term satisfies
(
x Xe(t)]l max ||ey|l v MSE -
C1/Da
Combining both, we have
b - h S |wvll (0
N Cl/D.
[|W + -(L) |W Well 1
CL/D,
which simplifies
N C1/Da
[|,
blll 0
This concludes the proof.
B Additional Clustering Visualization
We show the clustering visualization when the inputs are the intermediate representation.
Intermediate
Cluster
representation
(16 Channels)
Fourier
Clusters
Non-Fourier
Fourier
Clusters
Non-Fourier
Chasters
Fourier Clusters
Non-Fourier
Clasters
Fourier
Clusters
Non-Fourier
Clusters
Figure 9: Visualization of the top 10 clusters for Fourier and non-Fourier merging on intermediate representations. Same setting as in Figure Fourier-bace clustering yields denser and more structured groupings.
C Additional Related Work
Numerous methods have been proposed to reduce the computational cost of transformer- -hased models
by minimizing loken-level redundancy [ 4
10
14
[5
17
29
30), Among the most prominent
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Token Merging (ToMe) (2). DynamicVi1 VIT [18), and AdaViT [[6], which dynamically prune merge tokens based on content similarity or learned importance. These approaches significantly improve inference efficiency. However, they operate exclusively in the time domain and thus do not exploit the latent redundancy that may be more readily revealed through frequency-domain transformations.
Token Reduction in Vision Transformers. Token-level acceleration strategies have received substantial attention recent years. ToMe [2] reduces the number of tokens by clustering and merging similar ones during inference. Beyond pruning, other notable works focus on reorganizing token representations for efficiency. For instance, TC-Former [31] introduces human-centric clustering transformer that merges tokens based on content importance, while EVIT [13] employs progressive token pruning or early exiting by leveraging attention rollout and token significance, Touvron et al. (24] introduce distillation-basee approach for Vision Transformers, where specialized token used distill and condense teacher-mon meacher-model model representations into task-specific embedding, while Patch Slimming
23
reduces spatial redundancy through aggressive downsampling.
Frequency-domalle Representations. Compared to the wealth of time-domain methods, only a few studies have explored frequency-domain approaches for model acceleration. FNet ([2) replaces attention mechanisms with Fourier transforms in NLP models, offering a lightweight alternative to self-attention. Spectral pooling [20] leverages low-pass filtering in the frequency domain compress CNN activations. GFNet [19] highlights the untapped potential of frequency representations by replacing self-attention with learnable global filters in the frequency domain.
Additional Comparison SOTA. We included the results of several SOTA baselines as follows. AT-EDM 25 is SOTA token pruning method that offers 1.13 speedup over ToMe but loses quality. Our method, at a similar 1.1x speedup, actually decreases FID (40.75 vs. 41.31), showing better queddt trade-off. Other token Downsampling method [22] helps only very high merging ratios (c.g., 0.89), leading to longer latency at lower ratios (c.g.. 0.75). contrast, our method consistently achieves speedups across various ratios, e.g., 1.3x x at 41.8 FID.
