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\section{Experiments}

\label{sec:experiments}



In this section, we provide extensive experiments to show
the superiority of our model and ample ablation studies to
demonstrate the effectiveness of components of our method.


\subsection{Results in Image Classification}
% {\bf Implementation details.} We perform image classification experiments on the  
% ImageNet-1K dataset,
\paragraph{Experimental Setup.}
We evaluate our models on the ImageNet-1K benchmark using
\begin{wraptable}{r}{.56\textwidth}
\vspace{-4mm}
\caption{Comparison of Efficient Models on ImageNet-1k. Latency is evaluated across various platforms.}
\vspace{-2mm}
\label{Main results}
\centering
\resizebox{0.55\textwidth}{!}{
    \begin{tabular}{lccccc}
        \toprule
       \multirow{2}{*}{ \textbf{Model}} & \textbf{Top-1} & \textbf{Params} & \textbf{FLOPs} & \multicolumn{2}{c}{\textbf{Latency (ms)}} \\
      %\multirow{2}{*}{ \textbf{Model}} & \multirow{2}{*}{ \textbf{Top-1 ($\%$)}}& \multirow{2}{*}{\textbf{Params (M)}} & \multirow{2}{*}{\textbf{FLOPs (G)}} & \multicolumn{2}{c}{\textbf{Latency (ms)}} &{\textbf{Input}}\\
        \cmidrule(lr){5-6}
        & (\%) & (M) & (G) & \textbf{GPU} & \textbf{Mobile} \\
        %&  &  &  & \textbf{GPU} & \textbf{Mobile} &\textbf{Size}\\
        \midrule
        FasterNet-T0 \cite{chen2023run} & 71.9 & 3.9 & 0.3  & 2.5 & 0.7 \\
        MobileV2-1.0 \cite{sandler2018mobilenetv2} & 72.0 & 3.4 & 0.3  & 1.7 & 0.9 \\
        ShuffleV2-1.5 \cite{ma2018shufflenet} & 72.6 & 3.5 & 0.3  & 2.2 & 1.3 \\
        %Mobileformer-52 & 68.7 & 3.6 & 52  & 8.3 & 26.0 \\
        EfficientFormerV2-S0 \cite{li2023rethinking} & 73.7 & 3.5 & 0.4 & 2.0 & 0.9 \\
        MobileNetv4-Conv-S \cite{qin2024mobilenetv4} & 73.8 & 3.8 & 0.2 & 2.2 & 0.9 \\
         StarNet-S2 \cite{ma2024rewrite} & 74.8 & 3.7 & 0.5 & 1.9 & 0.9 \\
         LSNet-T \cite{wang2025lsnet} & 74.9 & 11.4 & 0.3 & 2.9 & 1.8 \\
        \rowcolor{gray!30} GmNet-S1& \textbf{75.5} & 3.7 &0.6  & \textbf{1.6} & 1.0 \\
        \midrule
        
        {EfficientMod-xxs} \cite{ma2024efficient} &76.0 & 4.7 & 0.6 & 2.3 & 18.2 \\
        Fasternet-T1 \cite{chen2023run} & 76.2 & 7.6 & 0.9  & 2.5  & 1.0 \\
        EfficientFormer-L1 \cite{li2022efficientformer} & 77.2 & 12.3 & 1.3  & 12.1 & 1.4 \\
         StarNet-S3 \cite{ma2024rewrite} & 77.3 & 5.8 & 0.7  & 2.3 & 1.1 \\
          MobileOne-S2 \cite{vasu2023mobileone} & 77.4 & 7.8 & 1.3 & 1.9 & 1.0 \\
        RepViT-M0.9 \cite{wang2024repvit} & 77.4 & 5.1 & 0.8  & 3.0 & 1.1 \\
        %EdgeViT-XS \cite{pan2022edgevits} & 77.5 & 6.8 & 1.2  & 3.0 & 1.6 \\ 
        {EfficientFormerV2-S1} \cite{li2023rethinking}  & 77.9 & 4.5 & 0.7  & 3.4 & 1.1 \\
       
        \rowcolor{gray!30} GmNet-S2& \textbf{78.3} & 6.2 & 0.9  & \textbf{1.9} & 1.1 \\
        \midrule
        {EfficientMod-xs} \cite{ma2024efficient} &78.3 & 6.6 & 0.8 &2.9 & 22.7 \\
        
