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+# Spiking Token Mixer: An Event-Driven Friendly Former Structure for Spiking Neural Networks
+
+Shikuang Deng $^{1*}$ , Yuhang Wu $^{1*}$ , Kangrui Du $^{1,2}$ , Shi Gu $^{1,3\boxtimes}$
+
+<sup>1</sup>University of Electronic Science and Technology of China
+
+$^{2}$ College of Computing, Georgia Institute of Technology
+
+3 Shenzhen Institute for Advanced Study, UESTC
+
+dengsk@uestc.edu.cn, gus@uestc.edu.cn
+
+# Abstract
+
+Spiking neural networks (SNNs), inspired by biological processes, use spike signals for inter-layer communication, presenting an energy-efficient alternative to traditional neural networks. To realize the theoretical advantages of SNNs in energy efficiency, it is essential to deploy them onto neuromorphic chips. On clock-driven synchronous chips, employing shorter time steps can enhance energy efficiency but reduce SNN performance. Compared to the clock-driven synchronous chip, the event-driven asynchronous chip achieves much lower energy consumption but only supports some specific network operations. Recently, a series of SNN projects have achieved tremendous success, significantly improving the SNN's performance. However, event-driven asynchronous chips do not support some of the proposed structures, making it impossible to integrate these SNNs into asynchronous hardware. In response to these problems, we propose the Spiking Token Mixer (STMixer) architecture, which consists exclusively of operations supported by asynchronous scenarios, including convolutional, fully connected layers and residual paths. Our series of experiments also demonstrates that STMixer achieves performance on par with spiking transformers in synchronous scenarios with very low timesteps. This indicates its ability to achieve the same level of performance with lower power consumption in synchronous scenarios. The codes are available at https://github.com/brain-intelligence-lab/STMixer_demo.
+
+# 1 Introduction
+
+Spiking neural networks (SNNs) are bio-inspired neural networks that use spike signals as a carrier for inter-layer transmission. Because of the spiking mechanism, SNNs have the potential to turn high-precision multiplication into addition, which makes them a better energy-saving infrastructure [Maass, 1997, Roy et al., 2019]. However, such an advantage in efficiency is not guaranteed for typical hardware [Kim et al., 2019] but for neuromorphic chips [Akopyan et al., 2015, Davies et al., 2018b, Pei et al., 2019]. The computational energy of the synchronous neuromorphic hardware is affected by the amount of calculation, the spike frequency, and the SNN time steps [Rathi and Roy, 2020, Li et al., 2021]. And synchronous hardware provides support for nearly all matrix operations, even allowing for the execution of mixed networks that combine the ANN and SNN. In order to achieve low power consumption on synchronous hardware, SNNs must consider using
+
+shorter time steps. But too few time steps may potentially undermine the performance of SNNs. On the other hand, in theory, event-driven asynchronous hardware serves as an ideal implementation platform for such a mechanism. On the event-driven hardware, each neuron is configured as an independent entity driven by events rather than the high-frequency clock. Compared to synchronous scenarios, asynchronous scenarios impose more stringent constraints on network architecture. Hence, in asynchronous neuromorphic hardware, SNN usually utilizes the soft-reset Integrate-and-Fire (IF) model as the fundamental neuron model. The supported network operations in asynchronous scenarios adhere to the rule that matrix multiplication should take place between a spike matrix and a fixed float weight matrix. Consequently, due to the accumulation of membrane potential in the IF neuron, spike arrival timing errors do not significantly impact subsequent layers.
+
+Over the past years, there have been a lot of efforts dedicated to advancing the performance of SNNs across various datasets. These efforts have explored a multitude of perspectives, aiming to optimize the training pipeline [Neftci et al., 2019, Wu et al., 2018, Li et al., 2021, Deng et al., 2023], refine the optimization objectives [Deng et al., 2021, Guo et al., 2022b], and design the network structure [Fang et al., 2021a, Zhou et al., 2023b,a]. Recently, Spikformer [Zhou et al., 2023b,a] has achieved remarkable results by incorporating the Transformer into SNN, resulting in unprecedented performance gains across a wide range of datasets. Spikformer has designed a new attention module, SSA, which aligns Spikformer with the forward rules of SNN during the inference phase. Such progression in a Spikformer-like design theoretically eliminates the need for high-precision matrix multiplication. Spikformer-like models have indeed demonstrated significant performance improvements in synchronous hardware or GPU simulation. However, an easily overlooked point is their feasibility in implementing asynchronous hardware. In event-driven scenarios, the spike arrival times may not be as precise as in synchronous scenarios with hardware clocks. Calculating two spike matrix multiplications or a spike matrix passing through the max-pooling layer could result in significant differences in the output.
+
+To address these issues, we proposed a new architecture called Spiking Token Mixer (STMixer), which only uses asynchronous neuromorphic hardware-compatible operations, including convolutional, fully connected layers and residual paths. STMixer still adopts the holistic framework of spikformer, which consists of spiking patch splitting (SPS), encoders, and classification heads (Fig. 1 (A)). The encoder module in STMixer consists of two components: the token mixing module and the channel module. In the token mixer component of STMixer, the value matrix undergoes matrix multiplication with a trainable attention map weight matrix (Fig. 2 (A)). This new architecture eliminates the requirement of performing matrix multiplication between two spike matrices during the forward pass, thereby conforming to the event-driven situation. We also analyzed and enhanced the SPS module, ensuring its compliance with the requirements of an asynchronous environment while reducing information loss. Furthermore, we employed the proposed mixer architecture to optimize the components of the surrogate module [Deng et al., 2023] and used the surrogate module learning to assist the SNN training. The following summarizes our main contributions:
+
+- We examine problems with the SSA module in Spikformer in asynchronous scenarios and suggest a new module, the Spiking Token Mixing (STM) module, which consists solely of network components that cater to asynchronous environments.  
+- We investigate the SPS module's asynchronous support and the information loss problem. To address these concerns, we proposed the information protection spiking patch splitting (IPSPS) module, which removes the maxpool layer and adds a residual path to protect information.  
+- Through the network architecture search, we verified that the STMixer structure excels among all possible configurations in terms of performance and parameter quantity. Additionally, we have validated that STMixer achieves performance on par with or even surpasses existing Spikformer-like works in synchronous scenarios by using only a single time step across almost all datasets. This suggests that STMixer has the potential to deliver high performance with lower power consumption in synchronous scenarios, too.
+
+![](images/d396af1d31ec308bfa7b0f041ef5637632c642ee8d7f4fcad176f1bfcf8862aa.jpg)
+
+![](images/5b6e99ad7acf91f4556699ae1779e05916fc36e17e609389bae3dab403b9d06e.jpg)  
+Figure 1: The architecture of Spikformer-like networks and the issue of spiking matrix multiplication and max pooling layers (A) The overall architecture of STMixer and Spikformer. (B) The issue of the Max Pooling layer when there is a spike delay in asynchronous hardware. (C) The spike delay issue of spiking matrix multiplication. When both matrices involved in matrix multiplication are composed of spikes, the imprecise timing of spike arrival can result in inaccuracies in the computed product.
+
+# 2 Related Work
+
+# 2.1 Transformer-like architectures in artificial neural networks
+
+Convolutional networks, such as ResNet and VGG, have historically been the most popular architectures in the realm of computer vision [Zheng et al., 2021, Rathi and Roy, 2020]. Some works have tried to add attention mechanisms to visual tasks [Yao et al., 2023, Zhu et al., 2024] after being inspired by the amazing results of transformers in NLP [Vaswani et al., 2017, Devlin et al., 2019, Floridi and Chiriatti, 2020]. Especially, the Vision Transformer (VIT) [Dosovitskiy et al., 2020] has demonstrated that transformer architectures can achieve remarkable performance in computer vision tasks. VIT uses patch embedding as an input and multiple stacked transformer encoders as the backbone. Since then, a lot of research has been done to improve performance and efficiency by making changes to how images are stored [Parmar et al., 2018, Chen et al., 2020, Liu et al., 2021], how attention is calculated [Shen et al., 2021, Bolya et al., 2022], and other methods [Touvron et al., 2021, Zhang and Sabuncu, 2020]. Meanwhile, some works have revealed that just employing MLPs as the token mixer can also lead to good performance[Tolstikhin et al., 2021, Touvron et al., 2022, Guo et al., 2022a]. Recently, a fresh perspective has been proposed by Metaformer[Yu et al., 2022], suggesting that the success of Transformer in visual tasks stems from the holistic encoder architecture, where token mixer and channel mixer interact. Interestingly, Metaformer even achieved competitive performance by employing a simple pooling layer as the token mixer. Their findings offer us novel insights into how to modify Spikformer.
+
+# 2.2 Spiking neural networks
+
+The forward process of SNN encompasses the following rule: all high-precision matrix multiplications should be capable of degrading to matrix additions, eliminating the need for high-precision matrix multiplication operations after implanting in hardware [DeBole et al., 2019, Davies et al., 2018b]. However, this rule restricts the flexibility of SNNs to incorporate various operations, limiting them to a few substructures such as the convolution layer, the fully connected layer, the pooling layer, and so on. The majority of SNN architectural designs are indeed based on these substructures, such as the advancements of ResNet structures like SEW-ResNet [Fang et al., 2021a] and MS-ResNet [Hu et al., 2024], as well as the structure search efforts like AutoSNN [Na et al., 2022] and SNASNet [Kim et al., 2022]. Recently, Spikformer [Zhou et al., 2023b] has achieved remarkable success by integrating the transformer architecture into SNNs, attaining SOTA performance across various datasets. The authors proposed the SSA module to ingeniously realize attention computation in SNN while adhering to the aforementioned forward rule. At present, Spikformer and its subsequent series [Zhou et al., 2023a, Yao et al., 2024] of improvements are steadily pushing the boundaries of SNN performance. Nevertheless, though SSA exhibits notable performance in clock-driven hardware, it
+
+may encounter significant discrepancies in event-driven hardware due to the fact that the spikes may not arrive at the same time.
+
+# 3 Preliminaries
+
+# 3.1 Spiking neural model
+
+The spike neuron serves as the fundamental unit of SNN, wherein their somatics harbor membrane potential that integrates information from the synaptic according to prescribed model principles. The leaky integrate-and-fire (LIF) model [Lapicque, 1907] is currently the most prevalent SNN neuron model, which describes the membrane potential update rule as follows:
+
+$$
+H [ t ] = \frac {1}{\tau} V [ t - 1 ] + X [ t ]
+$$
+
+$$
+S [ t ] = \Theta (H [ t ] - V _ {\mathrm {t h}}), \tag {1}
+$$
+
+$$
+V [ t ] = H [ t ] (1 - S [ t ]) + V _ {\text {r e s e t}} S [ t ],
+$$
+
+where  $V[t]$  is the membrane potential,  $\tau$  is the membrane time constant, and  $S[t]$  is the output in time  $t$ , it would be equal to 1 or 0, which means if there is a spike or not.  $\Theta(\cdot)$  means the Heaviside step function, which equals 1 when  $H[t]$  exceeds the threshold value  $V_{\mathrm{th}}$ . Furthermore, we employ the hard reset mechanism, which means the membrane potential will reset to the reset potential when the neuron fires a spike.
+
+# 3.2 Spikformer
+
+We use the Spikingformer [Zhou et al., 2023a] architecture as the foundational framework of our network. Spikingformer addresses the issue of residual connections in Spikformer by using membrane potential residual instead of spiking residual. The overall structure of Spikingformer, as depicted in Fig. 1 (A), comprises three main components: Spiking Patch Splitting (SPS), an encoder module, and a classification head. The SPS consists of five convolutional layers, where the first four layers are followed by a max pooling layer. The function of SPS is to perform convolution processing and downsampling on input images, resulting in an output matrix with the shape  $D \times h \times w$ . Eventually, this matrix is flattened into  $N$  ( $N = h \times w$ ) patches, each with a  $D$ -dimensional vector. The classification head consists of an adaptive average pooling layer, which is used for downsampling, and a fully connected layer, responsible for obtaining the network's final output. The classification head will introduce floating-point matrix multiplication, so it needs to be deployed on the CPU. Because the calculation amount of the classification head is very small, it will not cause much energy overhead or inference delay but will greatly improve SNN performance [Zhou et al., 2023a].
+
+The most important component of Spikformer is its encoder module, which consists of two cells and residual connections (Fig. 1 (A)). Cell 2 corresponds to the Feed Forward Network (FFN) in VIT. It consists of two fully connected layers designed for channel mixing. Cell 1 refers to the Spiking Self-Attention (SSA) module proposed by the Spikformer. SSA has been demonstrated to be more suitable for SNN compared to vanilla self-attention (VSA). In SSA, when the input is  $X \in \mathbb{R}^{N \times D}$ , the query ( $Q \in \mathbb{R}^{N \times D}$ ), key ( $K \in \mathbb{R}^{N \times D}$ ), and value ( $V \in \mathbb{R}^{N \times D}$ ) are computed using the following equations:
+
+$$
+Q = \mathcal {S N} _ {Q} (\mathrm {B N} (X W _ {Q})), K = \mathcal {S N} _ {K} (\mathrm {B N} (X W _ {K})), V = \mathcal {S N} _ {V} (\mathrm {B N} (X W _ {V})) \tag {2}
+$$
+
+where  $W_{Q}, W_{K}, W_{V} \in \mathbb{R}^{D \times D}$  are learnable weight matrices,  $\mathcal{SN}$  means spiking neural (LIF model) activation, and BN denotes the batch normalization layer. The authors of Spikformer argue that in ANN, the presence of outliers or negative values in the query and key makes the attention map require a softmax function. However, since the output of eqn. 2 is a binary matrix that only consists of 0 and 1, there are no outliers or negative values. Therefore, they directly perform matrix multiplication on the three matrices to obtain the output of SSA:
+
+$$
+\operatorname {S S A} (Q, K, V) = \mathcal {S N} \left(Q K ^ {\mathrm {T}} V * s\right) \tag {3}
+$$
+
+where  $s = 0.125$  is a constant factor.
+
+# 4 Methodology
+
+# 4.1 Event-driven Scenarios
+
+In ideal asynchronous scenarios, each neuron is an autonomous entity solely focused on updating its membrane potential upon receiving spike signals, then firing a spike once the membrane potential exceeds the threshold. Since the model in asynchronous scenarios lacks the specific hardware clock to control the spike firing and arriving time, it is inevitable to encounter temporal errors in the spike arrival time. In the extreme case, it is conceivable to assume that each neuron fires spikes at an individual time (the SNN time step T approaches infinity, and each spike fires at an individual time step). When there are errors in the arrival time of spikes  $(t = t + \sigma)$ , the output sum of asynchronous scenarios supported operations like convolution, fully connected layer, and average pooling will not change. However, some operations that are commonly observed in synchronous scenarios may have output errors due to input spike arrival timing error.