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Reviews:
[{"review_id": "TsrLqsxHIN", "reviewer": "Reviewer_4fDD", "summary": "A novel Fourier-based token merging method is proposed in the context of efficient image generation. Instead of conducting the cluster assignment based on spatial token features, the input tokens are transformed via DFT to the Fourier domain, truncated, and subsequently matched using bipartite soft matching. The cluster assignments are then used to combine the spatial features. \n\nThe authors find that by truncating the high-frequencies in Fourier space better clusters are achieved, due to focusing on coarse-grained elements of the tokens. Due to the fine-grainedness of the tokens being affected both by the denoising timestep and the layer of the network, the truncation rate is formulated as a linear interpolation between these.\n\nThe authors also offer an error analysis of the prior Token Merging (ToMe) method as well as an empirical analysis of ToMe and the proposed Fourier ToMe method, where it is demonstrated that the proposed method has more coherent clusters and in terms of member similarity and distances.", "questions": "1. Has the authors considered using the proposed methods on other image recognition tasks such as image classification? A direct replica of the ToMe off-the-shelf experiments using SWAV, MAE, and ViT could demonstrate how well the proposed method generalizes.\n\n2. Does the authors have any idea of why the oscillation in Figure 2 happens? I cant determine if this is related to the network depth, or other factors. Any insights into this would be beneficial in better understanding the comparison.\n\n3. If the authors can provide justification for only using the Real-coefficients of the Fourier tokens or a comparison with the full complex tokens, this would in my opinion strengthen the methodology significantly.\n\n4. It is unclear why the assumption of linear attention is made on line 117. To me this seems like a major assumption which does not correspond to the actual operations of Tome. If a justification of this can be provided the proof would in my opinion be stronger.", "limitations": "The authors state on line 306 that \"One limitation is that it is currently limited to transformer-based diffusion models\". I believe this limitation is incorrect, as the proposed method is not inherently limited to diffusion models. Instead I would suggest that the manuscript is currently limited in its evaluation due to only comparing on a single method (Stable Diffusion v1.5).", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The proposed method is very simple to implement and clearly explained. This is extremely important as it allows for the proposed method to be easily adopted by future researchers, and applicable in real life applications.\n\n2. Interesting theoretical and empirical results are provided demonstrating the effectiveness compared to standard ToMe.\n\n3. The method appears to outperform the baseline across all tested metrics (FID, Latency, MS-SSIM, and LPIPS). This indicates that not only is it faster than the baseline, but also produces better quality images. Relevant ablation studies (ie effect of merging ratio and comparison to a Wavelet approach) are also conducted and demonstrate the effectiveness of the method.\n\n\nWeaknesses:\n\n1. In the experimental result section is unclear whether the baseline in Figure 5 is the original Stable Diffusion v1.5 or the ToMeSD adapted model. This makes it hard to correctly judge the effectiveness of the method. It is also unclear why the latency of the plotted points differ across the three subplots of Figure 5.\n\n2. There is a stark lack of visual examples of the generated images. This would in my opinion be an essential part of a paper focused on image generation.\n\n3. It is not motivated why only the Real component of the Fourier tokens are used. It is to me not clear why the Fourier tokens could not be matched based on cosine similarity of the complex feature values.\n\n4. The method appears to be relevant to not just image generation, but also tasks such as image classification. This omission, while not critical, does make me wonder if it generalizes beyond image generation tasks.", "was_revised": true, "final_justification": "The authors addressed all of my concerns and have demonstrated an extra effort in showcasing the effectiveness of the proposed model across tasks and models. I therefore wholeheartedly recommend the paper for acceptance.", "rebuttal": "Dear Reviewer,\n\nThank you for the insightful comments. We address the following concerns:\n\n> **Comment 1: In the experimental result section is unclear whether the baseline in Figure 5 is the original Stable Diffusion v1.5 or the ToMeSD adapted model. This makes it hard to correctly judge the effectiveness of the method. It is also unclear why the latency of the plotted points differ across the three subplots of Figure 5.**\n\nResponse: To clarify, the baseline in Figure 5 is the ToMeSD adapted model, not the original Stable Diffusion v1.5. Our goal was to directly compare our Fourier Token Merging (FTM) method against ToMeSD, which is a state-of-the-art token merging technique for Stable Diffusion.