        
        %MobileV2-1.4 \cite{sandler2018mobilenetv2} & 74.7 & 6.9 & 0.6  & 1.6 & 1.3 \\
        %ShuffleV2-2.0 \cite{ma2018shufflenet} & 74.9 & 7.4 & 0.6  & {2.0} & 1.5 \\
        StarNet-S4 \cite{ma2024rewrite} & 78.4 & 7.5 & 1.1  & 3.3 & 1.1 \\
        SwiftFormer-S \cite{shaker2023swiftformer} & 78.5 & 6.1 & 1.0 & 3.8 & 1.1 \\
        RepViT-M1.0 \cite{wang2024repvit} & 78.6 & 6.8 & 1.2  & 3.6 & 1.1 \\
        UniRepLKNet-F \cite{ding2024unireplknet} & 78.6 & 6.2 & 0.9 & 3.1 & 3.5 \\
       
        \rowcolor{gray!30} GmNet-S3& \textbf{79.3} & 7.8 & 1.2  &  {2.1} & 1.3 \\
        \midrule
        
        RepViT-M1.1 \cite{wang2024repvit} & 79.4 & 8.3 & 1.3  & 5.1 & 1.2 \\
        MobileOne-S4 \cite{vasu2023mobileone} & 79.4 & 14.8 & 2.9 & 2.9 & 1.8 \\
        FastViT-S12 \cite{vasu2023fastvit} & 79.8 & 8.8 & 1.8  & 5.3 & 1.6 \\
        %FastViT-SA12 \cite{vasu2023fastvit} & 80.6 & 10.9 & 1.9  & 6.0 & 1.7 \\
        MobileNetv4-Conv-M \cite{qin2024mobilenetv4} & 79.9 & 9.2 & 1.0 & 9.2 & 1.4 \\
        LSNet-B \cite{wang2025lsnet} & 80.3 & 23.2 & 1.3 & 6.2 & 3.6 \\
        EfficientFormerV2-S2 \cite{li2023rethinking} & 80.4 & 12.7 &1.3 & 5.4 & 1.6\\
        %FastViT-S12  & 80.6 & 8.8 & 226  & 5.1 & 9.4 \\
        %MobileFormer-508M\\
        %EdgeViT-S\\
        
        %PoolFormer-s12 & 77.2 & 11.9 & 1.8 &11.9& 1.8 \\
        
        EfficientMod-s \cite{ma2024efficient} & 81.0 & 12.9 & 1.4 & 4.5 & 35.3 \\
        RepViT-M1.5 \cite{wang2024repvit} & 81.2 & 14.0 & 2.3 & 6.4 & 1.7 \\
        
        LeViT-256 \cite{graham2021levit} & 81.5 & 18.9  & 1.1 & 6.7 & 31.4\\
        \rowcolor{gray!30} GmNet-S4 & \textbf{81.5} & 17.0 & 2.7  & \textbf{2.9} & 1.9 \\
        \bottomrule
    \end{tabular}
    }
    \vspace{-5mm}
\end{wraptable}
an input resolution of $224 \times 224$ for both training and inference. To construct different GmNet variants, we vary the stage depth, embedding width, and expansion ratio while preserving the overall architectural design. Detailed configurations of each model scale are provided in the appendix.
All networks are optimized from random initialization for $300$ epochs using the AdamW optimizer with an initial learning rate of $3 \times 10^{-3}$. The global batch size is set to $2048$. Additional training hyperparameters, including data augmentation and regularization settings, are summarized in the supplementary material.
For latency evaluation, we export the trained PyTorch models to ONNX format and measure inference time on both an NVIDIA A100 GPU and a mobile device (iPhone 14). To ensure fair deployment comparison, the mobile benchmarks are conducted via CoreML conversion. Importantly, no re-parameterization, distillation, or architecture search techniques are applied; all reported results correspond to standard end-to-end training.

\noindent{\bf Compared with the state-of-the-art.} The experimental results are presented in Table \ref{Main results}. 
Without any strong training strategy, GmNet delivers impressive performance compared to many state-of-the-art lightweight models. 
With a comparable latency on GPU, GmNet-S1 outperforms MobileV2-1.0 by $3.5\%$. 
Notably, GmNet-S2 achieves $78.3\%$ with only $1.9$ms on the A100
 which is a remarkable achievements for the models under 1G FLOPS. 
GmNet-S3 outperforms RepViT-M1.0 and StarNet-S4 
by $1.9\%$ and $0.9\%$ in top-1 accuracy with $1.1$ ms and 
\begin{wrapfigure}{r}{0.4\textwidth}
\centering
\vspace{-2mm}
  \includegraphics[width=0.4\textwidth]{imgs/acc_latency3.pdf}
  