+
+Maxpool layer. As shown in Fig. 1 (B), a spike at a specific location arrives one time step later in the max pooling layer, resulting in a noticeable change in the output. This output error will further affect the result of the subsequent convolution layer, as well as the membrane potential accumulation of neurons and their output spikes. In the extreme case where each spike has an independent arrival time, all the spikes will pass through the max pooling layer. Therefore, the max pooling layer needs a clock to control the spike arrival time and cannot be simply deployed in event-driven scenarios. To address this problem, we can simply remove the max pooling layer and change the stride of the convolutional layer following the max pooling layer to 2 for downsampling. After this modification, when the arrival spikes have temporal discrepancies, the cumulative membrane potential of the post-convolution spike neurons will remain relatively stable.
+
+Spike matrix multiplication. On the synchronization and clock-driven hardware, the query, key, and value spike matrices can arrive on time. This guarantees that the forward SSA results on this hardware correspond to the GPU simulation results. However, on event-driven hardware, the absence of clock constraints often makes it challenging for spikes to arrive at precise timings. If there is any spike loss or similar occurrence during matrix multiplication in SSA, a significant error in the SSA output will occur. Here is a simple example that shows how the computed attention map matrix changes significantly in SSA when there is a spike delay at a certain point in both the query and key matrices (Fig. 1 (C)). This example shows how hard it is to make sure that the forward process with two spike matrix multiplications runs smoothly on event-driven neuromorphic hardware.
+
+# 4.2 Clock-driven Scenarios
+
+According to Rathi and Roy [2020], Hu et al. [2024], Zhou et al. [2023a], the theoretical computational energy consumption of SNNs on clock-driven scenarios can be defined as:
+
+$$
+E _ {\text {S N N}} = E _ {A C} \times \sum_ {l = 2} ^ {L - 1} S O P _ {C o n v} ^ {l} + E _ {M A C} \times \left(F L O P _ {C o n v} ^ {1} + F L O P _ {\text {L i n e a r}} ^ {L}\right), \tag {4}
+$$
+
+where  $E_{AC} = 0.9pJ$  and  $E_{MAC} = 4.6pJ$  are the assumed energy consumption of accumulate and multiply-accumulate operations, respectively, of 45nm CMOS hardware [Horowitz, 2014],  $SOP_{Conv}^{l}$  denotes the number of spike-based accumulate operations, and  $FLOP_{Conv}^{1}$  means the computation of the first layer that encodes static RGB images into a spike matrix [Zhou et al., 2023a],  $FLOP_{Linear}^{L}$  means the computation of the classification head. And  $SOP_{Conv}^{l} = FLOP_{Conv}^{l} \cdot fr^{l}$ .  $T$  is affected by the SNN fire rate  $fr^{l}$  and time step  $T$ . However, recent works [Dampfhoffer et al., 2022, Bhattacharjee et al., 2024] have pointed out that Eqn. 4.2 neglects the overhead of memory scheduling  $E_{Mem}$ , which is much larger than  $E_{MAC}$ .
+
+On each time step of SNN forward inference, hardware will deal with memory scheduling, so  $E_{MAC}$  is also affected by the time step  $T$ . And SNNs with a large number of time steps may have a large energy overhead in synchronous scenarios, even potentially surpassing that of ANNs under the same structure. Therefore, we must closely examine the performance of the new SNN architecture at extremely low time steps to ensure its adaptability to low-power scenarios.
+
+![](images/82f5d5ec581203181c06b5979659ba65f3e6757e4faec41664c6fe24cd41001e.jpg)  
+A  
+B
+
+![](images/9ee882c694b11c7f3823a4cf8dd60d7cb5f86fb7c4d7eae95be9dd92a331436a.jpg)  
+Figure 2: The architecture of (A) Spiking token mixing (STM) module and (B) Information protect Spiking Patch Splitting (IPSPS) module.
+
+# 4.3 STMixer
+
+To address the aforementioned requirements in the two scenarios, we present the STMixer network architecture. STMixer solely consists of asynchronously supported operations and performs well at extremely low time steps  $(T = 1)$ . STMixer has the same overall structure as Spikformer but uses STM as the token mixing module and IPSPS to replace the SPS module (Fig. 1 (A)). STMixer is controlled by three hyper-parameters:  $L$ ,  $D$ , and  $H$ , which respectively control the number of encoders, the encoding dimension, and the number of multi-heads of STM. We use the naming convention "STMixer-L-D-H" to represent the three parameters, which is similar to Spikformer-like works. Next, we will go into detail about the important modules in STMixer.
+
+# 4.3.1 Spiking Token Mixing (STM)
+
+As shown in Fig.2 (A), we use the STM module to realize the token mixing function. This module utilizes a weight matrix  $(W_{\mathrm{STM}} \in \mathbb{R}^{N \times N})$  to mix the token dimension information. Mathematically, its forward process can be expressed as follows:
+
+$$
+\operatorname {S T M} (X) = \mathcal {S N} \left(W _ {\mathrm {S T M}} V\right) = \mathcal {S N} \left(W _ {\mathrm {S T M}} \mathcal {S N} \left(\mathrm {B N} \left(X W _ {V}\right)\right)\right), \tag {5}
+$$
+
+Where  $X \in \mathbb{R}^{N \times D}$  is the input spiking matrix of STM,  $V \in \mathbb{R}^{N \times D}$  is the value matrix, and  $W_V \in \mathbb{R}^{D \times D}$  is the projection matrix to acquire the value matrix  $V$ . When we simplify the operations of the SSA module by removing the activations for query and key, the formula for SSA can be simplified as follows:
+
+$$
+\begin{array}{l} \operatorname {S S A} (Q, K, V) \rightarrow \mathcal {S N} \left(X W _ {Q} W _ {K} ^ {T} X ^ {T} V * s\right) \\ = \mathcal {S N} \left(X W X ^ {T} V * s\right). \tag {6} \\ \end{array}
+$$
+
+After this simplification,  $W_{Q}$  and  $W_{K}$  can converge into a single matrix, and the attention map is reduced to  $XWXT$ . So the objective of  $W_{\mathrm{STM}}$  is to gradually fit the attention map during the training process. And some studies also suggest that linear token mixing can capture external attention between samples in the dataset [Guo et al., 2022a].
+
+Furthermore, we evenly divide the value matrix  $V \in \mathbb{R}^{N \times D}$  into  $H$  groups based on dimension  $D$ , allocating a weight matrix  $W_{\mathrm{STM}}^h \in \mathbb{R}^{N \times N}$  to each group. This method employs different attention weights for various patch encoding groups  $V^h \in \mathbb{R}^{N \times D / H}$ . It facilitates a finer granularity in token mixing, leading to enhanced information mixing and integration among tokens. In addition, due to the relatively modest number of tokens in the vision task, implementing multi-head in this manner would not significantly increase the parameters and maintain the same level of computational complexity.
+
+# 4.3.2 Information Loss in Spiking Patch Splitting
+
+In a neural network, as the input  $X$  undergoes successive forward propagation layers, the inherent information gradually dissipates. Mathematically this phenomenon can be described as:
+
+$$
+I (X, X) \geq I (X, f (X)) \geq I (X, g (f (X))) \dots \geq I (X, Y), \tag {7}
+$$
+
+where  $I$  means the mutual information,  $f$  and  $g$  are the operations of network layers, and  $Y$  is the network output. An SNN layer may lose more information than an ANN layer because the layer output is a spike matrix, which inherently contains significantly less information than a floating-point
+
+![](images/bedd35fd1bd42358aff95226a39a0e2e6bad5c264e54cf2b4fbd5fa8fbe0cf51.jpg)  
+Figure 3: Visualization results of random initial weight output feature maps of the SPS module. We present the feature maps of the 100th channel and the average of all channels, and in SNN, the feature map is the mean feature map across all time steps. When using the LIF neuron model as the activation function for SPS, it is obvious that there is a significant information loss in the output feature map compared to the input image. Employing the IPSPS can alleviate this information loss.
+
+matrix. The Spiking Patch Splitting (SPS) module comprises five layers of convolutional operations and spiking neural activation. After passing through a total of five convolutional layers and spiking neural, the feature map may have already lost a significant amount of information contained in the input images (Fig. 3).
+
+Since Spikformer-like SNNs rely on convolutional layers in the SPS module for information extraction, reducing the number of layers in the SPS module will compromise the SNN's ultimate performance. To address this information loss problem, we add an additional path that directly uses a convolutional layer for input image encoding (Fig. 2 (B)). In our configuration, the encoding outputs from the auxiliary pathway account for  $1/8$  of the entire SPS output feature map. This new encoding module is called information protected spiking patch splitting (IPSPS).
+
+There are two advantages to IPSPS: (1) IPSPS does not reduce the convolutional layers in the original pathway; it only decreases the number of feature map channels by  $1/8$  for the last convolutional layer. Theoretically, this approach should not significantly impair the SNN performance. Simultaneously, the new pathway can provide feature maps with varying convolutional depths for subsequent encoders, enriching the SPS output's feature representation. (2) The output of the additional pathway in IPSPS can be regarded as a linear transformation applied to the input image. The input feature map for the first encoder from the additional pathway is essentially the linearly transformed input image passed through an SNN neural layer, which maximally preserves the input image's information.
+
+# 4.4 STM on surrogate module learning
+
+Even if we reduce the information loss of the encoder's input, the accumulation of gradient errors caused by surrogate gradient may still prevent SNN from achieving better performance with low time steps. As a result, we use the surrogate module learning (SML) method [Deng et al., 2023]. The SML is an exceptionally effective approach proposed to alleviate the issue of gradient errors in SNN by adding an auxiliary pathway during the training phase. However, the use of large kernel convolutions results in significant computational overhead during the training phase. In this paper, we employ the STMixer encoder to replace the computationally intensive convolutional encoder, thereby greatly reducing the computational overhead of SML. The STM in the surrogate module uses the GeLU activation function and employs the same number of heads as the STM in the main structure. Finally, we use a classification head for the output. Starting from index 0, we place a surrogate module after every two encoders for surrogate module learning. And the number of encoders in the surrogate module is half of the count of encoders after the place point in the main pathway.
+
+# 5 Experiments
+
+# 5.1 Comparison to exiting works
+
+CIFAR. In IPSPS, we only set the stride to 2 for the last two convolutional layers, similar to Spikformer. This implies that IPSPS downsamples the input images to a resolution of  $8 \times 8$  and
+
+subsequently obtains 64 patches through flattening. Here, STM in STMixer uses 32 heads. And we use the same training script as Spikeformer, the learning rate is set to 0.005, the batch size is 128, and training total 300 epochs. As shown in Tab. 1, STMixer achieves state-of-the-art performance both on the CIFAR-100 dataset and considering models with comparable parameter numbers and flops. However, on the CIFAR-10 dataset, the SML method slightly outperforms the results reported by us. This could be attributed to the larger parameter count of the structure they used or the structure being more suitable for the CIFAR-10 dataset. These results demonstrate the potential of STMixer to achieve excellent performance for SNN on simple static datasets, even with only one time step. It also suggests that STMixer can serve as a competitive alternative to Spikformer-like works.
+
+ImageNet. The IPSPS module of STMixer downsamples the input images of  $224 \times 224$  resolution to  $16 \times 16$  through four successive convolutional layers, ultimately obtaining 196 patches. We employ the same training script as Spikeformer, set the batch size to 256, and set the training total epochs to 300. All the STMixer use 16 heads for STM in ImageNet experiments. We set the time step to 1, aiming to verify its performance under low energy consumption. The energy consumption of synchronous hardware estimation is computed by Eqn. 4.2. As shown in Tab. 1, we even use a much lower time step (much lower energy consumption), our STMixer-8-768-16 performs better than Spikeformer and achieves SOTA accuracy on ImageNet. This observation highlights the substantial potential of STMixer in synchronous and low energy scenarios.
+
+Table 1: Performance comparison of our method with existing methods on CIFAR10/100 and ImageNet. Param means the number of parameters.  