\n\nRegarding the different latencies across the three subplots in Figure 5, these variations are intentional. \nEach subplot represents a different merging and/or truncation ratio applied by our Fourier Merging method.\nThe latency ranges from 2 s/im (with no merging) to as low as 1.3 s/im and we selects those Pareto optimals to give a direct comparison.\nLarger merging or truncation ratios inherently lead to lower latencies due to the reduction in computational load.\nBy showcasing these different ratios, we demonstrate the trade-off between inference speed and image quality (FID score) for both our method and the ToMeSD baseline. This allows for a more comprehensive comparison across a range of operational points.\n\nFor instance, Figure 5 clearly illustrates that for the same level of image quality (e.g., around 41.5 FID), our FTM method achieves approximately 1.5$\\times$ speedup compared to ToMeSD. Conversely, if we target the same inference speed (e.g., around 1.38 seconds per image), our method yields a 0.5 lower FID score, indicating superior image quality.\n\n> **Comment 2: There is a stark lack of visual examples of the generated images. This would in my opinion be an essential part of a paper focused on image generation.**\n\nResponse: We'll address this by adding a dedicated section with visual comparisons of images generated using our method at various merging ratios, along with those from the original Stable Diffusion and the ToMeSD adapted model. It will clearly illustrate the quality retention of our method, even with higher merging ratios.\n\n> **Comment 3: Has the authors considered using the proposed methods on other image recognition tasks such as image classification? A direct replica of the ToMe off-the-shelf experiments using SWAV, MAE, and ViT could demonstrate how well the proposed method generalizes.**\n\nResponse: We experimented our Fourier Token Merging in ViT models.\nResults show that our method generalizes well on image classification tasks.\nFor ImageNet val-1k dataset on the 4090 machine, with the same level of throughput (e.g. around 2000 im/s), our method increases the accuracy by 0.7 point (from 83.0\\% to 83.79\\%).\nWith the same level of accuracy, our method achieves 1.2$\\times$ speedups.\n\n> **Comment 4: Does the authors have any idea of why the oscillation in Figure 2 happens? I cannot determine if this is related to the network depth, or other factors. Any insights into this would be beneficial in better understanding the comparison.**\n\nResponse: These oscillations directly relate to the network depth of the Stable Diffusion model. As network depth increases, token dimensions decrease, leading to greater feature abstraction. Both ToMe and our method show this. The UNet's architecture, with its varying representational changes at different depths, causes these fluctuations in similarity scores, reflecting the changing redundancy and information content of tokens throughout the network.\n\n> **Comment 5: If the authors can provide justification for only using the Real-coefficients of the Fourier tokens or a comparison with the full complex tokens, this would in my opinion strengthen the methodology significantly.**\n\nResponse: Since input tokens are real-valued, their Fourier Transform has conjugate symmetry, meaning the imaginary part contains redundant information. We hypothesized the real part alone could adequately capture essential frequency-domain information for token merging.\nFurthermore, discarding the imaginary part significantly boosts speed and simplifies implementation by reducing computational and memory demands.\nOur experiments also confirmed this choice. Using only the real part consistently yielded superior or comparable performance. For instance, it achieved 84.33\\% accuracy, slightly outperforming the absolute value (84.29\\%) and full complex representation (84\\%).\n\n> **Comment 6: It is unclear why the assumption of linear attention is made on line 117. To me this seems like a major assumption which does not correspond to the actual operations of Tome. If a justification of this can be provided the proof would in my opinion be stronger.**\n\nResponse: While ToMe uses softmax attention, we made this simplification for tractable and interpretable approximation error analysis. Softmax's non-linearity makes error decomposition difficult. Linear attention allows us to directly expose second-order token interaction influence. It also clearly decomposes approximation error after token clustering, offering theoretical insight into how merging impacts self-attention.\nOur goal isn't exact ToMe characterization, but to provide theoretical understanding through an analyzable surrogate, a common approach in approximate attention research (e.g., Performer, Linformer).\n\n> **Comment 7: The authors state on line 306 that ``One limitation is that it is currently limited to transformer-based diffusion models\". I believe this limitation is incorrect, as the proposed method is not inherently limited to diffusion models. Instead I would suggest that the manuscript is currently limited in its evaluation due to only comparing on a single method (Stable Diffusion v1.5.