| \section{Revisiting Gating Mechanisms from A Frequency View} | |
| \label{sec:analysis} | |
| % \begin{wrapfigure}{r}{0.4\textwidth} | |
| % \centering | |
| % \includegraphics[width=1.0\linewidth]{imgs/variants1.png} | |
| % %\includegraphics[width=0.8\linewidth]{egfigure.eps} | |
| % \vspace{-3mm} | |
| % \caption{Block design of different variants of ResNet18 where $\odot$ represents the element-wise product and $\sigma$ means the activation function. | |
| % In this paper, we investigate GLUs using ReLU6 and GELU as activation functions where corresponding variants can be named as Res18-Gate-ReLU6/GELU. | |
| % %Base on the activation function used in Res18-Gate, | |
| % %the variant can be named as Res18-Gate-ReLU or Res18-Gate-GELU. | |
| % } | |
| % \label{fig-variants} | |
| % \end{wrapfigure} | |
| %To better understand how gating mechanisms work, this paper investigates the behavior of a convolutional neural net168 work (CNN) in classification tasks from a frequency perspective. | |
| % \paragraph{Frequency Decomposition.} | |
| % To examine spectral characteristics, we first transform the input image $x$ into the frequency domain. Let | |
| % \begin{equation} | |
| % z = \mathcal{F}(x), | |
| % \end{equation} | |
| % where $\mathcal{F}(\cdot)$ denotes the two-dimensional discrete Fourier transform (DFT). | |
| % We partition the spectrum into two complementary components based on a predefined cutoff radius $r$. Specifically, the frequency representation $z$ is decomposed into a low-frequency part $z_l$ and a high-frequency part $z_h$, satisfying | |
| % \begin{equation} | |
| % z = z_l + z_h, \quad (z_l, z_h) = \Theta(z; r), | |
| % \end{equation} | |
| % where $\Theta(\cdot; r)$ represents a radial masking operator that separates spectral coefficients according to their distance from the origin in the frequency plane. | |
| % In this formulation, coefficients located within radius $r$ are assigned to $z_l$, while those outside the radius are assigned to $z_h$. This decomposition enables us to analyze model behaviors under controlled spectral partitions. | |
| We begin by defining the components associated with different frequency bands and outlining the | |
| details of our experimental setup. | |
| With decomposing the raw data $\mathbf{z}$ into the high-frequency part $\mathbf{z}_h$ | |
| and the low-frequency part $\mathbf{z}_l$ where $ \mathbf{z} = \mathbf{z}_h + \mathbf{z}_l$. | |
| We partition the spectrum into two complementary components based on a predefined cutoff radius $r$. Specifically, the frequency representation $z$ is decomposed into a low-frequency part $z_l$ and a high-frequency part $z_h$, satisfying | |
| \begin{equation} | |
| \label{decompose} | |
| z = z_l + z_h, \quad (z_l, z_h) = \Theta(z; r), | |
| \end{equation} | |
| where $\Theta(\cdot; r)$ represents a radial masking operator that separates spectral coefficients according to their distance from the origin in the frequency plane. | |
| In this formulation, coefficients located within radius $r$ are assigned to $z_l$, while those outside the radius are assigned to $z_h$. This decomposition enables us to analyze model behaviors under controlled spectral partitions. | |
| % Denoting a threshold $r$ and an image $\mathbf{x}$, we have the following equations: | |
| % \begin{equation} | |
| % \label{decompose} | |
| % \mathbf{z}=\mathcal{F}(\mathbf{x}), \quad \mathbf{z}_{h}, \mathbf{z}_{l} =\theta(\mathbf{z} ; r). | |
| % \end{equation} | |
| % Here, $\mathbf{z}=\mathcal{F}(\mathbf{x})$ represents the two-dimensional discrete Fourier transform of the input $\mathbf{x}$. | |
| % % The operator $\theta(\cdot; r)$ applies a radial cutoff that partitions $\mathbf{z}$ into low- and high-frequency components, controlled by the radius parameter $r$. | |
| % where $\mathbf{z}=\mathcal{F}\left(\mathbf{x}\right)$ is the 2D Discrete Fourier Transform of $\mathbf{x}$ and $\theta(·; r)$ denotes a thresholding function that separates | |
| % the low and high frequency components from $z$ according | |
| % to a hyperparameter, radius $r$. | |