+
+<table><tr><td>Dataset</td><td>Methods</td><td>Architecture</td><td>Param (M)</td><td>Time Steps</td><td>Energy (mJ)</td><td>Accuracy (%)</td></tr><tr><td rowspan="8">CIFAR-10</td><td>tdBN [Zheng et al., 2021]</td><td>ResNet-19</td><td>12.57</td><td>4</td><td>-</td><td>92.92</td></tr><tr><td>TET [Deng et al., 2021]</td><td>ResNet-19</td><td>12.57</td><td>4</td><td>-</td><td>94.44±0.07</td></tr><tr><td>AutoSNN [Na et al., 2022]</td><td>AutoSNN (C=128)</td><td>21</td><td>8</td><td>-</td><td>93.15</td></tr><tr><td>SNASNet [Kim et al., 2022]</td><td>SNASNet-BW</td><td>-</td><td>8</td><td>-</td><td>94.12±0.25</td></tr><tr><td>SpikeDHS[Deng et al., 2022]</td><td>SpikeDHS-CLA (n3s1)</td><td>14</td><td>6</td><td>-</td><td>95.36±0.01</td></tr><tr><td>SML [Deng et al., 2023]</td><td>ResNet-18</td><td>11.22</td><td>4</td><td>-</td><td>96.04±0.10</td></tr><tr><td>Spikformer [Zhou et al., 2023b]</td><td>Spikformer-4-384</td><td>9.32</td><td>4</td><td>-</td><td>95.19</td></tr><tr><td>Spikingformer [Zhou et al., 2023a]</td><td>Spikingformer-4-384</td><td>9.32</td><td>4</td><td>-</td><td>95.61</td></tr><tr><td></td><td>ours</td><td>STMixer-4-384-32</td><td>8.29</td><td>1</td><td>-</td><td>95.49±0.13</td></tr><tr><td></td><td></td><td></td><td></td><td>4</td><td>-</td><td>96.01±0.11</td></tr><tr><td rowspan="8">CIFAR-100</td><td>tdBN [Zheng et al., 2021]</td><td>ResNet-19</td><td>12.57</td><td>4</td><td>-</td><td>70.86</td></tr><tr><td>TET [Deng et al., 2021]</td><td>ResNet-19</td><td>12.57</td><td>4</td><td>-</td><td>74.47±0.28</td></tr><tr><td>AutoSNN [Na et al., 2022]</td><td>AutoSNN (C=128)</td><td>21</td><td>8</td><td>-</td><td>69.16</td></tr><tr><td>SNASNet [Kim et al., 2022]</td><td>SNASNet-BW</td><td>-</td><td>8</td><td>-</td><td>73.04±0.36</td></tr><tr><td>SpikeDHS[Deng et al., 2022]</td><td>SpikeDHS-CLA (n3s1)</td><td>14</td><td>6</td><td>-</td><td>76.25±0.10</td></tr><tr><td>SML [Deng et al., 2023]</td><td>ResNet-18</td><td>11.22</td><td>4</td><td>-</td><td>79.49±0.11</td></tr><tr><td>Spikformer [Zhou et al., 2023b]</td><td>Spikformer-4-384</td><td>9.32</td><td>4</td><td>-</td><td>77.86</td></tr><tr><td>Spikingformer [Zhou et al., 2023a]</td><td>Spikingformer-4-384</td><td>9.32</td><td>4</td><td>-</td><td>79.09</td></tr><tr><td></td><td>ours</td><td>STMixer-4-384-32</td><td>8.29</td><td>1</td><td>-</td><td>80.00±0.21</td></tr><tr><td></td><td></td><td></td><td></td><td>4</td><td>-</td><td>81.87±0.16</td></tr><tr><td rowspan="10">ImageNet</td><td>tdBN [Zheng et al., 2021]</td><td>ResNet-34</td><td>21.79</td><td>6</td><td>-</td><td>63.72</td></tr><tr><td>SEW-ResNet [Fang et al., 2022]</td><td>SEW-ResNet34</td><td>21.79</td><td>4</td><td>-</td><td>67.04</td></tr><tr><td>TET [Deng et al., 2021]</td><td>SEW-ResNet34</td><td>21.79</td><td>4</td><td>-</td><td>68.00</td></tr><tr><td>SpikeDHS[Deng et al., 2022]</td><td>SpikeDHS-CLA-large</td><td>58M</td><td>6</td><td>-</td><td>68.64</td></tr><tr><td>SML [Deng et al., 2023]</td><td>ResNet-34</td><td>21.79</td><td>4</td><td>-</td><td>68.25</td></tr><tr><td>Spikformer [Zhou et al., 2023b]</td><td>Spikformer-6-512</td><td>23.37</td><td>4</td><td>9.41</td><td>72.46</td></tr><tr><td></td><td>Spikformer-8-512</td><td>29.68</td><td>4</td><td>11.57</td><td>73.38</td></tr><tr><td></td><td>Spikformer-10-512</td><td>36.01</td><td>4</td><td>13.89</td><td>73.68</td></tr><tr><td>Spikingformer [Zhou et al., 2023a]</td><td>Spikingformer-8-512</td><td>29.68</td><td>4</td><td>7.46</td><td>74.79</td></tr><tr><td></td><td>Spikingformer-8-768</td><td>66.34</td><td>4</td><td>13.68</td><td>75.85</td></tr><tr><td></td><td>ours</td><td>STMixer-6-512-16</td><td>23.63</td><td>1</td><td>1.96</td><td>73.31</td></tr><tr><td></td><td></td><td>STMixer-8-512-16</td><td>30.12</td><td>1</td><td>2.20</td><td>73.82</td></tr><tr><td></td><td></td><td>STMixer-10-512-16</td><td>36.61</td><td>1</td><td>2.48</td><td>74.91</td></tr><tr><td></td><td></td><td>STMixer-8-768-16</td><td>61.16</td><td>1</td><td>4.45</td><td>76.68</td></tr></table>
+
+# 5.2 Exploration of the architecture of Encoders
+
+In this section, we endeavor to validate the effect of different encoder structures (the structures of Cell 1 and Cell 2 in Fig. 1). Additionally, we also want to figure out whether the metaformer-like structure is a superior architecture in SNN. As depicted in Fig. 4, each cell consists of three positions, with each position having five possible structures (STM and SSA only use a single head). Disregarding any duplicates generated by the identity structure, there are a total of  $5^{6}$  different substructures. Conducting a comprehensive validation of all these cases is impractical. Inspired by the concept of a one-shot NAS, we train a supernet that encompasses all possible substructures. The supernet is composed of three main components: the SPS, the classification head, and four identical encoders. Each encoder consists of two cells that are used for network search. This supernet enables us to efficiently evaluate the performance of each substructure. The CIFAR training dataset is divided
+
+![](images/444c5a1e0b1836d8df7231c1b4f93ff2606831489e7fcbe498c23b570e56e439.jpg)  
+Figure 4: Neural architecture search space and search results. Left: Search space of each Cell. Mid: random sample result from supernet on CIFAR-100. Right: random sample result from supernet on CIFAR-10.
+
+Table 2: Effect of STMixer components and training methods  
+
+<table><tr><td>Modification</td><td>Accuracy (%)</td></tr><tr><td>STMixer</td><td>79.55</td></tr><tr><td>IPSPS →SPS</td><td>78.97</td></tr><tr><td>STM→SSA</td><td>78.33</td></tr><tr><td>SML(STM)→SML(Conv)</td><td>79.07</td></tr><tr><td>SML→SDT</td><td>77.17</td></tr></table>
+
+Table 3: Effect of number of STM heads  
+
+<table><tr><td>H</td><td>Accuracy (%)</td></tr><tr><td>1</td><td>79.02</td></tr><tr><td>2</td><td>79.07</td></tr><tr><td>4</td><td>79.39</td></tr><tr><td>8</td><td>79.51</td></tr><tr><td>16</td><td>79.55</td></tr><tr><td>32</td><td>80.15</td></tr></table>
+
+into training and validation sets in a ratio of 8.5:1.5. During each forward pass, the supernet will randomly select a path and train on the training set for 600 epochs. After the training is completed, we randomly sample 4000 distinct subnetworks from the supernet and evaluate their performance on the validation set. Additionally, we calculate the forward FLOPs for each of these subnetworks. Note that the FLOPs mentioned here refer to the estimated floating point operations on the training platform. The actual theoretical synaptic operations (SOPs) rely on FLOPs, the simulation time step, and spike frequency. We have categorized the subnetwork into four classes: (1) only convolutional layers; (2) combining SSA and convolutional layers; (3) incorporating STMixer and convolutional layers; and (4) simultaneously having STMixer and SSA. To draw ellipses, we use the mean and variance of accuracy and FLOPs, showcasing the relative merits of four different selection spaces. Furthermore, we have marked the positions of STMixer and Spikingformer. As shown in Fig. 4, considering the computational complexity and network performance, as well as the suitability of the components used in asynchronous scenarios, STMixer appears to be the most promising architecture (STMixer is located at the top-left corner of the result figure).
+
+# 5.3 Ablation studies
+
+To understand the effect of various components and training methods in STMixer, we conducted ablation experiments on CIFAR-100 with STMixer-4-384-16 ( $T = 1$ ). The summarized results are presented in Table 2. As we can see, SML is the most significant improvement method. Considering that STMixer-4-384-16 is indeed a 25-layer network, the training process affects the nonnegligible gradient error accumulation. SML is an effective approach to mitigating the gradient error problem. And the employment of the modified SML algorithm with STM modules enables a more efficient facilitation of training SNNs ( $+0.48\%$ ). Then, under the SML training method, the performance of the STM module surpasses that of the SSA module ( $+1.22\%$ ). Finally, by preserving more information from the input images by IPSPS, the SNN achieves better performance ( $+0.58\%$ ).
+
+Another parameter that significantly influences the STMixer performance is  $H$ , the number of heads in STM. Here, we increment the value of  $H$  from 1 to 32 and evaluate the performance of STMixer-4-384 on the CIFAR-100 dataset. As the number of heads ( $H$ ) increases, the STMixer accuracy on the test set has improved from  $79.02\%$  to  $80.15\%$ . This observation highlights the necessity of performing token mixing at a finer granularity.
+
+# 5.4 Benchmark on fully event-driven asynchronous scenario
+
+Besides the superior performance demonstrated on the aforementioned static datasets, our proposed STMixer model also exhibits significant adaptability for event-driven scenarios. In these scenarios,
+
+SNNs are deployed on asynchronous chips to receive and process event streams from DVS cameras, thereby forming a system that integrates sensing and computation. This integration leverages the sparse nature of spike activity, allowing SNNs to achieve ultra-low energy consumption compared to their ANN counterparts. To validate this advantage, we conducted additional experiments on fully asynchronous frameworks. Since the current commercial asynchronous chips (e.g., Speck of the Synsense [Richter et al., 2024] and Loihi [Davies et al., 2018a, Lines et al., 2018] of Intel) are limited in capacities and lack support for various operators such as shortcuts, we tested our models on a home-made  $\mathrm{C + + }$  simulator that reproduces the computational principles of the Speck DevKits (see Synsense Sinabs).
+
+Table 4: STMixer vs Spikformer under large T and event-driven scenario on CIFAR10-DVS.  
+
+<table><tr><td>Model</td><td>w/ bias</td><td>T=25</td><td>T=40</td><td>T=80</td><td>T=160</td><td>T=320</td><td>event-driven</td></tr><tr><td>STMixer(T=25)</td><td>66.73</td><td>64.31</td><td>65.32</td><td>63.91</td><td>61.79</td><td>58.37</td><td>50.70</td></tr><tr><td>Spikformer(T=25)</td><td>64.21</td><td>56.05</td><td>53.83</td><td>34.58</td><td>22.78</td><td>19.15</td><td>16.60</td></tr><tr><td>Model</td><td>w/ bias</td><td>T=40</td><td>T=80</td><td>T=160</td><td>T=320</td><td>T=640</td><td>event-driven</td></tr><tr><td>STMixer(T=40)</td><td>75.91</td><td>71.77</td><td>71.57</td><td>70.77</td><td>67.74</td><td>64.62</td><td>54.10</td></tr><tr><td>Spikformer(T=40)</td><td>67.74</td><td>10.08</td><td>10.08</td><td>10.08</td><td>10.08</td><td>10.08</td><td>10.00</td></tr></table>
+
+We compared STMixer to a structurally identical SpikFormer, created by replacing the STM component in STMixer with SSA on CIFAR10-DVS. Due to the prohibition of clock-driven bias in a fully event-driven scenario, we implemented a two-phase training pipeline: initially training a standard SNN, then absorbing BN layers and eliminating all bias terms through additional fine-tuning, while the side effect is a slight performance degradation (see the w/ bias column and  $\mathrm{T} = 25(40)$  column in Table. 4). The training of SpikFormer becomes extremely unstable during the bias-removal phase when  $\mathrm{T} = 40$  and fails to converge even with the learning rate reduced to  $10^{-5}$ .
+
+We first deployed the two models on our event-driven simulator and evaluated their test accuracy. As shown in Table. 4) event-driven column, Spikformer undergoes significant performance degradation after deployment because events fail to meet each other and end in SSA, whereas STMixer maintains high performance and demonstrates strong compatibility with the event-driven framework. We further assessed their event-driven adaptability by inference performance at large T. When T is considerably large, events rarely coincide within the same frame, effectively approximating an event-driven scenario. According to Table. 4), as T increases, Spikformer's performance deteriorates sharply, whereas STMixer exhibits only a modest accuracy decrease. These results are consistent with our analysis in Section 4.1. Further details are provided in the appendix.
+
+# 6 Conclusion
+
+The advantage of spiking neural networks over artificial neural networks is their efficient and low-power inherencing capabilities once implanted in neuromorphic hardware. Within neuromorphic hardware, synchronous hardware uses a hardware clock to control when the spikes fire and arrive. The power consumption of such hardware is influenced by the time step length (T), whereby a smaller T typically results in lower power consumption. Asynchronous hardware has a lower energy consumption potential than synchronous hardware, but it has limitations on the network structure and only supports some specific matrix operations due to the lack of the hardware clock. For the aforementioned reasons, SNN must use asynchronously supported operators to build new architectures (for asynchronous hardware) or complete the performance challenge with an extremely low time step (for synchronous hardware). In this paper, we propose a new architecture named STMixer that satisfies the above challenges. STMixer is completely composed of asynchronous scenario supported components, including convolutional layers, fully connected layers, and residual paths, and has excellent performance when the time step  $T = 1$ . Our work will provide ideas for the SNN architecture, which works on both event-driven friendly and clock-driven efficient SNNs.
+
+# 7 Acknowledgment
+
+This project is supported by NSFC Key Program 62236009, Shenzhen Fundamental Research Program (General Program) JCYJ 20210324140807019, NSFC General Program 61876032, and Key Laboratory of Data Intelligence and Cognitive Computing, Longhua District, Shenzhen.
+
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+Man Yao, Guangshe Zhao, Hengyu Zhang, Yifan Hu, Lei Deng, Yonghong Tian, Bo Xu, and Guoqi Li. Attention spiking neural networks. IEEE transactions on pattern analysis and machine intelligence, 45(8):9393-9410, 2023.  
+Man Yao, Jiakui Hu, Zhaokun Zhou, Li Yuan, Yonghong Tian, Bo Xu, and Guoqi Li. Spike-driven transformer. Advances in neural information processing systems, 36, 2024.  
+Weihao Yu, Mi Luo, Pan Zhou, Chenyang Si, Yichen Zhou, Xinchao Wang, Jiashi Feng, and Shuicheng Yan. Metaformer is actually what you need for vision. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 10819-10829, 2022.  
+Zhilu Zhang and Mert Sabuncu. Self-distillation as instance-specific label smoothing. Advances in Neural Information Processing Systems, 33:2184-2195, 2020.  
+Hanle Zheng, Yujie Wu, Lei Deng, Yifan Hu, and Guoqi Li. Going deeper with directly-trained larger spiking neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 11062-11070, 2021.  
+Chenlin Zhou, Liutao Yu, Zhaokun Zhou, Han Zhang, Zhengyu Ma, Huihui Zhou, and Yonghong Tian. Spikingformer: Spike-driven residual learning for transformer-based spiking neural network. arXiv preprint arXiv:2304.11954, 2023a.  
+Zhaokun Zhou, Yuesheng Zhu, Chao He, Yaowei Wang, Shuicheng YAN, Yonghong Tian, and Li Yuan. Spikformer: When spiking neural network meets transformer. In The Eleventh International Conference on Learning Representations, 2023b. URL https://openreview.net/forum?id=frE4fUwz_h.  
+Rui-Jie Zhu, Malu Zhang, Qihang Zhao, Haoyu Deng, Yule Duan, and Liang-Jian Deng. Tcja-snn: Temporal-channel joint attention for spiking neural networks. IEEE Transactions on Neural Networks and Learning Systems, 2024.
+
+![](images/8a48c05628af9744425223e2f65ee634dd7db6c44b905caa2392518f1952c224.jpg)  
+Figure 5: The STM attention weights of different encoders. Each row illustrates the 16 attention weights for each encoder.
+
+# A Attention Weight Visualization
+
+Here we show the token mixing attention weights of the encoders from the well-trained STMixer-8-768-16 on the ImageNet dataset. These weights provide insights into the blending process of tokens, revealing how they are mixing information. As shown in Fig. 5, most of the weight matrices exhibit a pronounced regularity: there are some parallel lines near the symmetry axis. This observation implies two insights. While the attention weight of the STM enables the extraction of global information, it still tends to favor the extraction of local information. In visual tasks, the correlation between each patch and its neighboring patches is maximized. Moreover, it can be observed that the 16 attention weight matrices in each row are different, indicating that each head of the multi-head STM is capable of extracting distinct token mixing patterns.