    \caption{Trade-off between Top-1 accuracy and latency on A100. 
    }
   \label{latency_acc}
   \vspace{-4mm}
    %\hfill
\end{wrapfigure} 
$1.4$ ms faster on the GPU latency, respectively. The improvements on
% \setlength{\intextsep}{-2pt}   
the speed are over $30\%$.
Additionally, with similar latency, GmNet-S3 delivers a $1.7\%$
improvement on the accuracy over MobileOne-S4. 
GmNet-S4 achieves 2x faster compared to RepViT-M1.5 on the GPU and it
surpasses MobileOne-S4 of $2.1\%$ under the similar latencies of both GPU and Mobile. LeViT-256 \cite{graham2021levit} matches the accuracy of GmNet-S4 but runs twice as slow on a GPU and 16 times slower on an iPhone 14
%These results clearly demonstrate that pure lightweight CNNs, with efficient architectural designs, can outperform the leading lightweight ViTs on mobile devices.
The strong performance of GmNet can be largely attributed to the clear insights of gating mechanisms  and simplest architectures.
Fig. \ref{latency_acc} further illustrates the latency-accuracy trade-off across different models.  
GmNet variants achieve substantially lower latency compared to related works, while maintaining competitive or superior Top-1 accuracy.
More comparisons and results can be found in supplementary.
%Notably, since the distillation has been demonstrated by some existing methods as an effective training strategy for fairness, results are compared with methods trained without the distillation.



\subsection{Ablation Studies}
%To evaluate the contribution of each component in our model and provide more evidences on our study on frequency learning, we conducted a series of ablation studies. 
%\\



\noindent{\bf More studies on different activation functions.} To further explore the effect of different activation functions, we trained various GmNet-S3 variants
%only with raw data of 
on ImageNet-1k.
%to avoid a heavy computational burden. 
As illustrated in Fig. \ref{arch-main}, we replaced ReLU6 with GELU, ReLU or remove the activation function. 
%Considering the higher resolutions of testing images, which implies more complex spectral information,
To better reflect the differences between different models, 
we set the radii to a larger range/ 
%For $ \mathbf{z}_i \in (0, r)$, we recognize it as the low-frequency component while $ \mathbf{z}_i \in (r, +\infty)$ represents the high-frequency component \cite{xu2019frequency, wang2020high}. 
\begin{wraptable}{r}{.6\textwidth}
%\vspace{-.2in}
\caption{The accuracies of classifying the raw data and their low-/high-frequency components under different activation functions on ImageNet-1k. We gradually increase the radii by a step of $12$. This result is the average of five testings. }
\vspace{1mm}
\label{eff-act}
\centering
\resizebox{0.59\textwidth}{!}{
\begin{tabular}{lcccccccc}
        \toprule
        Activation  & \multicolumn{2}{c}{Identity} & \multicolumn{2}{c}{ReLU} & \multicolumn{2}{c}{GELU} & \multicolumn{2}{c}{ReLU6} \\
        \midrule
        Raw data & \multicolumn{2}{c}{70.5} & \multicolumn{2}{c}{78.3} & \multicolumn{2}{c}{78.4} & \multicolumn{2}{c}{79.3} \\
        \cmidrule(lr){1-1} \cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7} \cmidrule(lr){8-9}
        Frequency & Low & High & Low & High & Low & High & Low & High \\
        \cmidrule(lr){1-1} \cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7} \cmidrule(lr){8-9}
        $r = 12$  &9.79 & 12.6  &12.0 &45.9 & 12.7&41.5 & 14.8&{51.7} \\
        $r = 24$  &  38.1 & 1.7 &38.6 &13.5  &40.0 & 9.4 &41.6&{12.1} \\
        $r = 36$  & 52.9 & 0.7 &56.2&4.9 &{58.7} & 3.9 & 55.2&4.7 \\
        $r = 48$  &63.2 & 0.5& 64.5&2.3 & {66.1} & 2.1 & 64.4&2.5 \\
        $r = 60$  &66.6 & 0.9& 69.4&1.0 & 70.7 & 1.1 & 71.1&1.4 \\
        \bottomrule
    \end{tabular}}
\end{wraptable}
%This change reduces computational complexity while still capturing the essential spectral properties without requiring further decomposition.
As shown in the Table. \ref{eff-act}, we can find that, the increases on classifying the high-frequency components are significant comparing models using and not using the activation functions. 
For example, comparing results of `Identity' and `ReLU' with the improvement of $11\%$ on the raw data, improvement on high-frequencies is over 3 times on average. `GELU' and 'ReLU' shows advances on low-/high- frequency components respectively compared to each other. This aligns with our understanding of how different types of activation functions impact frequency response.
%On the other hand, the closer performances on low-frequencies between `Identity' and `ReLU'/`GELU' indicate the low-frequency bias of conv-based networks.
Notably, the closer performance of models with Identity and ReLU/GELU at low frequencies suggests the low-frequency bias of convolution-based networks.