| We select three vision backbones including ResNet-18 \cite{he2016deep}, MobileNetv2 \cite{sandler2018mobilenetv2} and EfficientFormer-v2 \cite{li2023rethinking} as representations to demonstrate the drawbacks of the CNN networks and transformer-based architectures on capturing the high frequency information and how the gating mechanism improve the capability of learning high-frequency components. | |
| Modifications to the network blocks of ResNet-18 are depicted in Fig. \ref{fig-variants}. | |
| %All networks are trained on the CIFAR-10 dataset using CrossEntropyLoss, with the SGD optimizer set to a learning rate of $0.1$, implemented in PyTorch \cite{paszke2019pytorch}. | |
| %All experiments in this section utilize ResNet18 \cite{he2016deep} as the backbone, implemented in PyTorch \cite{paszke2019pytorch}. Modifications to the network blocks are depicted in Fig. \ref{fig-variants}. | |
| %To deliver a better comparison on high-frequency, the decompositions are also used in the training. | |
| We evaluate the classification performance on different frequency components of the input images at each training epoch. | |
| %Since directly classifying the high-frequency components is hard, the model is also trained with different frequency components. | |
| Changes in accuracy over time provide insights into the learning dynamics within the frequency domain \cite{wang2020high}. | |
| To avoid the occasionality, we calculated the average over three training runs. | |
| %We begin by defining the components associated with different frequency bands and outlining the details of our experimental setup. | |
| %The classification accuracy on the original data across different models remains comparable. We think analyzing training dynamics becomes more reliable when networks exhibit similar overall performance. | |
| %All related experiments in this section employ the ResNet18 as the backbone based on PyTorch[]. Different modification on the block can be found in Fig.XX. We train the network on CIFAR-10 with a CrossEntropyLoss and use the Adam as the optimizer with a learning rate of 1e-4. | |
| %We test of classifying different frequency components of input images which is defined in Eq. \ref{decompose} for each epoch. The change accuracy can indicate the learning dynamic in the frequency domain []. | |
| %Notably, the accuracy of classifying on raw data across different models are close. The study of the training dynamic is more reliable under networks with the similar overall performance. | |
| %XXX the overall performance of different models are close. It is important to study | |
| \begin{figure*} | |
| \centering | |
| \includegraphics[width=1.0\textwidth]{imgs/res18_r_10.png} | |
| %\caption{Comparison among Res18, Res18-Ewp. Res18-Gate-ReLU and Res18-Gate-GELU.} | |
| \vspace{-2mm} | |
| \caption{Comparison among Res18, Res18-Ewp, Res18-Gate-ReLU6 and Res18-Gate-GELU. | |
| The $r$ represents the threshold of determining the boundary between low-frequency and high-frequency. | |
| We plot the learning curves of Resnet18 and its variants for $100$ epochs, together plotted with the accuracy of different frequency components $\mathbf{z}_i$. We set $r$ to $10$. All curves of $\mathbf{z}$ are from the test set. The legends can be found in the top of the figure. | |
| We also provide more results with different $r$ and different settings in the appendix. } | |
| \label{bpgg_resnet} | |
| \end{figure*} | |
| \subsection{Effect of Element-wise Product} | |
| Inspired by the \textit{convolution theorem}, we first give an insight into why element-wise product can encourage NNs to learn on various frequency components from a frequency view. The convolution theorem states that for two functions $u(x)$ and $v(x)$ with Fourier transforms $U$ and $V$, | |
| \begin{equation} | |
| (u \cdot v)(x) = \mathcal{F}^{-1}(U \ast V), | |
| \end{equation} | |
| where $\cdot$ and $\ast$ denote element-wise multiplication and convolution respectively, and $\mathcal{F}$ is the Fourier transform operator defined as $\mathcal{F}[f(t)] = F(\omega) = \int_{-\infty}^{+\infty} f(t)e^{-j\omega t}\,dt$. This indicates that element-wise multiplication in the spatial domain corresponds to convolution in the frequency domain. | |