+
+# B Experiments
+
+# B.1 Experiments settings
+
+Training for comparison with existing works All experiments on CIFAR were conducted on two 4090 GPUs, while experiments on ImageNet were performed on four 4090 GPUs. We employed the same training script as Spikingformer [Zhou et al., 2023a] and employed identical data augmentation techniques. In the CIFAR experiments, we modify the learning rate to be 5e-3 and the minimum learning rate to 1e-7. In the ImageNet experiments, we modify the learning rate to be 3e-4 and the minimum learning rate to 1e-5.
+
+Training for event-driven experiments Models for event-driven experiments are all trained by our two-phase training pipeline. Specifically, we first train an SNN with bias and BN layers. Before further fine-tuning, we absorb BN layers and set all bias terms to 0. Finally, we fine-tune the model for more epochs with bias frozen. For the first phase, we use the AdamW optimizer and set the learning rate to 0.001 and weight decay to 0.02. For the second phase, we use the AdamW optimizer and set the learning rate to 0.001 and the weight decay to 0.005. For Spikformer and STMixer with  $T = 25$ , a time step of  $T = 10$  is used for the first phase over 300 epochs.  $T$  is then set to 25 for the bias-removal phase, during which models are fine-tuned for an additional 30 epochs. For Spikformer and STMixer with  $T = 40$ , a consistent  $T = 40$  is used throughout both phases. Models are trained for 100 epochs during the first phase, followed by 30 epochs of fine-tuning during the bias-removal phase. Notably, we found that MultiSpike IF neurons with soft reset (see Synsense Sinabs) are optimal for training models intended for event-driven deployment, and this configuration was applied in all event-driven experiments.
+
+Backbone for event-driven experiments We employed the modified STMixer-2-256-8 as the backbone for all event-driven experiments, incorporating several adjustments to enhance its suitability for fully event-driven deployment. First, the non-spiking membrane shortcut was replaced with a standard spiking shortcut [Zheng et al., 2021] for event-driven implementation. Second, the first two convolution layers of the SPS module are replaced with a single  $4 \times 4$  Avgpool layer. Building upon this foundation, we replace the STM module with SSA to construct the Spikformer model.
+
+# B.2 Compare with existing works on neuromorphic datasets in synchronous scenarios
+
+Table 5: Performance comparison of our method with existing methods on CIFAR10-DVS and DVS128 Gesture.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">CIFAR10-DVS</td><td colspan="2">DVS128 Gesture</td></tr><tr><td>Time step</td><td>Acc</td><td>Time step</td><td>Acc</td></tr><tr><td>LIAF-Net Wu et al. [2021]</td><td>10</td><td>70.4</td><td>60</td><td>97.6</td></tr><tr><td>TA-SNN Yao et al. [2021]</td><td>10</td><td>72.0</td><td>60</td><td>98.6</td></tr><tr><td>tdBN Zheng et al. [2021]</td><td>10</td><td>67.8</td><td>40</td><td>96.9</td></tr><tr><td>PLIF Fang et al. [2021b]</td><td>10</td><td>74.8</td><td>20</td><td>97.6</td></tr><tr><td>Spikformer Zhou et al. [2023b]</td><td>10</td><td>78.6</td><td>10</td><td>95.8</td></tr><tr><td>Spikingformer Zhou et al. [2023a]</td><td>10</td><td>79.9</td><td>10</td><td>96.2</td></tr><tr><td>ours</td><td>10</td><td>82.67</td><td>10</td><td>97.19</td></tr></table>
+
+CIFAR10-DVS is a neuromorphic dataset that is obtained from the CIFAR-10 dataset through a DVS camera. There are 10k images in CIFAR10-DVS, and we split them into 9k training images and 1k test images. We downsample the image resolution from  $128 \times 128$  to  $48 \times 48$  and input them to STMixer-2-512-8 structure SNN. We don't use the IPSPS module here because we find it does not enhance the performance on the neuromorphic dataset too much. For the training setting, we employ the AdamW optimizer with a learning rate of 0.01 and a weight decay of 0.02. The training is conducted for 300 epochs. Meanwhile, we employ the same data augmentation methods as previous works Li et al. [2021], Deng et al. [2021].
+
+As depicted in Table. 5, our architecture achieves the best performance on CIFAR10-DVS, demonstrating the suitability of STMixer for this neuromorphic dataset.
+
+DVS128 Gesture Amir et al. [2017] is a gesture recognition dataset that consists of 11 hand gesture classes performed by 29 individuals under 3 different lighting conditions. We employ the network architecture STMixer-2-512-8 and apply the same augmentation techniques from Spikingformer except cutmix. We don't use the IPSPS module here because we find it does not enhance the performance on neuromorphic dataset too much. We train the network for 200 epochs with TET loss Deng et al. [2021] and the SML method. STMixer also achieves the best performance on this dataset.
+
+# NeurIPS Paper Checklist
+
+The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.
+
+Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:
+
+- You should answer [Yes], [No], or [NA].  
+- [NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available.  
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+
+The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.
+
+The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes]" is generally preferable to "[No]", it is perfectly acceptable to answer "[No]" provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No]" or "[NA]" is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.
+
+IMPORTANT, please:
+
+- Delete this instruction block, but keep the section heading "NeurIPS paper checklist",  
+- Keep the checklist subsection headings, questions/answers and guidelines below.  
+- Do not modify the questions and only use the provided macros for your answers.
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: We have stated in the abstract and introduction that the model structure we proposed can support asynchronous hardware and has low energy consumption on synchronous hardware.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: In the 4.2 Clock-driven Scenarios section, we mentioned that the current SNN energy cost calculation equation is incorrect and ignores the cost of memory calls. Furthermore, the real energy cost is still proportional to the time step, so SNN needs to use a lower time step to reduce potential energy consumption. Constructing an accurate energy consumption calculation equation is beyond the scope of this paper. In addition, due to current asynchronous hardware scale limitations (e.g., Speck 2e only supports 9-layer SNNs), we cannot implant large-scale SNNs into the hardware for testing. Therefore, we can only theoretically build models using the asynchronous hardware supported operators.
+
+# Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory Assumptions and Proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [Yes]
+
+Justification: We provide the explanation of why the multi-head STM module is effective from the perspective of fitting the attention map, and explaion the effectiveness of IPSPS from the perspective of information loss.
+
+Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental Result Reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+# Answer: [Yes]
+
+Justification: All the experiments in this paper use the open-source dataset. And we have provided the code that reproduces the results of the paper in the supplementary.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.
+
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
+
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
+
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
+
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
+
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+# Answer: [Yes]
+
+Justification: we have provided the code that reproduces the results of the paper in the supplementary.
+
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+
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+
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+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental Setting/Details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: We use the same training script as the open source work and provide reproducible training code.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment Statistical Significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [Yes]
+
+Justification: In comparing the results with existing work, in addition to the Imagenet results, we provide the mean and standard deviation of the results of three different seed experiments.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments Compute Resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: We provide the above information in the appendix.
+
+Guidelines:
+
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+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code Of Ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
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+
+# 10. Broader Impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [Yes]
+
+Justification: We have discussed it in the conclusion part.
+
+Guidelines:
+
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+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
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+
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
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+Justification: The paper poses no such risks.
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+Guidelines:
+
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+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
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+
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+
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+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New Assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [Yes]
+
+Justification:
+
+Guidelines:
+
+- The answer NA means that the paper does not release new assets.
+
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+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and Research with Human Subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing nor research with human subjects. Guidelines:
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+
+# 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
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+Answer: [NA]
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+Justification:
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+Guidelines:
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+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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2024/Spiking Transformer with Experts Mixture/full.md

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+# Spiking Transformer with Experts Mixture
+
+Zhaokun Zhou $^{1,2*}$ , Yijie Lu $^{1*}$ , Yanhao Jia $^{3,7}$ , Kaiwei Che $^{1,2}$ , Jun Niu $^{1}$ , Liwei Huang $^{4,2}$ , Xinyu Shi $^{5,4}$ , Yuesheng Zhu $^{1}$ , Guoqi Li $^{6,2}$ , Zhaofei Yu $^{5,4}$ , Li Yuan $^{1,2\dagger}$
+
+$^{1}$ School of Electronic and Computer Engineering, Shenzhen Graduate School, Peking University  $^{2}$ Peng Cheng Laboratory
+
+<sup>3</sup> College of Computing and Data Science, Nanyang Technological University
+
+$^{4}$ School of Computer Science, Peking University
+
+$^{5}$ Institute for Artificial Intelligence, Peking University
+
+$^{6}$ Institute of Automation, Chinese Academy of Sciences
+
+<sup>7</sup>Deep NeuroCognition Lab, I2R and CFAR, Agency for Science, Technology and Research
+
+# Abstract
+
+Spiking Neural Networks (SNNs) provide a sparse spike-driven mechanism which is believed to be critical for energy-efficient deep learning. Mixture-of-Experts (MoE), on the other side, aligns with the brain mechanism of distributed and sparse processing, resulting in an efficient way of enhancing model capacity and conditional computation. In this work, we consider how to incorporate SNNs' spike-driven and MoE's conditional computation into a unified framework. However, MoE uses softmax to get the dense conditional weights for each expert and TopK to hard-sparsify the network, which does not fit the properties of SNNs. To address this issue, we reformulate MoE in SNNs and introduce the Spiking Experts Mixture Mechanism (SEMM) from the perspective of sparse spiking activation. Both the experts and the router output spiking sequences, and their element-wise operation makes SEMM computation spike-driven and dynamic sparse-conditional. By developing SEMM into Spiking Transformer, the Experts Mixture Spiking Attention (EMSA) and the Experts Mixture Spiking Perceptron (EMSP) are proposed, which performs routing allocation for head-wise and channel-wise spiking experts, respectively. Experiments show that SEMM realizes sparse conditional computation and obtains a stable improvement on neuromorphic and static datasets with approximate computational overhead based on the Spiking Transformer baselines.
+
+# 1 Introduction
+
+The spiking neural networks (SNNs) are regarded as the third generation of neural networks [1], distinguished by biological plausibility [2], spike-driven characteristic, and low power consumption. SNNs emulate the dynamics of biological neurons at a microscopic level, utilizing asynchronous binary spikes for information transmission. The membrane potential of spiking neurons in SNNs is only updated upon the arrival of spikes, avoiding calculations of zero values. The inherent features make SNNs promising candidates for low-energy consumption on neuromorphic hardware, such as TrueNorth [3] and Loihi [4]. There are lots of architectures in SNNs include Spiking Recurrent Neural Networks [5], ResNet-like SNNs [6-9], Spiking Graph Neural Networks [10], and Spiking Transformers [11-13]. Spiking Transformers stand at the forefront. Spikformer [11] introduces Spiking Self-Attention (SSA). The Spike-Driven Transformer [12] introduces Spike-driven Self
+
+Attention. Other works explore the Spiking Transformer in terms of structural improvements [14-17], training methods [18], and different tasks [19], respectively.
+
+Mixture-of-Experts (MoE) [20, 21] is known for allowing each expert to learn specific tasks or features, showing better performance, conditional computing and dynamic adaptability, which are crucial features in the brain mechanism [22, 23]. In this work, we are committed to exploring the effective integration of MoE and Spiking Transformer. As shown in Fig.1(a), MoE introduces a large number of parameters based on the original Transformer, and the conditional computation is achieved by calculating the routing probability of each token on each expert through the softmax function. Selecting Top-K experts based on the routing probability, MoE achieves hard sparsification. However, SNN calculations need to avoid multiplication and cannot use complex softmax functions. The features in SNNs are dynamically sparse and do not require additional TopK. All the parameters of the expert must be loaded, which aggravates the difficulty of neuromorphic chip deployment. These factors make porting MoE to SNNs non-trivial. To tackle this problem, we develop the Spiking Experts Mixture Mechanism (SEMM), as shown in Fig.1(b), a universal SNN-MoE paradigm with the following three main features. 1) The SEMM is spike-driven. The outputs of the expert and the router are spike sequences, and the element-wise operation between them conforms to the SNN characteristics, i.e., avoiding multiplication. 2) SEMM leverages the sparse spiking activation of SNNs to achieve dynamic conditional computation of MoEs, which is more flexible than the fixed hard sparsification of Artificial Neural Network MoE (ANN-MoE). 3) With reasonable parameter count settings, SEMM enables Spiking Transformers to achieve stable performance gains with negligible overhead.
+
+Based on SEMM, we modify the Spiking Self-attention (SSA) and Multi-layer Perceptron (MLP) of Spiking Transformers to obtain the Experts Mixture Spiking Attention (EMSA) and the Experts Mixture Spiking Perceptron (EMSP). EMSA treats each head of SSA as an expert, computes respective attention, and employs a temporal-aware router to integrate attention. As for Multi-layer Perceptron (MLP), it is common to substitute the entire MLP with MoE [24], which leads to better performance but a significant increase in the overall parameter number. EMSP implements a channel-wise MoE within MLP to
+
+![](images/61eca0afc910772ee730a4e73e1e18a69a282f10376ca32b2b010c1f46d0d1b0.jpg)  
+Figure 1: ANN-MoE and Spiking Experts Mixture Mechanism (SEMM).  $N$  denotes the length of image patches.
+
+overcome the shortcoming. EMSA and EMSP can be inserted directly and seamlessly into existing Spiking Transformer variants [11, 14, 12]. Our work contributes in three main aspects:
+
+- We introduce the Spiking Experts Mixture Mechanism (SEMM), a universal SNN-MoE paradigm. SEMM is spike-driven, capable of efficient dynamic sparse conditional computation.  
+- Based on SEMM, we develop the Experts Mixture Spiking Attention (EMSA), whose information from all head-wise experts is selectively integrated through a temporal-aware router. We restructure MLP by a channel-level spiking-sparse SEMM, named the Experts Mixture Spiking Perceptron (EMSP). They can seamlessly replace self-attention and MLP in Spiking Transformers.  
+- Extensive experiments demonstrate the stable performance improvement of SEMM on both static and neuromorphic datasets. Notably, Spike-driven Transformer-8-512 with SEMM achieves a remarkable  $76.62\%$  accuracy on ImageNet with 4 time steps, surpassing the baseline  $(74.57\%)$ .
+
+# 2 Related Work
+
+Deep Spiking Neural Networks and Spiking Transformers. Spatio-temporal backpropagation(STBP) [25] directly trains SNNs by performing backpropagation on both spatial and temporal domains. Temporal backpropagation [26] computes the gradients of the timings of existing spikes for the membrane potential at the spike timing. Threshold-dependent batch normalization (tdBN) [8] is used to extend the network depth. SEW-ResNet [7] proposed the spiking element-wise residual for SNNs. Spikformer [11] firstly converts all the components of Vision Transformer (ViT) into
+
+![](images/5ce64783e780ee866024411ee1fffa16a872eb817d69888b462827cb180ea367.jpg)
+
+![](images/02f9d8b5119d9bd262ac9c9d2e9a9232c55db2ed4bef93eecb432d6740cb535e.jpg)  
+(a) Spiking Transformer  
+(b) EMSA
+
+![](images/05a6105d4f6f6a00acae66e16a2e997d75e23082476b1691bef143248ba1f673.jpg)  
+(c) EMSP  
+Figure 2: (a) The overview of Spiking Transformer. (b) The Experts Mixture Spiking Attention (EMSA). (c) The Experts Mixture Spiking Perceptron (EMSP). EMSA and EMSP can directly replace SSA and MLP in (a).