%of performs better that `ReLU' on low-frequency components especially  when $r=12$.

Moreover, even considering the improvements on the raw data, model using the ReLU6 shows obvious increase on the high-frequency components compared to the model using GELU especially when we set $r$ to $12$, $24$, $36$. 
Compared to the model with ReLU, ReLU6 is more effective in preventing overfitting to high-frequency components since it has better performance on low-frequencies.
Considering performances of ReLU, GELU, and ReLU6, we can observe that achieving better performance on high frequencies at the expense of lower frequencies does not necessarily lead to overall improvement, and vice versa.
To get a better performance on the raw data, it is essential to enhance the model's ability to learn various frequency signals.
%Also, comparing 'Identity', 'GELU' and 'ReLU6', it can also indicate that learning on the high-frequency components can help the model improve the performance.
%For our lightweight model, ReLU6 is one of the most effective choices. 
%However, it does not mean it is suitable for all models and all tasks.
%In this section, our purpose is to prove a solid and insightful angle to explain the training dynamic of GLUs. 
%We hope readers can get some enlightenment of designing their own models.
\\



\noindent{\bf Comparison with existing methods from the frequency perspective.} As addressed in Table \ref{eff-act}, 
a model should achieves strong
performance across different frequency components to deliver a better overall performance.
However, 
both pure convolutional architectures and transformers exhibit a low-frequency bias, as discussed in \cite{bai2022improving, tang2022defects}. 
Therefore, enhancing the performance of a lightweight model depends on its ability to more effectively capture high-frequency information.
%As shown in Tab. \ref{eff-act}, we can find that it is not necessarily leading to a better overall performance if a model performs better on high-frequencies but worse on low-frequencies (ReLU vs GELU). However, both pure 
%convolutional  architectures and transformers have a low-frequency biased property  which has been addressed in \cite{bai2022improving, tang2022defects}. 
%Therefore, the key to improve the performance of a lightweight is related to better capture the high-frequency information. 
To address the advantages of GmNet on overcoming the low-frequency bias, 
we test some existing models on different frequency components of different 
\begin{wraptable}{r}{.6\textwidth}
% \vspace{-3mm}
\centering
     \caption {Comparison with recent methods. We test models on the high-/low-frequency components
     %with different radii 
     on the ImageNet-1k. The highest values of each columns are highlighted.}
     \vspace{-1.5mm}
\resizebox{0.6\textwidth}{!}{
    \begin{tabular}{lccccccc}
        \toprule
       \multirow{2}{*}{ {Methods}} & {Top-1}  & \multicolumn{2}{c}{$r = 12$}& \multicolumn{2}{c}{$r = 24$}& \multicolumn{2}{c}{$r = 36$} \\
       %\cmidrule{lr}{3-4} \cmidrule{lr}{5-6} \cmidrule{lr}{7-8}
        & (\%) & {High}& {Low}& {High}& {Low}& {High}& {Low} \\
        \midrule
        
        MobileOne-S2 \cite{vasu2023mobileone} & 77.4 & 35.0&11.6&6.5&36.9&2.4&53.5\\
        
       EfficientMod-xs \cite{ma2024efficient} & 78.3 & 45.4&12.9&9.4&40.6&3.5&54.6 \\ 
       
       StarNet-S4 \cite{ma2024rewrite}& 78.4 & 43.3&13.8&9.4&41.3&3.4 &54.8\\
         GmNet-S3 & \textbf{79.3} & \textbf{51.7} & \textbf{14.8} & \textbf{12.1} & \textbf{41.6} & \textbf{4.7} & \textbf{55.2} \\