| To see its implication more clearly, consider the simplest situation: the self-convolution of a function. If the support set of $\mathcal{F}(\omega)$ is $[-\Omega, \Omega]$, then the support set of $\mathcal{F}\ast \mathcal{F}(\omega)$ will expand to $[-2\Omega, 2\Omega]$. In other words, self-convolution broadens the frequency spectrum. With this enriched frequency content, neural networks have more opportunities to capture and learn from both high-frequency and low-frequency components. | |
| % Inspired by the \textit{convolution theorem}, we first give a theoretical insight of why element-wise product can encourage NNs to learn on various frequency components from a frequency view. | |
| % %we reveal how the element-wise multiplication works from a frequency view. | |
| % \begin{theorem}[\textit{Convolution Theorem}] | |
| % \label{conv} | |
| % Consider two functions $u(x)$ and $v(x)$ with their Fourier transforms $U$ and $V$: | |
| % $${(u\cdot v)}(x) = \mathcal{F}^{-1}(U\ast V),$$ | |
| % \end{theorem} | |
| % where $\cdot$ and $\ast$ means the element-wise product and convolution respectively. $\mathcal{F}$ refers to the Fourier transform operator where $\mathcal{F}[f(t)] = F(\omega) =\int_{-\infty}^{+\infty}f(t)e^{-j\omega t}dt$. | |
| % The convolution theorem indicates that the element-wise multiplication in the spatial domain equals to the convolution operation in the frequency domain. Let's consider the simplest situation which is the self-convolution of a function. Denoting the support set of $\mathcal{F}(\omega)$ is $[-\Omega, \Omega]$, the support set of $\mathcal{F}\ast \mathcal{F}(\omega)$ will be $[-2\Omega, 2\Omega]$. | |
| % It means that self-convolution resulting in a broader frequency spectrum. With richer frequency information, NNs have more chances to | |
| % learn both high-frequency and low-frequency components. | |
| %To verify this insight, we conduct experiments on the Res18 and Res18-Ewp to find the effect of element-wise product. | |
| %As shown in Fig. \ref{bp}, the accuracy curve of Res18-Ewp | |
| %visibly surpass that of Res18 for decomposed parts in ranges of $(0, r_1)$ and early epochs in ranges of $(r_1, r_2)$. | |
| %The results demonstrate that broader spectral ranges enable the network to learn faster and better across multiple frequency components. Empirically, it is usually the lower-frequency components. | |
| %For example, for $\mathbf{z}_i \in (0, r_1)$) and $\mathbf{z}_i \in (r_1, r_2)$ before epoch $60$, the accuracy curve of Res18-Ewp is visibly above that of Res18. | |
| %From the performance on very high-frequency components (for $\mathbf{z}_i \in (r_3, +\infty)$), it is evident that neural networks face challenges in directly classifying these components, which are nearly imperceptible to the human eye. | |
| %The accuracy for very high-frequency components, such as those for $\mathbf{z_i} \in (r_e, +\infty)$, remains consistent and is similar to that of random selection, indicating that these high-frequency components are challenging for the network to learn effectively. | |
| %However, examining the accuracy curve for components within the range $(r_2, r_3)$ reveals a significant drop in accuracy growth after $40$ epochs compared to the base model. | |
| %From the performance on higher-frequency components, we can observe that it is challenging for NNs to classify those components directly which is almost imperceptible to human. | |
| %It proves that the boarder spectral ranges encourage the network learn faster on multi frequency components, especially of some lower frequencies. The accuracy of very high-frequency e.g. for $\mathbf{z_i} \in (r_e, +\infty)$ shows consistent performance which is same the random selection. That indicates thar very high-frequency components are not be learned easily. | |
| %However, looking at the curve of components in the range of $(r_2, r_3)$, we can find the growth of accuracy is apparently dropped significantly after the $40$ epochs compared to that of base model. For the accuracy under $20\%$, we regard it as the random selection which means NNs almost cannot learn anything form that frequency range. | |