**\n\nResponse: Aside from the image classfication experiment, we will also include results for Diffusion Transformer (DiT) models in the revision.\nSpecifically, we have done experiments on DiT models, tailoring FTM to their specific transformer blocks. Results are promising: even with a large merging ratio (0.75), we maintain high image quality (1.83 FID increase) while achieving over 20\\% latency reduction.\nThis suggests FTM's strong generalization capabilities to diverse diffusion models, including DiT-like architectures, distinguishing our approach and highlighting its potential for current foundation models.\n\n> **Comment 8: Paper Formatting Concerns.**\n\nResponse: We're grateful for these precise suggestions, as they significantly improve the accuracy and clarity of our paper."}, {"review_id": "zzGqbFAstu", "reviewer": "Reviewer_87Fh", "summary": "This paper introduces Fourier Token Merging (FTM), a training-free, model-agnostic method for accelerating vision transformer inference by reducing token redundancy in the frequency domain. Unlike prior approaches that operate in the spatial domain, FTM uses a 2D FFT to identify and prune low-energy frequency components, which correspond to less informative spatial tokens. The method applies a single FFT and IFFT without modifying model weights or requiring retraining, and can be inserted at various layers during inference. Empirical results show that FTM achieves up to 2× speedup with minimal accuracy drop across multiple ViT architectures (e.g., DeiT, MAE, DINOv2) and tasks including classification, detection, and segmentation. It consistently outperforms baseline token reduction techniques like ToMe and UTS in terms of the accuracy–FLOPs trade-off, and generalizes well across models and datasets. Overall, FTM offers a simple, effective, and hardware-friendly solution for efficient ViT inference.", "questions": "See weaknesses.", "limitations": "N/A", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. FTM is a training-free and model-agnostic method that can be applied to a wide range of ViT architectures without retraining or modifying model weights. It leverages the frequency domain via FFT to identify and prune redundant tokens more effectively than spatial-domain methods.\n\n2. The approach is computationally efficient, requiring only a single FFT/IFFT pair, and achieves up to 2× speedup with minimal accuracy degradation. It outperforms existing token reduction methods (e.g., ToMe, UTS) in FLOP–accuracy trade-offs.\n\nWeaknesses:\n\n1. The method is evaluated on Stable Diffusion, which is relatively outdated. Based on prior experience, token merging techniques often do not transfer well to more recent diffusion transformer models such as DiT, PixArt, FLUX, or modern video generation models. These newer models tend to suffer significant accuracy drops under small merging ratios, casting doubt on the applicability of this line of work to current foundation models. Demonstrating that FTM generalizes effectively to DiTs would constitute a meaningful contribution; otherwise, the method appears to offer only incremental gains over existing token merging approaches.\n\n2. The use of frequency-domain operations, although efficient, may not be well-optimized or readily supported on all deployment hardware, potentially limiting practical benefits.\n\n3. The merging strategy is static and hand-crafted, lacking the adaptability of learnable or data-driven approaches that can tailor token reduction more effectively to specific tasks or inputs.\n\n4. Missing SOTA baselines such as token pruning [1,2,3] or token downsampling works [4].\n\n[1] Attention-Driven Training-Free Efficiency Enhancement of Diffusion Models, CVPR'24\n[2] Layer- and Timestep-Adaptive Differentiable Token Compression Ratios for Efficient Diffusion Transformers, CVPR'25\n[3] Accelerating Diffusion Transformers with Token-wise Feature Caching, ICLR'25\n[4] ToDo: Token Downsampling for Efficient Generation of High-Resolution Images, IJCAI'24", "was_revised": true, "final_justification": "The paper could make a significant contribution if it convincingly addresses the challenges of applying previous token merging techniques to diffusion transformers. However, based on the current limited results and unclear experimental settings, it is difficult to draw a definitive conclusion. More comprehensive evaluations and clearer descriptions are needed to support the claims. The merging strategy remains heuristic; a more adaptive or learnable approach would further strengthen the method.\n\nI am currently in a borderline position and believe that additional results (comparison to baselines; analysis on what and why make the differences) on diffusion transformers would help substantiate the method's effectiveness and improve the overall strength of the paper.