+
+spike-form and pioneers the field of SNN-Transformer. Spike-driven Transformer [12] goes further by introducing linear spike-driven self-attention. Spikingformer [14] proposes a hardware-friendly spike-driven residual learning architecture. Besides, Masked Spiking Transformer [27] combines SNNs and Transformers from the perspective of the ANN-to-SNN conversion. However, the SNN Transformer architecture that can bring SNNs' superiority into full play is still ongoing research.
+
+Mixture-of-Experts. The Mixture-of-Experts (MoE) [28, 29] combines the predictions of multiple specialized experts, which is effective in handling high-dimensional data and complex problems. Researchers explore diverse gating mechanisms [30, 31], optimizing expert allocation strategies [32, 33], and enhancing the scalability of MoE models [34, 35]. Sparsely-Gated Mixture-of-Experts [30] adopted MoE into architectures such as the Long Short-Term Memory (LSTM) [36], showcasing effectiveness in Language Modeling. The Transformer also benefits from MoE with the substitution of the Multi-layer Perceptron (MLP) [24, 37]. Switch Transformer [35] has scaled the models with trillions of parameters. Currently, there is no existing work on MoE with Spiking Transformers.
+
+# 3 Methodology
+
+# 3.1 Preliminaries and Overall Architecture
+
+Spiking neuron is the basic unit of SNNs. For the dynamics of the Leaky Integrate-and-Fire (LIF) neuron used in this work, the  $t - th$ -time-step membrane potential  $U[t]$  is equal to the sum of the state potential  $H[t - 1]$  at the previous time step and the input  $X[t]$ . When membrane potential exceeds the threshold  $u_{th}$ , the neuron will fire a spike, otherwise, it remains inactive. Consequently, the output  $S[t]$  only contains binary values, either 1 or 0.  $\mathrm{Hea}(\cdot)$  is a Heaviside function that satisfies  $\mathrm{Hea}(x) = 1$  when  $x \geq 0$ , otherwise  $\mathrm{Hea}(x) = 0$ .  $H[t]$  is the temporal state output, and  $V_{reset}$  denotes the reset potential after a spiking event.  $\beta < 1$  determines the rate of decay. If the neuron remains inactive, the potential  $U[t]$  decays towards  $H[t]$  over time. LIF can be described as:
+
+$$
+U [ t ] = H [ t - 1 ] + X [ t ], \tag {1}
+$$
+
+$$
+S [ t ] = \operatorname {H e a} \left(U [ t ] - u _ {t h}\right), \tag {2}
+$$
+
+$$
+H [ t ] = V _ {\text {r e s e t}} S [ t ] + (\beta U [ t ]) (1 - S [ t ]). \tag {3}
+$$
+
+Spiking Transformer[11, 14, 12] baselines contain Patch Splitting (PS), Relative Position Embedding (RPE), Spiking Self Attention (SSA) (e.g., Spiking Self-Attention [11] and Spike-driven Self-Attention[12]), MLP and linear classification head, as shown in Fig. 2(a). Given a 2D image sequence  $\mathbf{I} \in \mathbb{R}^{T \times C \times H \times W}$ , where  $T$ ,  $C$ ,  $H$ , and  $W$  denote time-step, channel, height and width, the PS splits it into a sequence of  $N$  flattened spike patches  $\mathbf{x}$  with  $D$  dimensional channel. A Convolution-BatchNorm-LIF block generates Relative Position Embedding (RPE):
+
+$$
+\mathbf {X} _ {0} = \operatorname {P S} (\mathbf {I}) + \operatorname {R P E}, \tag {4}
+$$
+
+$$
+\mathbf {X} _ {l} ^ {\prime} = \operatorname {S S A} \left(\mathbf {X} _ {l - 1}\right) + \mathbf {X} _ {l - 1}, \tag {5}
+$$
+
+$$
+\mathbf {X} _ {l} = \operatorname {M L P} \left(\mathbf {X} _ {l} ^ {\prime}\right) + \mathbf {X} _ {l} ^ {\prime}, \tag {6}
+$$
+
+$$
+\mathbf {Y} = \operatorname {C H} \left(\operatorname {G A P} \left(\mathbf {X} _ {L}\right)\right), \tag {7}
+$$
+
+where the  $\mathbf{X}_0$  is fed into the  $L$ -blocks and each block consists of a SSA and a MLP.  $\mathbf{X}_l^{\prime}, \mathbf{X}_l$  are spike sequences and  $l = 1\dots L$  is layer. A Global Average-pooling (GAP) is utilized on the  $\mathbf{X}_L$  and the linear Classification Head (CH) to output the prediction  $\mathbf{Y}$ . See Appendix. A for more details.
+
+# 3.2 Spiking Experts Mixture Mechanism
+
+Before exploring the adaptation of spiking MoE in SNNs, we first review MoE in ANNs. An ANN-MoE layer typically comprises a set of  $m$  experts  $\mathbf{E}_{\mathrm{A}} = \{\mathbf{E}_1,\mathbf{E}_2,\dots,\mathbf{E}_m\}$ , along with a router  $\mathcal{R}_{\mathrm{A}}$  for selecting the corresponding experts. Given an input sequence  $\mathbf{X}_{\mathrm{A}}$ , the resulting output can be expressed as the sum of the Top-K selected experts from  $m$  candidates using a router:
+
+$$
+\mathbf {y} = \sum_ {k = 1} ^ {K} \mathcal {R} _ {k} \left(\mathbf {X} _ {\mathrm {A}}\right) \cdot \mathbf {E} _ {k} \left(\mathbf {X} _ {\mathrm {A}}\right), \tag {8}
+$$
+
+$$
+\mathcal {R} (\mathbf {x}) = \operatorname {T o p K} \left(\operatorname {s o f t m a x} \left(\mathbf {X} _ {\mathrm {A}} \mathbf {W} _ {\mathcal {R}}, K\right)\right), \tag {9}
+$$
+
+$$
+\operatorname {T o p K} (\boldsymbol {v}, K) = \left\{ \begin{array}{l l} \boldsymbol {v} & \text {i f} \boldsymbol {v} \text {i s i n t h e t o p} K \text {e l e m e n t s} \\ 0 & \text {o t h e r w i s e} \end{array} \right. \tag {10}
+$$
+
+where  $\mathbf{W}_{\mathcal{R}}$  is the router weight matrix. The  $\mathrm{TopK}(\cdot, K)$  together with the  $\mathrm{softmax}(\cdot)$  sets all elements of the routing vector to zero except the largest top  $K$  values. The sparse conditional computing is obtained from the selecting of TopK and the different routing weights of the softmax, while  $K$  is usually taken as  $0.5 * m$ . We argue that the ANN-MoE is not suitable for SNN for two main reasons. First, the float-point routing-expert and the softmax which involve exponentiation and division, do not adhere to the computation principles of SNNs. Second, SNN experts are inherently highly sparse, so the hard sparsification approach of additional TopK is unnecessary. An event-triggered spike-based sparse conditional computation on asynchronous neuromorphic chips is needed more than TopK. To bridge these gaps, we introduce a generalized representation of the Spiking Experts Mixture Mechanism (SEMM), which is as follows,
+
+$$
+\operatorname {S E M M} (\mathbf {E}, \mathcal {R}, \mathrm {F} (\cdot)) = \mathrm {F} (\mathbf {E}, \mathcal {R}), \tag {11}
+$$
+
+where  $\mathbf{E} = \{\mathbf{E}_1, \mathbf{E}_2, \dots, \mathbf{E}_m\}$  represents the spiking sequence of  $m$  spiking experts in the Spiking Transformer, and  $\mathcal{R} \in \{0, 1\}^{T \times N \times m}$  represents the spiking router for allocating computation.  $\mathrm{F}(\cdot)$  denotes the element-wise form of the operation between the router and the expert output spiking sequence, i.e., Hadamard product and addition.
+
+As a MoE mechanism specifically designed for SNNs, SEMM has the following three significant advantages. i) Spike-driven. The SEMM is spike-driven, which is important for SNNs. Due to the spike-driven computation mode of experts, i.e., Spiking Self-attention and Spiking-MLP, SEMM computations are triggered by sparse spikes of experts and require only synaptic manipulation. For example, the Hadamard product between the spiking signals  $\mathcal{R}$  and  $\mathbf{E}$  is equivalent to mask. ii) Sparse-spiking conditional computation. The SEMM subtly utilizes the sparse activation of the spiking routers for the conditional computation of MoE. SEMM has a variable sparsity when dealing with different data. Additionally, SEMM does not suffer from the load imbalance problem in ANN-MoE, i.e., TopK selects fixed number of experts. The sparse conditional computation is distributed to each expert. iii) Efficient computation. Unlike loading with multiple heavy expert modules of ANN-MoE, the SEMM has comparable parameters and operations to the previous SSA and MLP of Spiking Transformers. These are further discussed in Sec. 3.5. Without loss of generality, we use the mainstream architecture of Spiking Transformer for SEMM embedding in Sec. 3.3 and Sec. 3.4.
+
+# 3.3 Experts Mixture Spiking Attention
+
+We begin by reviewing the processing of Spiking Self-Attention (SSA). Different from vanilla ANN-Transformers [38], it discards the softmax normalization for the attention map. The SSA mechanism
+
+can be described by the following equation:
+
+$$
+\mathbf {Q} = \mathcal {S N} _ {\mathbf {Q}} (\mathrm {L} - \mathrm {B N} _ {\mathbf {Q}} (\mathbf {X})), \mathbf {K} = \mathcal {S N} _ {\mathbf {K}} (\mathrm {L} - \mathrm {B N} _ {\mathbf {K}} (\mathbf {X})), \mathbf {V} = \mathcal {S N} _ {\mathbf {V}} (\mathrm {L} - \mathrm {B N} _ {\mathbf {V}} (\mathbf {X})), \tag {12}
+$$
+
+$$
+\mathbf {A} = \operatorname {S S A} (\mathbf {Q}, \mathbf {K}, \mathbf {V}) = \left\{ \begin{array}{l} \mathcal {S N} \left(\mathbf {Q} \mathbf {K} ^ {\mathrm {T}} \mathbf {V} * s\right), \text {f o r S p i k i n g S e l f - A t t e n t i o n} \\ \mathcal {S N} \left(\operatorname {S U M} _ {\mathrm {c}} \left(\mathbf {Q} \odot \mathbf {K}\right)\right) \odot \mathbf {V}, \text {f o r S p i k e - D r i v e n S e l f - A t t e n t i o n} \end{array} \right. \tag {13}
+$$
+
+where  $\mathcal{SN}$  represents the spiking neuron and L-BN represents that the features pass sequentially through Linear and BatchNorm.  $s$  is the scaling factor and  $\mathbf{Q},\mathbf{K},\mathbf{V}\in \{0,1\}^{T\times N\times D}$  are in spikeform.  $\mathbf{A}\in \{0,1\}^{T\times N\times D}$  is the spiking output of SSA.  $\odot$  is the Hadamard product, and  $\mathrm{SUM}_{\mathrm{c}}(\cdot)$  denotes the sum of each column. As shown in Fig 2(b), the Experts Mixture Spiking Attention (EMSA) based on SEMM and SSA is formulated as:
+
+$$
+\mathbf {A} _ {\mathbf {m}} = \operatorname {S S A} \left(\mathbf {Q} _ {\mathbf {m}}, \mathbf {K}, \mathbf {V}\right), \tag {14}
+$$
+
+$$
+\mathbf {E} = \left\{\mathbf {A} _ {1}, \mathbf {A} _ {2}, \dots , \mathbf {A} _ {m} \right\}, \tag {15}
+$$
+
+$$
+\mathcal {R} = \mathcal {S N} (\mathrm {B N} (\mathbf {X W} _ {\mathcal {R}})) = \left\{\mathbf {r} _ {1}, \mathbf {r} _ {2}, \dots , \mathbf {r} _ {m} \right\}, \tag {16}
+$$
+
+$$
+\operatorname {S E M M} (\mathbf {E}, \mathcal {R}, \mathrm {F} (\cdot)) = \sum_ {i = 1} ^ {m} \mathbf {r} _ {i} * \mathbf {A} _ {i}, \tag {17}
+$$
+
+$$
+\operatorname {E M S A} = \mathcal {S N} (\operatorname {B N} (\operatorname {S E M M} (\mathbf {E}, \mathcal {R}, \mathrm {F} (\cdot))) \mathbf {W} _ {o}), \tag {18}
+$$
+
+where  $\mathbf{A}_m\in \{0,1\}^{T\times N\times d}$  is the output of each SSA experts, and  $\mathbf{W}_{\mathcal{R}}\in \mathbb{R}^{D\times m}$  is the router weight matrix.  $\mathcal{R}\in \{0,1\}^{T\times N\times m}$  is the routing sequence. We set  $d\leq D$  by default to avoid introducing too many parameters. The operation  $\mathrm{F}(\cdot)$  between expert-router is computed by masking the expert using router pair-pair and then summing them up. The output of EMSA is obtained by performing matrix multiplication between  $\mathrm{SEMM}(\mathbf{E},\mathcal{R},\mathrm{F}(\cdot))$  and synaptic weight  $\mathbf{W}_o\in \mathbb{R}^{d\times D}$ , the same as the last layer of SSA block in Spiking Transformers.