| %It indicates that although element-wise product can encourage the model learn faster on some frequency components, newly introduced high-frequencies can become the noise to disturb the effective learning on high-frequencies. | |
| %Therefore, the activation layer is necessary to regulate the newly generated frequencies, minimizing the impact of high-frequency noise. | |
| \subsection{How Activation Function Works?} | |
| \label{act_analysis} | |
| % We begin with a brief mathematical analysis illustrating how the differential properties of activation functions influence the frequency characteristics of GLUs. The \textit{differential properties of the Fourier transform} state that if a function $f(t)$ is continuous in $(-\infty, +\infty)$, then | |
| We begin by analyzing how an activation function's smoothness influences the frequency characteristics of the features it produces. | |
| There is a well-established principle in Fourier analysis that connects a function's smoothness to the decay rate of its Fourier transform's magnitude. | |
| For a function $f(t)$ that is sufficiently smooth (i.e., its $n$-th derivative $f^{(n)}(t)$ exists and is continuous), the magnitude of its Fourier transform, $|F(\omega)|$, is bounded and decays at a rate proportional to $1/|\omega|^n$ | |
| for large $\omega$. | |
| This is a direct consequence of the differentiation property of the Fourier transform: | |
| % \begin{equation} | |
| % \mathcal{F}[f^{\prime}(t)] = j\omega F(\omega), | |
| % \end{equation} | |
| % where the notation is consistent with convolution theorem. More generally, if the $n$-th derivative $f^{(n)}(t)$ exists and is continuous, we have | |
| \begin{equation} | |
| \label{n-dif} | |
| \mathcal{F}[f^{(n)}(t)] = (j\omega)^n F(\omega), | |
| \end{equation} | |
| This property implies that the smoother a function is (i.e., the more continuous derivatives it has), the more rapidly its high-frequency components decay. | |
| Conversely, functions with discontinuities or sharp "corners" where derivatives are undefined (such as the kink in ReLU-like activations) are known to possess significant high-frequency energy. | |
| These sharp features require a broad spectrum of high-frequency sinusoids to be accurately represented. | |
| This leads to a Fourier transform that decays much more slowly. | |
| For example, a function with a simple discontinuity will have a spectrum that decays only at a rate of $1/|\omega|$. | |
| Therefore, we hypothesize that non-smooth activation functions will encourage the network to retain and utilize more high-frequency information compared to their smooth counterparts like GELU and Swish, which is infinitely differentiable. | |
| % which can be easily proved by mathematical induction. From Eq.~\ref{n-dif}, we can deduce that $|F(\omega)| \leq \frac{C}{|\omega|^n}$, where $C$ is a sufficiently large constant. This implies that if higher-order derivatives exist and are continuous, the high-frequency components of the Fourier transform decay more rapidly at a rate of $|\omega|^n$. Conversely, if higher-order derivatives do not exist or are discontinuous, the decay of the high-frequency components slows down. | |
| % Notably, for smooth activations such as $\operatorname{GELU}(x)=x \cdot \frac{1}{2}[1+\operatorname{erf}(x / \sqrt{2})]$, the function is infinitely differentiable across the entire real domain, with all orders of derivatives continuous and finite. This continuity and boundedness of higher-order derivatives cause the high-frequency components of the Fourier transform to decay quickly, leading to fewer retained high-frequency components. | |
| To validate this hypothesis, we conduct an experiment to compare the frequency learning of a representative smooth activation (GELU) against a non-smooth one (ReLU6) within a ResNet18 architecture. | |
| As shown in Fig. \ref{bpgg_resnet}, the model using the non-smooth ReLU6 activation consistently outperforms the GELU variant in learning from high-frequency components across different thresholds. | |
| This result supports our hypothesis, illustrating a clear practical difference between these two activation types. | |