\n\nThat said, I appreciate the \"Theoretical Approximation Error Analysis\" section—it provides a solid theoretical foundation for the proposed approach.", "rebuttal": "Dear Reviewer,\n\nThank you for the insightful comments. We address the following concerns:\n\n> **Comment 1: The method is evaluated on Stable Diffusion, which is relatively outdated. Based on prior experience, token merging techniques often do not transfer well to more recent diffusion transformer models such as DiT, PixArt, FLUX, or modern video generation models. These newer models tend to suffer significant accuracy drops under small merging ratios, casting doubt on the applicability of this line of work to current foundation models. Demonstrating that FTM generalizes effectively to DiTs would constitute a meaningful contribution; otherwise, the method appears to offer only incremental gains over existing token merging approaches.**\n\nResponse: While we primarily evaluated on Stable Diffusion, our Fourier Token Merging method aims for broader applicability. Existing token merging methods often struggle with DiT's specialized architecture and RoPE embeddings.\nWe have done experiments on DiT models, tailoring FTM to their specific transformer blocks. Results are promising: even with a large merging ratio (0.75), we maintain high image quality (1.83 FID increase) while achieving over 20\\% latency reduction.\nThis suggests FTM's strong generalization capabilities to diverse diffusion models, including DiT-like architectures, distinguishing our approach and highlighting its potential for current foundation models. We will add the results into the final version.\n\n> **Comment 2: The use of frequency-domain operations, although efficient, may not be well-optimized or readily supported on all deployment hardware, potentially limiting practical benefits.**\n\nResponse: Thanks for highlighting the hardware optimization aspect of frequency-domain operations.\nOur method relies on Fast Fourier Transform (FFT), which is highly optimized on GPUs: Modern GPUs excel at parallel FFTs. Libraries like cuFFT (NVIDIA) provide massive speedups, making our operations highly efficient and well-supported on GPU-accelerated platforms.\nEven constrained MCUs with DSP extensions or modern cores often have hardware accelerators or optimized DSP libraries (e.g., CMSIS-DSP) for efficient FFTs.\n\n> **Comment 3: The merging strategy is static and hand-crafted, lacking the adaptability of learnable or data-driven approaches that can tailor token reduction more effectively to specific tasks or inputs.**\n\nResponse:  Currently we support adaptive token merging by scheduling merging rates for different layers and diffusion steps.\nWe agree that incorporating more adaptability could be a future step to take to further develop this line of work.\nOne way is to allow dynamic merging ratios based on features of the input image (e.g., complexity, content density) or intermediate activations, similar to how some adaptive token pruning methods operate.\n\n> **Comment 4: Missing SOTA baselines such as token pruning or token downsampling works.**\n\nResponse:  Thanks for suggesting comparisons with SOTA token pruning and downsampling methods. %We've included the baselines.\nWe included the results of those baselines as follows.\n\nToken pruning: AT-EDM offers 1.13$\\times$ speedup over ToMe but loses quality. Our method, at a similar 1.1$\\times$ speedup, actually decreases FID (40.75 vs. 41.31), showing better quality-speed trade-off.\n\nToken Downsampling: This method helps only at very high merging ratios (e.g., 0.89), leading to longer latency at lower ratios (e.g., 0.75). In contrast, our method consistently achieves speedups across various ratios, e.g., 1.3$\\times$ at 41.8 FID.\n\nThe comparisons suggest that our method offers competitive or superior performance, especially in maintaining quality while providing consistent speedups. We'll include a comprehensive comparison in the revised manuscript."}, {"review_id": "RKOeJQwr3N", "reviewer": "Reviewer_kCHM", "summary": "This paper proposes Fourier Token Merging to speed up image generation. It transforms tokens into the frequency domain and clusters them using low-frequency information. This makes the tokens more compact and semantically aligned. The Fourier Token Merging reduces errors in self-attention and experiments on Stable Diffusion show up to 25% lower latency without losing image quality. The theoretical analysis is sufficient and the experiments show that the method is lightweight and easy to use with existing models.", "questions": "Will the code for this paper be open-sourced? It is recommended to do so, as it would greatly benefit and advance the community.", "limitations": "Please refer to the weakness part for rebuttal.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\n(1) The idea of performing token merging in the frequency domain is well-motivated. It extends beyond traditional spatial clustering by exploiting the decorrelation properties of DFT.\n\n(2) The formal error bound analysis for token merging is good and the reasons about why frequency-domain clustering reduces approximation error make sense.\n\n(3) In the experiment part, the method shows consistent improvements across FID, LPIPS and MS-SSIM metrics with significant inference speedups.\n\nWeaknesses:\n\n(1) Figure 1 needs to be refined. The clarity and aesthetics of this figure could be improved.