+
+# 3.4 Experts Mixture Spiking Perceptron
+
+As illustrate in Fig. refig:method, the Experts Mixture Spiking Perceptron (EMSP) launches the channel-wise SEMM within MLP. The architecture can be written as follows:
+
+$$
+\mathbf {E} = \mathcal {S N} (\operatorname {D W C} (\mathcal {S N} (\operatorname {B N} (\mathbf {X W} _ {1})))) \in \{0, 1 \} ^ {T \times N \times m} \tag {19}
+$$
+
+$$
+\mathcal {R} = \mathcal {S N} \left(\mathrm {B N} \left(\mathbf {X W} _ {\mathcal {R}}\right)\right) \in \{0, 1 \} ^ {T \times N \times m}, \tag {20}
+$$
+
+$$
+\operatorname {S E M M} (\mathbf {E}, \mathcal {R}, \mathrm {F} (\cdot)) = \mathbf {E} \odot \mathcal {R}, \tag {21}
+$$
+
+$$
+\operatorname {E M S P} = \mathcal {S N} (\operatorname {B N} ((\operatorname {S E M M} (\mathbf {E}, \mathcal {R}, \mathbf {F} (\cdot)) \mathbf {W} _ {o})) \in \{0, 1 \} ^ {T \times N \times D}, \tag {22}
+$$
+
+where  $\mathbf{W}_1, \mathbf{W}_{\mathcal{R}} \in \mathbb{R}^{D \times m}$  is the weight matrix of first layer in MLP and router, respectively. We integrate a  $3 \times 3$  Depth Wise Convolution layer DWC to capture local features on each channel expert, which has fewer parameters and is computationally efficient compared to original convolution.  $\mathbf{W}_o \in \mathbb{R}^{m \times D}$  is the wight matrix of output layer. The original Spiking Transformer's MLP would have a  $D$  to  $4 * D$  channel dimension increase after the first layer and the second layer would reduce the dimension back to  $D$ . In EMSP we set  $m$  to  $(8 // 3) * D$  to try to match the parameter number of the original MLP. EMSP integrates sparse routing of multiple channel-wise experts. It can also be viewed as the channel-wise gating, which is suitable for temporal information processing, allowing information to flow unimpeded through potentially many time steps [39]. The gate (router) branch and expert branch can be also regarded as incorporating the Spike Element-Wise (SEW) [7] residual block into EMSP. Channel-wise convolution and element-wise (Hadamard) products only introduce a minor increase in computational cost. By appropriately configuring the number and dimensions of expert networks, our EMSP achieves more efficient computation compared to the original spiking MLP. More details of the computational overhead are given in the next sub-section.
+
+# 3.5 Characteristics of SEMM
+
+We explain each of the three advantages of SEMM, i.e., Spike-driven, Sparse-spiking conditional computation and efficient computation.
+
+Spike-driven has the following formal definition, meaning that the gathering of input current is initiated by sparse spikes released from presynaptic neurons:
+
+Definition 1. In SNN, the operation is spike-driven if the input currents satisfy the following form,
+
+$$
+I _ {i} [ t ] = \sum_ {j} w _ {i, j} s _ {j} [ t ] = \sum_ {j, s _ {j} [ t ] \neq 0} w _ {i, j}, \tag {23}
+$$
+
+where  $I_{i}[t]$  is the input current of the  $i$ -th postsynaptic neuron at time step  $t$ ,  $s_{j}[t] \in \{0,1\}$  is the spike output of the  $j$ -th pre-synaptic neuron,  $w_{i,j}$  is the weight of the synaptic connection from  $j$  to  $i$ .
+
+For EMSA, the input current for the LIF in Eq. 18 at a specific time step  $t$  is given by:
+
+$$
+\mathbf {I} [ t ] = \operatorname {S E M M} (\mathbf {E} [ t ], \mathcal {R} [ t ], \mathrm {F} (\cdot)) \mathbf {W} _ {o} = \sum_ {i = 1} ^ {m} r _ {i} [ t ] * \mathbf {A} _ {i} [ t ] \mathbf {W} _ {o}, \tag {24}
+$$
+
+$$
+I _ {p, q} [ t ] = \sum_ {i = 1} ^ {m} r _ {i} [ t ] \sum_ {l} a _ {i, p, l} [ t ] w _ {l, q} = \sum_ {\substack {i, l \\ (r _ {i} [ t ] \wedge a _ {i, p, l} [ t ]) \neq 0}} w _ {l, q}, \tag{25}
+$$
+
+where  $\mathbf{I}[t]$  has a dimension of  $P\times Q$ , with  $p$  and  $q$  represent the  $p$ -th row and  $q$ -th column, respectively. EMSA is essentially spike element-wise addition after masking operation on SSA, consistent with the characteristics of spike-driven. For EMSP, the operation between two spiking sequences  $\mathbf{E}$  and  $\mathcal{R}$  corresponds to the logical AND function [7], which also conforms to the spike driven,
+
+$$
+\mathbf {I} [ t ] = \operatorname {S E M M} (\mathbf {E} [ t ], \mathcal {R} [ t ], \mathbf {F} (\cdot)) \mathbf {W} _ {o} = (\mathbf {E} [ t ] \odot \mathcal {R}) [ t ] \mathbf {W} _ {o}, \tag {26}
+$$
+
+$$
+I _ {p, q} [ t ] = \sum_ {l} e _ {p, l} [ t ] r _ {p, l} [ t ] w _ {l, q} = \sum_ {\substack {l \\ (e _ {p, l} [ t ] \wedge r _ {p, l} [ t ]) \neq 0}} w _ {l, q}. \tag{27}
+$$
+
+Sparse-spiking conditional computation means using spiking routers to dynamically allocate the computation in temporal and spatial dimension. More analysis is detailed in experiments.
+
+Efficient computations means that SEMM approximates SSA and MLP in terms of number of parameters and theoretical synaptic operations. Tab. 1 demonstrates the computation load for EMSA, EMSP versus SSA and MLP. In terms of parameter number, despite having a routing layer  $mD$ , EMSA has a smaller number of parameters than SSA when the number of experts is within a reasonable range  $(\geq 2)$ . EMSP additionally introduces a  $3 \times 3$  depthwise convolution, and the number is slightly higher than MLP. Due to the similarity of the actual spiking rates, EMSA is smaller than SSA on the  $TND^{2}$  term, while the calculation of the additional introduced by SEMM is on the  $TND$  term (much smaller than  $TND^{2}$ ) and therefore can be ignored. The situation is similar on EMSP, with depth-wise convolution adding a slight computational
+
+![](images/236d4365793988bc9f78e3209e17a06cbcd6b93826425ce1eb6c08ad96756c34.jpg)  
+Figure 3: Comparison of parameters-accuracy for different MoEs. Methods of each line from left to right correspond to: 1. the baseline, 2. EMSP, 3. ANN-router with four heavy MLP experts, 4. Spiking-router with four heavy MLP experts, respectively. The last two use softmax and TopK  $(K = 2)$ .
+
+overhead. The design of EMSP is different from ANN-MoE, which selects multiple heavy MLPs as experts. We demonstrate it's validity in Fig. 3, i.e., it performs better while the number of parameters is much smaller than ANN-MoE.
+
+# 4 Experiments
+
+# 4.1 Sparse Conditional Computation Analysis
+
+We analyze the average spiking rate (ASR) of routers for EMSA and EMSP on the ImageNet validation set, which is shown in Tab. 2. The SD-Transformer-8-512 is used for the analysis. The ASR of EMSA is around 0.5, which is comparable to the regular TopK setting of ANN-MoE. The ASR of EMSP is low and gradually decreases as the block deepens, compared to the fixed TopK hard sparse in ANN-MoE, SEMM fully reflects the advantage of SNN dynamic sparse conditional computation. To further verify the spiking router, we ablate it, i.e., cancel it in EMSA and EMSP, as
+
+Table 1: Number of parameters and theoretical synaptic operations.  $\overline{R}$ ,  $\widehat{R}$ ,  $\tilde{R}$  and  $R$  denote the average spike firing rates (the proportion of non-zero elements in the spike matrix) in various spike matrices.  $T$ ,  $N$ , and  $D$  are the time step, sequence length, and channel dimension of the input features, respectively.  $d$  is the channel dimension of  $\mathbf{A}_m$  and  $m$  is the number of experts. The details are provided in the Appendix. B.  
+
+<table><tr><td></td><td>Param</td><td>OPs</td></tr><tr><td>SSA</td><td>4D2</td><td>4RTND2 + 2RHTN2D</td></tr><tr><td>EMSA</td><td>(1 + 1/m)D2 + (2d + m)D</td><td>(1 + 1/m)RTND2 + (2d + m)RFTND + (D + md)RTN2</td></tr><tr><td>MLP</td><td>8D2</td><td>8RTND2</td></tr><tr><td>EMSP</td><td>8D2 + 24D</td><td>8RTND2 + 24RFTND</td></tr></table>
+
+![](images/a83f27a86c8f3c2527e6acb9fc470214dff23b588606fab1d11e714c807868c7.jpg)  
+Figure 4: Ablation study on the EMSA and EMSP router.
+
+![](images/08d9408e49bab8ab7bf060c7e0e8753ddca3cbc47fb3b7c507eeed5545c3624e.jpg)
+
+shown in Fig. 4, the accuracy decreases significantly after canceling the router, and it is even lower than the baseline model on the Spikingformer. Directly analyzing the conditional computation of SEMM is challenging, we therefore use visualizations to illustrate this intuitively. As show in Fig. 5, for the same image, each expert's router assigns a different computational region, e.g., the third router filters the background of the expert's features, while the second router assigns the computation to the foreground target. The computation allocation of the spiking router to the irregular object "snake" can be seen that the router is highly dynamic and effective. See Appendix. D for more samples. In addition, as shown in Fig. 6, we also report the ASR of spatial-temporal locations of routers in different kinds of images. As can be seen by the difference in average firing rates, spiking router has a dynamic adjustment of ASR processing different kinds of images, further illustrating its data-dependent conditional computation property.
+
+Table 2: The average spike rate of EMSA and EMSP router in 8 blocks testing on the ImageNet.  
+
+<table><tr><td></td><td>Block0</td><td>Block1</td><td>Block2</td><td>Block3</td><td>Block4</td><td>Block5</td><td>Block6</td><td>Block7</td></tr><tr><td>EMSA</td><td>0.64</td><td>0.68</td><td>0.49</td><td>0.52</td><td>0.47</td><td>0.43</td><td>0.51</td><td>0.50</td></tr><tr><td>EMSP</td><td>0.25</td><td>0.30</td><td>0.33</td><td>0.27</td><td>0.20</td><td>0.10</td><td>0.05</td><td>0.02</td></tr></table>
+
+# 4.2 Results on various Datasets
+
+We conduct experiments on static datasets, i.e., ImageNet [40] and CIFAR [41], and neuromorphic datasets, i.e., CIFAR10-DVS [42], DVS128 Gesture [43] to verify the effectiveness of SEMM. See Appendix. C for more details. ImageNet results are shown in Tab. 3 which mainly compares SEMM with the Spiking Transformer baselines. At slightly lower model parameter counts, SEMM is steadily superior to baselines. For instance, Spikformer-8-512 with SEMM is  $2.55\%$  higher than the baseline with  $28.22M$  parameters, Spike-driven Transformer-8-384 with SEMM obtain  $2.05\%$  improvement. SEMM on Spikingformer presents similar findings.
+
+![](images/f655fc2e5dccf0381956bc009c976852d5b0b032a7ca977a2165813b627ebcd2.jpg)  
+Figure 8: Accuracy with different experts number.
+
+![](images/79af3dcbf6da61f4de877b45733605e296a263becb994bd0c75738877a09a3c6.jpg)
+
+![](images/30bc8090dab3b3152f10a929058d31ef932fa09e096e8a58d005d72d3679b1f5.jpg)
+
+![](images/c0a5827f56b987229753a7f2764c22db7d2d35729710568c8a6276bfb4992ffe.jpg)  
+Figure 5: Visualization of routers as masks. The mask position (black) indicates router of 0 here and the background image is the same for each subplot. We show the dynamic sparsity of spiking router for different experts (horizontal direction) and time steps (vertical direction).
+
+![](images/e7adb3aab3964027eaff585c10c9c8d18c80652a0b5c3a67e2819768701fdc35.jpg)
+
+![](images/f3fb1a14ec9067a60f572c227b2dc15adb11fcc9465c9bc97659f846952eed35.jpg)
+
+![](images/fa66797c482e897a66e87d121629b2998574c82d3d58c7e4c794910635a45309.jpg)  
+(a)
+
+![](images/3cdedcc4fe409c0e6ccb140c0036f3f13526d8ab528f10521e46967d9e4677d8.jpg)
+
+![](images/e32418162906d82124bccadf6d3f5807920e4c05aaa30e9d3601c52e79fad62d.jpg)  
+(b)
+
+![](images/8dd52a1eacbcc482dd289f86d87757dbb8252e2ae19faf09bf95efa3f5114aa0.jpg)
+
+![](images/3e08515013dae105789012139301a9bb75b47c65464f10be6c33b9847b4cca8d.jpg)  
+Figure 6: Average spiking rate of different kinds of images in the ImageNet validation set in the spatial-temporal dimension. The height of the cube is the time step. (a) Japanese spaniel. (b) Volcano.
+
+![](images/7ef22c296d4f6c9d1e3aa41afb3c8fd3e0fb957053d56ca08db4659f42f2721d.jpg)  
+Figure 7: Ablation study of EMSA and EMSP module.
+
+The experimental results on CIFAR, CIFAR10-DVS, and DVS128 Gesture are shown in Tab. 4. These four datasets are relatively small. We basically keep the experimental setup in [11, 12, 14], including the network structure, training settings, etc., and details are given in the Appendix. C.1. SEMM has demonstrated stable performance improvements across various datasets when integrated
+
+Table 3: Results on ImageNet-1k. Model-L-D represents a model with  $L$  encoder blocks and  $D$  channels.  