| The superior performance of ReLU6 on high-frequency data suggests that the slow spectral decay associated with non-smooth functions can be beneficial for tasks requiring fine-grained detail. | |
| Conversely, the GELU variant shows a stronger relative performance on low-frequency components, indicating its suitability for capturing broad, structural patterns. | |
| While a more exhaustive study is needed, this experiment provides clear evidence for the link between activation smoothness and a model's spectral learning preferences. | |
| %To validate the above theoretical insights, we conduct experiments on ResNet18 and its variants to examine the impact of gating mechanisms and activation smoothness on frequency learning. As shown in Fig.~\ref{freq_comparison}, the testing curves of the model employing non-smooth activation functions (ReLU6) visibly surpass those of other models under different thresholds, demonstrating superior performance in learning from high-frequency components. This supports the hypothesis that reduced smoothness in the activation function enhances the model's ability to retain high-frequency information. Conversely, the variant using GELU exhibits stronger performance on low-frequency components, reflecting the advantage of smooth activations in modeling low-frequency signals. | |
| %As shown in Fig.X, we apply the full gating structure in the ResNet18. | |
| % In this paper, we mainly focus on three activation functions ReLU6, GELU and ReLU6 which are mostly used in the current lightweight models | |
| % based on their properties. | |
| % \begin{itemize} | |
| % \item Smooth activation: Smooth function means the differential coefficient is continues like GELU and Swish. This type of functions usually tends to produce lower frequency components. | |
| % \item Non-smooth activation: Functions like ReLU and ReLU6 have discontinuous derivatives, which can introduce more high-frequency components in gating mechanisms. | |
| % \end{itemize} | |
| % Specially, we select GELU and ReLU as representations to reveal how two kinds of activation functions control the new generated frequencies differently. | |
| % %\subsubsection{Theoretical Demonstrations} | |
| % We begin with a brief mathematical analysis illustrating how the differential properties of activation functions influence the frequency characteristics of GLUs. | |
| % \begin{theorem}[\textit{Differential Properties of the Fourier Transform}] | |
| % \label{dpft} | |
| % If a function $f(t)$ is continues in $(-\infty, +\infty)$, then | |
| % $$\mathcal{F}[f^{\prime}(t)] = j\omega {F}[\omega] $$ | |
| % \end{theorem} | |
| % % \begin{equation} | |
| % % \mathcal{F}\left[f^{\prime}(t)\right]=\int_{-\infty}^{+\infty} f^{\prime}(t) e^{-j \omega t} dt | |
| % % =\left.f(t) e^{-j \omega t}\right|_{-\infty} ^{+\infty}+j \omega \int_{-\infty}^{+\infty} f(t) e^{-j \omega t} d t | |
| % % =j \omega \mathcal{F}[f(t)] | |
| % % \end{equation} | |
| % The definitions of the symbols are same to Theorem \ref{conv}. | |
| % \begin{lemma} | |
| % If $f^{(n)}(t)$ is continues in $(-\infty, +\infty)$, then | |
| % \begin{equation} | |
| % \label{n-dif} | |
| % \mathcal{F}[f^{(n)}(t)] = (j\omega)^n {F}[\omega], | |
| % \end{equation} | |
| % \end{lemma} | |
| % \noindent where $f^{(n)}(t)$ represents the $n$-th derivative of $f(t)$. | |
| % The proof of Lemma \ref{n-dif} can be easily obtained by the mathematical induction based on Theorem \ref{dpft}. | |
| % From Eq. \ref{n-dif}, we can find that $|F(\omega)|\leq \frac{C}{|\omega|^n}$ where $C $ is a sufficiently large constant. | |
| % It indicates that if higher-order derivatives exist and are continuous, the high-frequency components of the Fourier transform decay more quickly with a rate of $|\omega|^n$. | |
| % Conversely, if higher-order derivatives do not exist or are discontinuous, the decay of the high-frequency components of the Fourier transform slows down. | |
| % Notably, for the smooth activation like $\operatorname{GELU}(x)=x \cdot \frac{1}{2}[1+\operatorname{erf}(x / \sqrt{2})]$, it is infinitely differentiable across the entire real domain, with all orders of derivatives being continuous and finite. | |