        \bottomrule
    \end{tabular}}
    \vspace{-3mm}
    \label{fre-methods}
\end{wraptable}
radii. 
We select three kinds of typical lightweight methods for comparison including pure conv-based model MobileOne-S2 \cite{vasu2023mobileone}, attention-based model EfficientMod-xs \cite{ma2024efficient} and model also employing GLUs-like structure 
StarNet-S4 \cite{ma2024rewrite}.
As shown in Table \ref{fre-methods}, accuracies of low-frequency components are close 
among different models considering the overall performance. 
However, 
it shows that GmNet-S3 clearly surpass the other models in high frequency components. For example, GmNet-S3 has a $6.3\%$ improvement compared to EfficientMod-xs when $r = 12$ and $2.7\%$ increase when $r = 24$. 
 For StarNet, which also uses a GLU-like structure with dual-channel FC, it struggles to effectively emphasize high-frequency signals.
The simplest GLUs design can achieve a better
balance between the efficiency and the effectiveness. 
%Although directly classifying the high-frequency components is very hard to models, it can still help us understanding how the performance improves from a frequency perspective.
%Moreover, in the supplementary materials, we provide additional comparisons of various methods.  More discussions are provided about the relationship between responses to different frequency components and overall performance.
\\
\begin{table*}[t]
    \centering
    \caption {Comparison of different GLU designs for GmNet-S3 on ImageNet-1K. Here, LN, DW, and Pool represent layer normalization, depth-wise convolution with a kernel size of 3, and average pooling with a 
    3×3 window, respectively. We underline all notable scores in classifying the different frequency decompositions. Considering gaps of overall performances, an improvement which is remarkable should exceed $1.0$. This result is the average of five testings.
    We also provide more variants of GLUs in the supplementary materials. }
  %  \tabcolsep=0.1cm
     \vspace{1mm}
     % \renewcommand{\arraystretch}{0.1}
     \resizebox{0.99\textwidth}{!}{
    \begin{tabular}{lccccccccccccc}
        \toprule
       \multirow{2}{*}{ {GLUs}} & {Top-1} & {Params} & {GPU} & \multicolumn{2}{c}{$r = 12$}& \multicolumn{2}{c}{$r = 24$}& \multicolumn{2}{c}{$r = 36$}& \multicolumn{2}{c}{$r = 48$}& \multicolumn{2}{c}{$r = 60$}\\
        %\cmidrule(lr){5-6}\cmidrule(lr){7-8}\cmidrule(lr){9-10}\cmidrule(lr){11-12}\cmidrule(lr){13-14}
        & (\%) & (M)  & (ms) & {Low} & High & {Low} & High & {Low} & High & {Low} & High & {Low} & High \\
        \midrule
        
        $\sigma(x)\cdot\text{LN}(x)$ & 78.9 & 7.8 & 2.9  & 12.1&47.6&41.6&10.9&56.4&\underline{5.2}&64.7&2.4&69.8&1.2 \\
        
       $\sigma(x)\cdot\text{DW}(x)$ & 79.0 & 8.0 & 2.4 & 12.3&49.0&\underline{42.7}&9.6&$\underline{58.1}$&4.6&\underline{65.7}&2.3&\underline{71.2}&1.1 \\
       
       {$\sigma( x)\cdot(x-\text{Pool}(x))$} & 78.6 & 7.8 & 2.4 & 14.2&50.1&42.3&10.8&55.8&4.9&63.8&2.7&69.9&1.3 \\
        $\sigma(x)\cdot\text{FC}(x)$ & 79.2 & 20.2 & 3.6 & 10.8&51.4&39.6&8.7&52.6&4.4&62.9&3.4&69.9&2.4 \\
         $\sigma(x)\cdot x$ & 79.3 & 7.8 & 2.1  & \underline{14.8} & \underline{51.7} & 41.6 & \underline{12.1}&55.2&4.7&64.4&2.5&71.1&1.4 \\
        \bottomrule
    \end{tabular}}
    \vspace{-3mm}
    \label{glu-design-main}
\end{table*}