\n\n(2) The paper does not provide any visual comparison of generated images between the proposed Fourier Token Merging and the baseline method. Since the primary goal is to preserve image quality while reducing computational cost, qualitative results are crucial to verify that the visual fidelity and semantics are indeed retained. Including such examples would strengthen the empirical claims.\n\n(3) The Fourier Token Merging indicates a plug-and-play solution, but the experiment is insufficient. It only conducts experiments based on stable diffusion model. In my opinion,  if a method clarifies that the method is a plug-and-play module, it needs to verify the proposed module into different backbones for a wider evaluation of the effectiveness.", "was_revised": true, "final_justification": "The author‘s rebuttal has addressed my concerns. The additional experiments to validate the generalizability of the proposed Fourier Token Merging (FTM) module are well-executed and convincing. I will keep my original rate.", "rebuttal": "Dear Reviewer,\n\nThank you for the insightful comments. We address the following concerns:\n\n> **Comment 1: Figure 1 needs to be refined. The clarity and aesthetics of this figure could be improved.**\n\nResponse: We will enhance the figure.\n\n> **Comment 2: The paper does not provide any visual comparison of generated images between the proposed Fourier Token Merging and the baseline method. Since the primary goal is to preserve image quality while reducing computational cost, qualitative results are crucial to verify that the visual fidelity and semantics are indeed retained. Including such examples would strengthen the empirical claims.**\n\nResponse: We'll address this by adding a dedicated section with visual comparisons of images generated using our method at various merging ratios, along with those from the original Stable Diffusion and the ToMeSD adapted model. It will clearly illustrate the quality retention of our method, even with higher merging ratios.\n\n> **Comment 3: he Fourier Token Merging indicates a plug-and-play solution, but the experiment is insufficient. It only conducts experiments based on stable diffusion model. In my opinion, if a method clarifies that the method is a plug-and-play module, it needs to verify the proposed module into different backbones for a wider evaluation of the effectiveness.**\n\nResponse: We applied our method to more models as follows.\nWe experimented our Fourier Token Merging in ViT models.\nResults show that our method generalizes well on image classification tasks.\nFor ImageNet val-1k dataset on the 4090 machine, with the same level of throughput (e.g. around 2000 im/s), our method increases the accuracy by 0.7 point (from 83.0\\% to 83.79\\%).\nWith the same level of accuracy, our method achieves 1.2$\\times$ speedups.\n\nAside from the image classfication experiment, we also include results for Diffusion Transformer (DiT) models.\nSpecifically, we have done experiments on DiT models, tailoring FTM to their specific transformer blocks. Results are promising: even with a large merging ratio (0.75), we maintain high image quality (1.83 FID increase) while achieving over 20\\% latency reduction.\nThis suggests FTM's strong generalization capabilities to diverse diffusion models, including DiT-like architectures, distinguishing our approach and highlighting its potential for current foundation models.\n\nWe will add those results into the final version.\n\n> **Comment 4: Will the code for this paper be open-sourced? It is recommended to do so, as it would greatly benefit and advance the community.**\n\nResponse: Yes. We will open source the code."}, {"review_id": "60lBBTIcGo", "reviewer": "Reviewer_qJuw", "summary": "The paper studies fourier token merging to accelerate image generation in diffusion models. It leverages frequency domain for cluster and token reduction. The proposed method results up to 25% reduction in generation latency with no loss in the performance. The author also provides analysis showing the insights of the proposed method.", "questions": "See Weaknesses above.", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths**:\n- The use of frequency domain for token merging is interesting.\n- The motivation is clear for the efficiency of generation process.\n- The proposed method is shown to be effective.\n\n**Weaknesses**:\n- The current method is evaluated only on diffusion models with U-Net architecture. Applicability to up-to-date architectures like Diffusion Transformer is not demonstrated.\n- The method introduces more hyper-parameters and the token merging process is highly affected by this hyper-parameters which may require more tuning during the training process.\n- The improvement is not significant. In Figure 5, The FID and latency improvement is not significant when compared to the baseline.", "was_revised": true, "final_justification": "Most of my concerns have been addressed in the rebuttal.", "rebuttal": "Dear Reviewer,\n\nThank you for the insightful comments. We address the following concerns:\n\n> **Comment 1: The current method is evaluated only on diffusion models with U-Net architecture. Applicability to up-to-date architectures like Diffusion Transformer is not demonstrated.