+
+<table><tr><td>Methods</td><td>Architecture</td><td>Param (M)</td><td>Time Step</td><td>Top-1 Acc (%)</td></tr><tr><td rowspan="3">SEW ResNet[7]</td><td>SEW-ResNet-34</td><td>21.79</td><td>4</td><td>67.04</td></tr><tr><td>SEW-ResNet-101</td><td>44.55</td><td>4</td><td>68.76</td></tr><tr><td>SEW-ResNet-152</td><td>60.19</td><td>4</td><td>69.26</td></tr><tr><td rowspan="3">MS-ResNet[9]</td><td>MS-ResNet-18</td><td>11.69</td><td>6</td><td>63.10</td></tr><tr><td>MS-ResNet-34</td><td>21.80</td><td>6</td><td>69.42</td></tr><tr><td>MS-ResNet-104*</td><td>77.28</td><td>5</td><td>76.02</td></tr><tr><td rowspan="2">Spikformer[11]</td><td>Spikformer-8-384</td><td>16.81</td><td>4</td><td>70.24</td></tr><tr><td>Spikformer-8-512</td><td>29.68</td><td>4</td><td>73.38</td></tr><tr><td rowspan="2">Spikformer + SEMM</td><td>Spikformer-8-384</td><td>16.05</td><td>4</td><td>72.86</td></tr><tr><td>Spikformer-8-512</td><td>28.22</td><td>4</td><td>75.93</td></tr><tr><td rowspan="2">Spike-driven Transformer[12]</td><td>SD-Transformer-8-384</td><td>16.81</td><td>4</td><td>72.28</td></tr><tr><td>SD-Transformer-8-512</td><td>29.68</td><td>4</td><td>74.57</td></tr><tr><td rowspan="2">SD-Transformer + SEMM</td><td>SD-Transformer-8-384</td><td>16.05</td><td>4</td><td>73.93</td></tr><tr><td>SD-Transformer-8-512</td><td>28.22</td><td>4</td><td>76.62</td></tr><tr><td rowspan="2">Spikingformer[14]</td><td>Spikingformer-8-384</td><td>16.81</td><td>4</td><td>72.45</td></tr><tr><td>Spikingformer-8-512</td><td>29.68</td><td>4</td><td>74.79</td></tr><tr><td rowspan="2">Spikingformer + SEMM</td><td>Spikingformer-8-384</td><td>16.05</td><td>4</td><td>73.58</td></tr><tr><td>Spikingformer-8-512</td><td>28.22</td><td>4</td><td>76.03</td></tr></table>
+
+Table 4: Results on CIFAR10-DVS, DVS128 Gesture, and CIFAR.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">CIFAR10-DVS</td><td colspan="2">DVS128 Gesture</td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td></tr><tr><td>T</td><td>Acc</td><td>T</td><td>Acc</td><td>T</td><td>Acc</td><td>T</td><td>Acc</td></tr><tr><td>tdBN [8]</td><td>10</td><td>67.80</td><td>40</td><td>96.90</td><td>6</td><td>93.20</td><td>-</td><td>-</td></tr><tr><td>PLIF [44]</td><td>20</td><td>74.80</td><td>20</td><td>97.60</td><td>8</td><td>93.50</td><td>-</td><td>-</td></tr><tr><td>DIET-SNN [45]</td><td>-</td><td>-</td><td>-</td><td>-</td><td>5</td><td>92.70</td><td>5</td><td>69.70</td></tr><tr><td>Dspike [46]</td><td>10</td><td>75.40</td><td>-</td><td>-</td><td>6</td><td>94.30</td><td>6</td><td>74.20</td></tr><tr><td>DSR [47]</td><td>10</td><td>77.30</td><td>-</td><td>-</td><td>20</td><td>95.40</td><td>20</td><td>78.50</td></tr><tr><td rowspan="2">Spikformer [11]</td><td>10</td><td>78.90</td><td>10</td><td>96.90</td><td>4</td><td>95.19</td><td>4</td><td>77.86</td></tr><tr><td>16</td><td>80.90</td><td>16</td><td>98.30</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">Spikformer + SEMM</td><td>10</td><td>82.32</td><td>10</td><td>97.56</td><td>4</td><td>95.78</td><td>4</td><td>79.04</td></tr><tr><td>16</td><td>82.90</td><td>16</td><td>98.63</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">Spike-Driven Transformer [12]</td><td>10</td><td>78.90</td><td>10</td><td>96.90</td><td>4</td><td>95.60</td><td>4</td><td>78.40</td></tr><tr><td>16</td><td>80.00</td><td>16</td><td>99.30</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">Spike-Driven Transformer + SEMM</td><td>10</td><td>81.10</td><td>10</td><td>97.56</td><td>4</td><td>96.12</td><td>4</td><td>80.26</td></tr><tr><td>16</td><td>82.42</td><td>16</td><td>99.30</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">Spikingformer [14]</td><td>10</td><td>79.90</td><td>10</td><td>96.20</td><td>4</td><td>95.81</td><td>4</td><td>79.21</td></tr><tr><td>16</td><td>81.30</td><td>16</td><td>98.30</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">Spikingformer + SEMM</td><td>10</td><td>80.70</td><td>10</td><td>96.88</td><td>4</td><td>96.16</td><td>4</td><td>80.24</td></tr><tr><td>16</td><td>82.10</td><td>16</td><td>98.56</td><td></td><td></td><td></td><td></td></tr></table>
+
+into different Spiking Transformer baselines. Specifically, SEMM achieves SOTA on CIFAR-10 (96.16%), CIFAR-100 (80.26%), CIFAR10-DVS (82.9%) and DVS128 Gesture (99.3%).
+
+# 4.3 Ablation Study and Hyperparameter Sensitivity
+
+Module ablation. EMSA and EMSP together improve the performance of the baseline, as shown in Fig. 7. The use of both EMSA and EMSP alone is better than baseline, illustrating their respective superiority.
+
+Experts number. We examine the utility brought by different numbers of experts of EMSA. The results of Spiking Transformer baselines on CIFAR are presented in Fig. 8. It indicates that within a certain range of expert numbers, the results can still be competitive and robust. Among these, we select 4 experts as the parameter setting for our final application.
+
+# 5 Conclusion
+
+In this work, we explored the feasibility of adapting MoE in Spiking Neural Networks and formulated an SNN-MoE paradigm named the Spiking Experts Mixture Mechanism. Unlike the vanilla MoE, which uses softmax and TopK hard sparse, SEMM implements dynamic conditional computation from the viewpoint of spiking sparse activation. With redesigned Router-Expert pairs and element-wise spike-driven operations, SEMM is computation-efficient and SNN-compatible. Embedded in Spiking Transformers, SEMM-based EMSA and EMSP can bring stable performance improvement on static and neuromorphic datasets. SEMM paradigm can inspire future exploration of high-performance, high-capacity Spiking Transformers. We hope SEMM can bring vitality to dynamic conditional computation and the design of next-generation architecture for SNNs. Future work will explore SEMM implementation in a wider range of tasks and larger SNN models.
+
+# 6 Acknowledgments and Disclosure of Funding
+
+We sincerely thank Dr. Wei Fang for his assistance with this work. This work is supported by grants from the National Natural Science Foundation of China (62088102, 62202014, 62332002, 62425101, 62236009, U22A20103, 62441606), Shenzhen Basic Research Program (No.JCYJ20220813151736001), the major key project of the Pengcheng Laboratory (PCL2021A13) and National Science Foundation for Distinguished Young Scholars (62325603). Computing support was provided by Pengcheng Cloud Brain.
+
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+[48] Filipp Akopyan, Jun Sawada, Andrew Cassidy, Rodrigo Alvarez-Icaza, John Arthur, Paul Merolla, Nabil Imam, Yutaka Nakamura, Pallab Datta, Gi-Joon Nam, et al. Truenorth: Design and tool flow of a 65 mw 1 million neuron programmable neurosynaptic chip. IEEE Transactions on Computer-aided Design of Integrated Circuits and Systems, 34(10):1537-1557, 2015.  
+[49] Ole Richter, Yannan Xing, Michele De Marchi, Carsten Nielsen, Merkourios Katsimris, Roberto Cattaneo, Yudi Ren, Qian Liu, Sadique Sheikh, Tugba Demirci, et al. Speck: A smart event-based vision sensor with a low latency 327k neuron convolutional neuronal network processing pipeline. arXiv preprint arXiv:2304.06793, 2023.
+
+# A Overall architecture of Spike-driven Transformer/Spikingformer with SEMM
+
+We provide a detailed description of the overall structure of Spike-driven Transformer with SEMM, and Spikingformer with SEMM. Given a 2D image sequence  $\mathbf{I} \in \mathbb{R}^{T \times C \times H \times W}$ , the Patch Splitting Module (PSM), consisted of three Convolution-Batch Normalization- $S\mathcal{N}$ -Maxpooling layers and one Convolution-Batch Normalization-Maxpooling layer, is used to projects and splits  $\mathbf{I}$  into a sequence of  $N$  flattened spike patches s with  $D$  channels. The Relative Position Embedding (RPE) module, a Convolution-BatchNorm layer, is used to generate position embedding to get  $\mathbf{U}_0$ .  $S\mathcal{N}(\cdot)$  means the spike neuron layer. Then the  $\mathbf{U}_0$  is passed to the  $L$ -blocks encoder. The encoder block consists of an Experts Mixture Spiking Attention (EMSA) and an Experts Mixture Spiking Perceptron (EMSP) block. Membrane-shortcut residual connections are applied in both the EMSA and EMSP blocks. A global average-pooling (GAP) is utilized on the processed feature from the encoder and outputs the  $D$ -dimension feature which will be sent to the fully-connected-layer classification head (CH) to output the prediction  $Y$ . The architecture can be written as follows:
+
+$$
+\mathbf {u} = \operatorname {P S M} (\mathbf {I}), \quad \mathbf {I} \in \mathbb {R} ^ {T \times C \times H \times W}, x \in \mathbb {R} ^ {T \times N \times D}, \tag {28}
+$$
+
+$$
+\mathbf {s} = \mathcal {S N} (\mathbf {u}), \quad \mathbf {s} \in \mathbb {R} ^ {T \times N \times D} \tag {29}
+$$
+
+$$
+\mathbf {R P E} = \operatorname {B N} (\operatorname {C o n v 2 d} (\mathbf {s})), \quad \mathrm {R P E} \in \mathbb {R} ^ {T \times N \times D} \tag {30}
+$$
+
+$$
+\mathbf {U} _ {0} = \mathbf {u} + \mathbf {R P E}, \quad \mathbf {U} _ {0} \in \mathbb {R} ^ {T \times N \times D} \tag {31}
+$$
+
+$$
+\mathbf {S} _ {0} = \mathcal {S N} (\mathbf {U} _ {0}), \quad \mathbf {S} _ {0} \in \mathbb {R} ^ {T \times N \times D} \tag {32}
+$$
+
+$$
+\mathbf {U} _ {l} ^ {\prime} = \operatorname {E M S A} \left(\mathbf {S} _ {l - 1}\right) + \mathbf {U} _ {l - 1}, \quad \mathbf {U} _ {l} ^ {\prime} \in \mathbb {R} ^ {T \times N \times D}, l = 1 \dots L \tag {33}
+$$
+
+$$
+\mathbf {S} _ {l} ^ {\prime} = \mathcal {S N} \left(\mathbf {U} _ {l} ^ {\prime}\right), \quad \mathbf {S} _ {l} ^ {\prime} \in \mathbb {R} ^ {T \times N \times D}, l = 1 \dots L \tag {34}
+$$
+
+$$
+\mathbf {S} _ {l} = \mathcal {S N} (\operatorname {E M S P} \left(\mathbf {S} _ {l} ^ {\prime}\right) + \mathbf {U} _ {l} ^ {\prime}), \quad \mathbf {S} _ {l} \in \mathbb {R} ^ {T \times N \times D}, l = 1 \dots L \tag {35}
+$$
+
+$$
+\mathbf {Y} = \operatorname {C H} \left(\operatorname {G A P} \left(\mathbf {S} _ {L}\right)\right), \tag {36}
+$$
+
+Following the Spiking Experts Mixture Mechanism and Spike-driven Self-Attention (SDSA) used in Spike-driven Transformer, the EMSA can be written as:
+
+$$
+\mathbf {Q} = \mathcal {S N} _ {\mathbf {Q}} (\mathrm {L} - \mathrm {B N} _ {\mathbf {Q}} (\mathbf {X})), \mathbf {K} = \mathcal {S N} _ {\mathbf {K}} (\mathrm {L} - \mathrm {B N} _ {\mathbf {K}} (\mathbf {X})), \mathbf {V} = \mathcal {S N} _ {\mathbf {V}} (\mathrm {L} - \mathrm {B N} _ {\mathbf {V}} (\mathbf {X})), \tag {37}
+$$
+
+$$
+\mathbf {A} = \operatorname {S D S A} (\mathbf {Q}, \mathbf {K}, \mathbf {V}) = \mathcal {S N} \left(\operatorname {S U M} _ {\mathrm {c}} (\mathbf {Q} \odot \mathbf {K})\right) \odot \mathbf {V}, \tag {38}
+$$
+
+The rest operates according to standard EMSA 18. The EMSP in Spike Driven Transformer is also much the same as the standard 22, except that no spiking neuron layer in the output layer.
+
+# B Computation Overhead Details of SEMM
+
+# B.1 EMSA
+
+The linear layers that generate  $m$  Query  $\mathbf{Q}_{\mathbf{m}}$  have  $m\left(\frac{D^2}{m}\right) = D^2$  parameters. The linear layer for Key  $\mathbf{K}$  has  $\frac{D^2}{m}$  parameters. Generating Value  $\mathbf{V}$  requires  $dD$  parameters. The output linear has  $dD$  parameters. The router linear layer transforms  $D$  to expert number  $m$ , and has  $mD$  parameters. The number of total parameters is  $(1 + 1/m)D^2 + (2d + m)D$ . For operations, obtaining Query and Key needs  $(1 + 1/m)\overline{R}TND^2$ . Computing Value or the output of EMSA requires  $d\tilde{R}TND$ . The number of operations between  $\mathbf{Q}_{\mathbf{m}}$ ,  $\mathbf{K}$ ,  $\mathbf{V}$  is  $(D + md)R_{\mathcal{R}}TN^2$ . The number of router matrix computation operations is  $m\tilde{R}TND$ . Due to the sparsity of SEMM and the smaller data dimensions, the number of element-wise operations for the router and expert is negligible.
+
+# B.2 EMSP
+
+The parameter number of  $\mathbf{W_1},\mathbf{W}_{\mathcal{R}},\mathbf{W_o}$  are all  $(8 / / 3)D^{2}$ . The number of parameters for  $3\times 3$  depthwise convolution is  $3\times 3\times (8 / / 3)D = 24D$ . The all parameter number is  $8D^{2} + 24D$ . The operation number of  $\mathbf{W_1},\mathbf{W}_{\mathcal{R}},\mathbf{W_o}$  are all  $(8 / / 3)TND^2$ . The number of operations for depthwise convolution is  $24\tilde{RT} ND$ .
+
+# C Experiment Details
+
+# C.1 Experimental Setup
+
+Our experimental framework closely aligns with the methodology outlined in [11]. All of Baseline's code is publicly available, and we abide by their credentials. For the ImageNet-1K, we utilize a fixed timestep count of  $T = 4$ . The optimizer is the AdamW, with a batch size of 128 or 256 over the course of 310 training epochs. The learning rate is governed by a cosine-decay schedule, starting from an initial value of 0.0004. We incorporate a suite of standard data augmentation techniques, including random augmentation, mixup, and cutmix, into our training. For four small datasets, we adapt the SEMM to a variety of baseline models, following the precedents set by [11, 12]. For CIFAR, we maintain a timestep count of  $T = 4$ . For the neuromorphic datasets, we increase this to  $T = 10$  and  $T = 16$ , respectively. Our experimental setup is consistent with each Spiking Transformer baselines, as detailed below.
+
+Spikformer with SEMM. The training epoch is set to 310 for CIFAR, 200 for DVS128 Gesture, and 106 for CIFAR10-DVS. The batch size is 128 for CIFAR, 16 for DVS128 Gesture and CIFAR10-DVS. The learning rate is initialized to 0.0005 for CIFAR10/100, 0.001 for DVS128 Gesture and CIFAR10-DVS. All of them are reduced with cosine decay. We follow [11] to apply data augmentation on DVS128 Gesture and CIFAR10-DVS. In addition, the network structures used in CIFAR-10, CIFAR10-DVS, and DVS128 Gesture are Spiking Transformer-4-384, Spiking Transformer-2-256 and Spiking Transformer-2-256.
+
+Spike Driven Transformer with SEMM. The training epoch is set to 210 for CIFAR10/100 datasets, 200 for DVS128 Gesture, and 106 for CIFAR10-DVS. The batch size is 32 for CIFAR10/100, 16 for DVS128 Gesture and CIFAR10-DVS. The learning rate is initialized to 0.0005 for CIFAR10/100, 0.0003 for DVS128 Gesture and 0.01 for CIFAR10-DVS. The rest is consistent with Spikformer.