\noindent{\bf Study on designs of  the GLU.} In GmNet, the gated linear unit adopts the simplest design, which can be defined as $\sigma(x)\cdot x$.
For comparison, we modify the GLU design and conduct experiments to test performance on raw data as well as on decompositions at different frequency levels.
%We design the variants with two main objectives: (1) smoothing the introduced frequency components, as in $\sigma(x)\cdot\text{LN}(x)$ and $\sigma(x)\cdot\text{DW}(x)$; (2) introducing more higher-frequency components like $\sigma( x)\cdot(x-\text{Pool}(x))$. LN, DW, and Pool denote layer normalization, depth-wise convolution, and 2D average pooling, respectively.
As shown in the Table \ref{glu-design-main}, the simplest design achieve the best performance both on effectiveness and efficiency for the overall performance. 
For the decomposed frequency components, we observe clear differences among various GLU designs. The GLU of $\sigma(x)\cdot x$ demonstrates significantly higher accuracy in classifying high-frequency components.
For example, for $r = 12$ and $r = 24$, the GLU with $\sigma(x)\cdot x$ shows an improvement of $4.1$ over the LN design and $2.5$ over the DW design. 
This indicates that the simplest GLU design is already effective at introducing reliable high-frequency components to enhance the model's ability to learn them. 
Designs aimed at smoothing information show a notable improvement in some low-frequency components. 
For instance, with similar overall performance, the GLUs using $\sigma(x)\cdot\text{DW}(x)$ and $\sigma(x)\cdot\text{LN}(x)$ achieve better results on low-frequency components when the radii are set to $24$, $36$, and $48$. 
The model using a linear layer in GLUs offers performance comparable to GmNet-S3 and is adept at learning low-frequency features.
However, its placement at a high-dimensional stage is problematic. This design choice leads to an excessive number of parameters and a significant increase in latency.
{Moreover}, depth-wise convolution is more effective than layer normalization in encouraging neural networks to learn from low-frequency components which is also more efficient. 
For the design with the average pooling, it does not perform better in classifying high-frequency signals. This may be because 
$x -\text{pool}(x)$ acts as an overly aggressive high-pass filter, which does not retain the original high-frequency signals in $x$ well and instead introduces more high-frequency noise. 
\\

\noindent{\bf Bandwidths analysis of convolution kernels.} As discussed in the \cite{tang2022defects}, the convolution layer may play roles of 'smoothing' the feature which means it has a low-frequency bias. Experiments on studying weights of the convolution layer is insightful to give more evidences of how GLUs effect the learning of different frequency components \cite{wang2020high, tang2022defects, bai2022improving}. 
%In there paper, they use the weight map to visualize the change of the kernal which is 
In this paper, we propose using the bandwidths of convolution kernels to represent their ability of responding to different frequency components. Specifically, a wider bandwidth indicates that the kernel can process a broader range of frequencies, allowing it to capture diverse frequency components simultaneously and thereby preserve rich information from the feature. 
As illustrated in Figure \ref{bandwidth}, the distributions of the ReLU model suggest that its convolution kernels tend to focus on a narrow range of frequency components leading to relatively lower bandwidths.
It indirectly reflects an overemphasis on high-frequency components. 
%This causes the convolution kernels to focus on a narrow range of frequency components, either high or low, leading to relatively lower bandwidths.
%As a smooth function, the model using GELU activation may tend to capture lower-frequency components compared to the model using ReLU6. 
%Compared to others, the improved bandwidth distribution of convolution kernels of the model using ReLU6 indicates that GLUs with both GELU and ReLU have a frequency bias while ReLU6 has better generalization in this task.
%Although the model using GELU performs a better distribution at the beginning , it still suffers from the low-frequency bias and lead to bad performance in the ending block. 
Although the model using GELU exhibits a better distribution in the top convolutional layers, it still has a low-frequency bias, leading to a distribution shift in the bottom convolutional layers.
Compared to other activations, the enhanced bandwidth distribution of the model using ReLU6 demonstrates better generalization for this task.
%encourage the model to better capture various frequency components, 
The properties of the convolution kernels align with results in Table \ref{eff-act}.
\\

\begin{figure}[t]
  \centering
    \includegraphics[width=0.49\linewidth]{imgs/ke_w_b.png}  
  \hfill
    \includegraphics[width=0.49\linewidth]{imgs/ke_w_e.png}
   %\includegraphics[width=0.8\linewidth]{egfigure.eps}
   \vspace{-2mm}
   \caption{The histogram illustrates the distribution of bandwidths of convolution kernels. Bandwidths represents the capability of  a convolution kernel for capturing various frequency information. 
   We use weights of the convolution layer which under
   the GLU in the first block (left) and the last block (right) of the GmNet-S3. All modals are trained on the raw data of the ImageNet-1k. In general, the further the distribution shifts to the right, the stronger the convolutional kernel's ability to capture signals of different frequencies.}
   \vspace{-2mm}
   \label{bandwidth}
\end{figure}
%The details of how to compute the bandwidths can be found in the supplementary. 
%By computing the frequency responses and calculating the bandwidths of the convolution kernels, we can quantitatively compare two sets of kernels in terms of their ability to capture different frequency information. 
%This method provides a clear and measurable way to assess and select kernels based on their frequency characteristics, which is crucial in applications such as image processing and signal analysis.