| % The continuity and finite values of higher-order derivatives cause the high-frequency components of the Fourier transform to decay quickly, resulting in fewer high-frequency components. | |
| % %On the other hand, discontinuity of higher-order derivatives can lead to generating more high-frequencies. Non-smooth activation may encourage remaining more higher frequency components in the gating mechanism. | |
| % %As shown in Fig. \ref{gp}, compared to the model without activation, the complete gating unit (using ReLU) learns better in higher frequency components compared to Res18-Ewp, like in $\mathbf{z}_i \in (r_2, r_3)$. It effectively avoids the disturbance of high-frequency noise. | |
| % %Moreover, we have conducted contrast experiments of using different kinds of activation functions. | |
| % To validate the above theoretical insights, we conduct experiments on ResNet18 and its variants to examine the impact of gating mechanisms and activation smoothness on frequency learning. | |
| % As shown in Fig.~\ref{freq_comparison}, the testing curves of the model employing non-smooth activation functions (ReLU6) shows | |
| % visibly surpass above the other models with different thresholds. | |
| % It demonstrate superior performance in learning from high-frequency components. | |
| % %As shown in Fig.~\ref{freq_comparison}, the testing curves of the model using the non-smooth activation function (ReLU6) clearly surpass those of the other models, demonstrating superior performance in capturing high-frequency components. | |
| % This supports the hypothesis that reduced smoothness in the activation function enhances the model's ability to retain high-frequency information. | |
| % Conversely, the variant using GELU exhibits stronger performance on low-frequency components, reflecting the strength of smooth activations in modeling low-frequency signals. | |
| % Furthermore, we observe that improvements in overall classification accuracy closely align with enhanced capability in capturing high-frequency information, underscoring the critical role of frequency-aware design in improving model performance. | |
| % Also, the performance of Res18-Gate-GELU on $(r_2, r_3)$ which is under $20\%$ also indicates the low-frequency bias for the smooth activation function. | |
| % Meanwhile, both models show similar learning curve for the lower-frequencies which might be caused by the fast | |
| % convergence of the model training. | |
| % Additionally, as shown in Fig. \ref{bpgg}, GELU and ReLU demonstrate their superiority among the variants in capturing low-frequency and high-frequency components, respectively. | |
| % It indicates that the model with a complete gating unit can better capture the information for various ranges of frequency based on their properties. | |
| % Also, the improvement of the overall accuracy aligns with their performance on increasing capturing the high-frequency information. | |
| %\subsubsubsection{D} | |
| %\noindent{\bf Discussions.} Different datasets and training strategies may lead to different dynamics which has been addressed in \cite{wang2020high, tang2022defects}. | |
| %For the small dataset, the convergence is faster which means NNs can easily capture some certain frequency components and then learning from other frequency components becomes very slow. | |
| %Therefore, the generated high-frequency noise has a greater impact on disturbing the learning from high-frequencies. | |
| %We will show more evidences and discussions of the difference between various activation functions in ablation studies on a larger dataset. | |
| %Moreover, from the performance on very high-frequency components (for $\mathbf{z}_i \in (r_3, +\infty)$), it is evident that neural networks face challenges in directly classifying these components, which are nearly imperceptible to the human eye. | |
| %However, NNs can be sensitive to the perturbation of high-frequency noise \cite{yin2019fourier}. This is also an interesting topic of revealing the robustness of NNs, but it is beyond the scope of this article. | |
| %For the point production only model, | |
| %we conduct experiments on recording the changes of the accuracy | |