**\n\nResponse: While we primarily evaluated on Stable Diffusion, our Fourier Token Merging method w for e applicability. Existing token merging methods often struggle with DiT's specialized architecture and RoPE embeddings.\nWe have done experiments on DiT models, tailoring FTM to their specific transformer blocks. Results are promising: even with a large merging ratio (0.75), we maintain high image quality (1.83 FID increase) while achieving over 20\\% latency reduction.\nThis suggests FTM's strong generalization capabilities to diverse diffusion models, including DiT-like architectures, distinguishing our approach and highlighting its potential for current foundation models. We will add the results into the final version.\n\n> **Comment 2: The method introduces more hyper-parameters and the token merging process is highly affected by this hyper-parameters which may require more tuning during the training process.**\n\nResponse: Our Fourier Token Merging method uses more hyperparameters, but this provides crucial control over the efficiency-quality trade-off, letting users customize for their needs (e.g., merging ratio).\nOptimal values for parameters like truncation ratios will be included in the config file.\nCrucially, as Fourier Token Merging is an off-the-shelf, training-free method, parameter tuning is highly efficient, allowing quick experimentation and rapid optimization for applications.\n\n> **Comment 3: The improvement is not significant. In Figure 5, The FID and latency improvement is not significant when compared to the baseline.**\n\nResponse: Our method not only reduces latency but also increases the image quality.\nFigure 5 clearly illustrates that for the same level of image quality (e.g., around 41.5 FID), our FTM method achieves approximately 1.5$\\times$ speedup compared to ToMeSD. Conversely, if we target the same inference speed (e.g., around 1.38 seconds per image), our method yields a 0.5 lower FID score, indicating superior image quality."}, {"review_id": "U8Zt3xjme9", "reviewer": "Reviewer_xhJ6", "summary": "The paper presents Fourier Token Merging, which improves image generation efficiency by using frequency domain clustering to reduce token redundancy. Unlike traditional methods in the time domain, this approach leverages Fourier transforms to better exploit redundancy, reducing latency by up to 25% in experiments with the Stable Diffusion model, while maintaining image quality. This contributes a novel technique for optimizing neural network computations in image generation.", "questions": "1. I think the amount of experiments in this work is not enough, especially more diffussion methods should be used for experiments instead of just one SD model.\n2. Figure 1 is not clear, and it is difficult to intuitively see the execution process of the Fourier Merging Module. It would be better to have a figure to explain the merging process in detail.\n3. In line 84, \"only the lowest τd frequency\", what is τd? It seems that there is a typo here.\n4. What is the connection between 3.2 and 3.1? A clear expression may be needed here.\n5. The writing of the Modular Integration section seemed confusing, and although there were descriptions of the advantages of the approach, I thought they were just common sight and did not help me understand how the proposed approach was effective.", "limitations": "1. Although this work contains a lot of quantitative results, it lacks image visualization comparison.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "1. By transforming tokens into the frequency domain, the method better leverages inherent redundancy after de-correlation, leading to more efficient token merging.\n2. Frequency domain clustering allows for more precise identification and merging of similar tokens, compared to traditional time domain methods.\n3. Experiments show up to a 25% reduction in latency during image generation processes.", "was_revised": true, "final_justification": "Thank you for authors' reply. I keep my original opinion.", "rebuttal": "Dear Reviewer,\n\nThank you for the insightful comments. We address the following concerns:\n\n> **Comment 1: I think the amount of experiments in this work is not enough, especially more diffussion methods should be used for experiments instead of just one SD model.**\n\nResponse: We applied our method to more models as follows.\nWe experimented our Fourier Token Merging in ViT models.\nResults show that our method generalizes well on image classification tasks.\nFor ImageNet val-1k dataset on the 4090 machine, with the same level of throughput (e.g. around 2000 im/s), our method increases the accuracy by 0.7 point (from 83.0\\% to 83.79\\%).\nWith the same level of accuracy, our method achieves 1.2$\\times$ speedups.\n\nAside from the image classfication experiment, we also include results for Diffusion Transformer (DiT) models.\nSpecifically, we have done experiments on DiT models, tailoring FTM to their specific transformer blocks. Results are promising: even with a large merging ratio (0.75), we maintain high image quality (1.83 FID increase) while achieving over 20\\% latency reduction.\nThis suggests FTM's strong generalization capabilities to diverse diffusion models, including DiT-like architectures, distinguishing our approach and highlighting its potential for current foundation models.\n\nWe will add those results into the final version.\n\n> **Comment 2: Figure 1 is not clear, and it is difficult to intuitively see the execution process of the Fourier Merging Module. It would be better to have a figure to explain the merging process in detail.**\n\nResponse: We will make clear the merging process through image illustration as detailed in Section 3.1.\n\n> **Comment 3: In line 84, ``only the lowest $\\tau d$ frequency\", what is $\\tau d$? It seems that there is a typo here.**\n\nResponse: As indicated earlier in the paragraph, $d$ is the dimension of the token **x** and $\\tau$ is the truncation ratio, so $\\tau d$ is $\\tau \\times d$ which refers to the dimension we keep after truncating on the frequency domain. We will add the clarification into the text to avoid any confusions.\n\n> **Comment 4: What is the connection between 3.2 and 3.1? A clear expression may be needed here.**\n\nResponse: Token Merging itself is an approximation computing method and brings in error for the image generation process.\nSection 3.1 discusses our main idea of introducing the Fourier Merging Module into the Token Merging pipeline. \nA theoretical interpretation and intuition behind the Fourier Token Merging is then explained in Section 3.2 and show the potential of achieving smaller approximation error compared to the original Token Merging.\n\n> **Comment 5: The writing of the Modular Integration section seemed confusing, and although there were descriptions of the advantages of the approach, I thought they were just common sight and did not help me understand how the proposed approach was effective.**\n\nResponse: Our idea is to cluster (merging) tokens in the frequency domain so as to maximize its decorrelation ability.\nWe show both theoretically that our method gives smaller approximation errors compared to the original method (Section 3.2) and empirically that our method merges tokens that have higher similarity scores (Section 3.3).\nSection 4 shows that Fourier-based clustering produces more structured and semantically coherent clusters.\nWe further evaluate end-to-end image generation and find our method not only increases the image generation quality but also achieves speedup compared to the original Token Merging method."}]

Author rebuttal:

Meta-review: This paper introduces Fourier Token Merging (FTM), a training-free method to accelerate inference in transformer-based image generation models. The core idea is to perform token merging in the frequency domain. By applying FFT, truncating high-frequency components, and then clustering the tokens, the method aims to merge tokens based on their underlying structural similarities, thereby reducing computational load more effectively than traditional spatial-domain approaches like Token Merging (ToMe).

The initial reviews were mostly in agreement, recognizing the method's simplicity, clear motivation, and strong theoretical and empirical support. However, multiple reviewers raised a significant and shared concern: the experiments were limited to the Stable Diffusion v1.5 model, which is a U-Net-based architecture. This raised serious questions about the method's generalizability to more modern and relevant architectures, particularly Diffusion Transformers (DiTs). Other common concerns included a lack of visual examples of generated images and the absence of comparisons to other state-of-the-art token reduction techniques.

The authors provided a thorough and convincing rebuttal that addressed every major point of criticism. They also conducted extensive new experiments that significantly strengthen the paper's contribution:
- Generalization to Diffusion Transformers (DiTs): They successfully applied FTM to the modern Flux.1-dev model, demonstrating a 23.4% speedup and a 50% memory reduction with a minor impact on FID. This result addressed the most critical weakness identified by the reviewers.
- Generalization to Other Tasks: They tested FTM on Vision Transformer (ViT) models for image classification, showing a 1.2x speedup or a 0.7 percentage point accuracy increase on ImageNet.
- SOTA Baseline Comparisons: They added new comparisons against recent token pruning and downsampling methods, demonstrating FTM's superior quality-speed trade-off.
- Additional Clarifications: They committed to adding visual comparisons and clarified other methodological points, such as the choice of baseline and the minimal overhead of the FFT operation (<0.4%).

The rebuttal convinced the reviewers. Reviewer 4fDD, who was already positive, became a strong advocate, recommending to accept the paper. Reviewer 87Fh, who was initially a "borderline reject," found the new DiT results convincing and stated they "resolved most of my concerns". The other reviewers also confirmed that their concerns were addressed. The review process worked as intended, with the authors' diligent response elevating a borderline paper to a clear accept.
Decision: Accept (poster)