+
+Spikingformer with SEMM The training epoch is set to 410 for CIFAR10/100 datasets, 200 for DVS128 Gesture, and 106 for CIFAR10-DVS. The batch size is 64 for CIFAR10/100, 16 for DVS128 Gesture and CIFAR10-DVS. The learning rate is initialized to 0.0005 for CIFAR10/100, 0.1 for DVS128 Gesture and CIFAR10-DVS. The rest is consistent with Spikformer.
+
+# C.2 Model Details
+
+In our experiments, we use 8 NVIDIA-4090 GPUs for ImageNet, and 1 NVIDIA-4090 GPU for other datasets. We adjust the value of membrane time constant  $\tau$  in spike neuron when training models on DVS datasets. In direct training SNN models with surrogate function,
+
+$$
+\operatorname {S i g m o i d} (x) = \frac {1}{1 + \exp (- \alpha x)} \tag {39}
+$$
+
+We select the Sigmoid function as the surrogate function with  $\alpha = 4$  in all experiments.
+
+# C.3 Additional Experiments
+
+Table 5: Ablation study results on the time step.  
+
+<table><tr><td>Datasets</td><td>Models</td><td>Time-Step</td><td>Top1-Acc(%)</td></tr><tr><td rowspan="12">CIFAR10/100</td><td rowspan="4">Spikformer + SEMM</td><td>1</td><td>94.11/74.67</td></tr><tr><td>2</td><td>94.87/78.49</td></tr><tr><td>4</td><td>95.78/79.04</td></tr><tr><td>6</td><td>95.99/79.29</td></tr><tr><td rowspan="4">Spike Driven Transformer + SEMM</td><td>1</td><td>94.97/76.77</td></tr><tr><td>2</td><td>95.58/78.49</td></tr><tr><td>4</td><td>96.12/80.26</td></tr><tr><td>6</td><td>96.48/80.87</td></tr><tr><td rowspan="4">Spikingformer + SEMM</td><td>1</td><td>94.54/77.16</td></tr><tr><td>2</td><td>95.42/78.85</td></tr><tr><td>4</td><td>96.16/80.24</td></tr><tr><td>6</td><td>96.67/81.22</td></tr></table>
+
+The accuracy regarding different simulation time steps of the spiking neuron is shown in Tab. 5. At all the time steps, our Spiking Transformer with SEMM shows competitive performance. When the
+
+time step is 1, our Spikformer with SEMM is  $1.7\%$  lower than the network with  $T = 4$  on CIFAR10, and our Spikformer with SEMM with 1 time step still achieves  $74.67\%$  on CIFAR100. The above results show that Spikformer with SEMM is robust under fewer time steps.
+
+In EMSP, the role of Deep Wise Convolution layer (DWC) is to independently extract features at the channel-level experts using a  $3 \times 3$  receptive field, thereby enhancing representational capacity. The ablation study about DWC in Tab. 6 demonstrates its effectiveness.
+
+Table 6: Ablation on Deep Wise Convolution layer.  
+
+<table><tr><td>Model</td><td>Module</td><td>CIFAR100</td><td>CIFAR10-DVS</td></tr><tr><td>Spikformer</td><td>EMSP</td><td>78.53</td><td>82.32</td></tr><tr><td>Spikformer</td><td>EMSP w/o DWC</td><td>78.17</td><td>81.30</td></tr><tr><td>Spike-driven Transformer</td><td>EMSP</td><td>79.81</td><td>81.10</td></tr><tr><td>Spike-driven Transformer</td><td>EMSP w/o DWC</td><td>78.95</td><td>80.51</td></tr><tr><td>Spikingformer</td><td>EMSP</td><td>79.81</td><td>81.95</td></tr><tr><td>Spikingformer</td><td>EMSP w/o DWC</td><td>79.44</td><td>81.20</td></tr></table>
+
+# D Visualization
+
+# D.1 Image Samples Visualization
+
+More dynamic allocation visualizations of the spiking router are given in Fig. 9
+
+# E Discussion on the Feasibility of Hardware Deployment for SEMM
+
+SEMM can theoretically be deployed on chips that are synchronous within layers and asynchronous between layers, such as TrueNorth [48]. Specifically, multiple spiking router matrices, i.e.  $\{r_1, r_2, \dots, r_m\}$ , can be implemented by one block (Linear-BatchNorm-SpikingNeuron), thus can be mapped to the one core of the neuromorphic chip for computation. Outputs from the same core are considered synchronous. The spiking Hadamard product and addition that occur in SEMM are analogous to the AND and ADD operations in SEW ResNet [7]. These element-wise spiking operations can be adapted on neuromorphics chips that support multiple branches and residuals, such as Speck V2 [49].
+
+![](images/fc99ba4ee208686f4397317bfb6de4c4d14b2ecb9e9aa74ef49e6d4ec812af5b.jpg)
+
+![](images/c7de9d4532bcfcd69624f914567c4414b690332dfd98847c6178cee7a940d0b3.jpg)
+
+![](images/5df62490141261717cd3a3bcbf2186a5ac819bb2e7e25f2ef97c3cf39bce788d.jpg)
+
+![](images/6f84314a86bcdd09679be2c7b44c3783709a3c1669c14523ac01b3df24cd3bc5.jpg)
+
+![](images/13531965e2f6dd6d219d36e2f5bf493344fb397d6ad5b4cf3776b69d85203d9a.jpg)
+
+![](images/fdf45f4370181e266a15d4e556f72a6e7a87f3fb78bad14bd8970c38155f8e50.jpg)
+
+![](images/db6ce832495ead533d8eb50296ed6508392cff32172f978db4ade22139fb5e3d.jpg)
+
+![](images/c9aef9f000aab0b8723412f0b9e2d81153024093a895475fe56e0006341a86e3.jpg)
+
+![](images/69376b409829c2d83a6d9b360f167c931e28c01ab288c2b2737d1c033d45e9a0.jpg)  
+Figure 9: More visualization samples of routers as masks.
+
+![](images/f878773962563a854ab3efea118d986743b3702eb7a74ed5034e237243494128.jpg)
+
+# NeurIPS Paper Checklist
+
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+
+Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:
+
+- You should answer [Yes], [No], or [NA].  
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+
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+
+The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While "[Yes]" is generally preferable to "[No]", it is perfectly acceptable to answer "[No]" provided a proper justification is given (e.g., "error bars are not reported because it would be too computationally expensive" or "we were unable to find the license for the dataset we used"). In general, answering "[No]" or "[NA]" is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.
+
+IMPORTANT, please:
+
+- Delete this instruction block, but keep the section heading "NeurIPS paper checklist",  
+- Keep the checklist subsection headings, questions/answers and guidelines below.  
+- Do not modify the questions and only use the provided macros for your answers.
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: Please see abstract.
+
+Guidelines:
+
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+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: Please see Section 5.
+
+# Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
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+
+# 3. Theory Assumptions and Proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
+
+Justification: The work does not involve proof of theory.
+
+Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
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+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental Result Reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer:[Yes]
+
+Justification: We detail all the techniques for reproducing the results of our work in Section 4 and Appendix. C.
+
+Guidelines:
+
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+
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
+
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+
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+
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+
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+Answer: [Yes]
+
+Justification: Please see the codes in the supplementary material.
+
+# Guidelines:
+
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+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
+
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental Setting/Details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+Answer: [Yes]
+
+Justification: Please see Appendix. C.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment Statistical Significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+Answer: [No]
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
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+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments Compute Resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: Please see Appendix C.2.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.
+
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code Of Ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: Our work conforms with the NeurIPS Code of Ethics.
+
+Guidelines:
+
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+- If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.  
+- The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).
+
+# 10. Broader Impacts
+
+Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
+
+Answer: [NA]
+
+Justification: Our work does not involve social impact.
+
+Guidelines:
+
+- The answer NA means that there is no societal impact of the work performed.  
+- If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.  
+- Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.  
+- The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.  
+- The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.  
+- If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
+
+# 11. Safeguards
+
+Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
+
+Answer: [NA]
+
+Justification: Our work does not release data or models with a high risk of misuse.
+
+Guidelines:
+
+- The answer NA means that the paper poses no such risks.
+
+- Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.  
+- Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.  
+- We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.
+
+# 12. Licenses for existing assets
+
+Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
+
+Answer: [Yes]
+
+Justification: We describe in Section C.1.
+
+Guidelines:
+
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+- If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.  
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+- If this information is not available online, the authors are encouraged to reach out to the asset's creators.
+
+# 13. New Assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
+
+Justification: Our work does not introduce new assets.
+
+Guidelines:
+
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+- Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.  
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+- At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
+
+# 14. Crowdsourcing and Research with Human Subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
+
+Answer: [NA]
+
+Justification: Our work does not involve crowdsourcing experiments and research with human subjects.
+
+# Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.  
+- According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
+
+# 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
+
+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: Our work does not study participants.
+
+# Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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+- For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

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+# Splatter a Video: Video Gaussian Representation for Versatile Processing
+
+Yang-Tian Sun $^{1*}$  Yi-Hua Huang $^{1*}$  Lin Ma Xiaoyang Lyu $^{1}$  Yan-Pei Cao $^{2}$  Xiaojuan Qi $^{1\dagger}$
+
+<sup>1</sup>The University of Hong Kong 2 VAST
+
+# Abstract
+
+Video representation is a long-standing problem that is crucial for various downstream tasks, such as tracking, depth prediction, segmentation, view synthesis, and editing. However, current methods either struggle to model complex motions due to the absence of 3D structure or rely on implicit 3D representations that are ill-suited for manipulation tasks. To address these challenges, we introduce a novel explicit 3D representation—video Gaussian representation—that embeds a video into 3D Gaussians. Our proposed representation models video appearance in a 3D canonical space using explicit Gaussians as proxies and associates each Gaussian with 3D motions for video motion. This approach offers a more intrinsic and explicit representation than layered atlas or volumetric pixel matrices. To obtain such a representation, we distill 2D priors, such as optical flow and depth, from foundation models to regularize learning in this ill-posed setting. Extensive applications demonstrate the versatility of our new video representation. It has been proven effective in numerous video processing tasks, including tracking, consistent video depth and feature refinement, motion and appearance editing, and stereoscopic video generation.
+
+# 1 Introduction
+
+Video processing, which encompasses a variety of tasks such as video editing, can enable numerous applications in fields like social media, filmmaking, and advertising [2, 50]. A video can be viewed as a collection of spatiotemporal pixels. However, processing a video directly in its pixel space, while maintaining temporal consistency, poses challenges due to the inherent complexities associated with appearance, motion, occlusions, and noise in the video data [14, 27]. Consequently, a robust video representation capable of abstracting and disentangling appearance and motion is crucial for facilitating various applications and overcoming these challenges.
+
+Existing research on video representation for processing has primarily focused on 2D/2.5D techniques, employing methods such as optical flow and tracking to associate pixels across frames [47, 14, 58]. These approaches often involve learning a canonical image [12, 33, 49, 30] or a layered atlas with persistent motion patterns [14, 24, 4, 9] to facilitate editing and then use optical flow or tracks to propagate edits throughout a video. The most recent work [33] utilizes hash grids combined with implicit functions to embed a video into a learned canonical image for appearance and a deformation field for motion. Despite achieving promising results in appearance editing tasks, these methods struggle to handle occlusions of objects (see Fig. 3), leading to erroneous propagation. Although layered 2.5D representation [14, 24, 4, 9] can mitigate this issue, they still face challenges with complex self-occlusions within a layer. Moreover, these techniques have limited or no capability in
+
+addressing processing tasks that require 3D information, such as video representation with complex occlusions, consistent depth prediction, and stereoscopic video generation.
+
+![](images/168438a1943e5f521a681b8049d9469f4ee812dc6f0c941a4959cf356455805e.jpg)  
+Figure 1: We propose an approach to convert a video into a Video Gaussian Representation (VGR), which can be used for versatile video processing tasks conveniently.
+
+Drawing inspiration from the fact that a video is essentially a projection of the dynamic 3D world onto the 2D image plane at different moments, we pose the question: is it possible to represent a video in its intrinsic 3D form? By doing so, we could potentially bypass the limitations of 2D representations, such as occlusions, reduce the complexity of motion modeling, and support processing tasks that require 3D information. Recent work [48] has explored 3D representations, which employ an implicit radiance field to model a canonical 3D space and leverage a bi-directional mapping network for associating 2D pixels with 3D representations. While this approach demonstrates promising performance in dense tracking, it falls short in faithfully representing video appearance, making it incapable of performing video processing tasks that require generating new videos, such as video editing. Moreover, its implicit nature limits its applicability to a variety of video processing tasks that require explicit content or motion manipulations, such as the removal or addition of objects and adjustments to the motion patterns of objects.
+
+In this paper, we introduce a novel explicit video Gaussian representation (VGR) based on 3D Gaussians [16]. Our core idea revolves around utilizing Gaussians in a canonical 3D space to model video appearance while associating each Gaussian with time-dependent 3D motion attributes to control its locations at different time steps for video motion. This 3D representation can then be employed to process and render videos effectively. The subsequent challenge lies in how to map a video onto such a 3D Gaussian representation. This is inherently difficult due to the loss of essential 3D information during 3D-to-2D projection, as well as the entanglement of motion and appearance in videos. However, recent advancements in large models have facilitated the acquisition of high-quality monocular priors from images and videos, such as optical flow [44, 11] and monocular depth [54, 15, 53]. While these 2D priors may not be perfect, they can serve as regularization for learning through knowledge distillation. Consequently, we propose leveraging these 2D priors in conjunction with our 3D motion regularization for learning. By doing so, we effectively lift 2D information—such as pixels, depth, and optical flow—into a unified and compact 3D representation.
+
+Upon learning, our video Gaussian representation can be used to support versatile video processing tasks, as shown in Fig. 1. Here, we showcase its efficacy in 7 video-processing tasks: Specifically, it can be used to obtain 1) dense tracking and 2) improve the consistency of monocular 2D prior across frames, leading to better video depth and feature consistency. Secondly, our representation facilitates a range of video editing tasks, including 3) geometry editing and 4) appearance editing. Thirdly, it also proves useful in video interpolation, allowing for 5) the generation of smooth transitions between frames. Finally, as our representation is inherently 3D, it opens up additional possibilities, such as 6) novel view synthesis (to a certain extent) and 7) the creation of stereoscopic videos.
+
+# 2 Related Work
+
+As our method utilizes dynamic 3D Gaussians to represent videos and supports versatile video processing, this section introduces related works on video editing, tracking, and dynamic Gaussian splatting. We briefly cover the most relevant works. For additional references, see Sec.A.7.
+
+![](images/64998164e9abf43db26ded4f46c469887112eed65f7e116f95bf6d50cacc5c6e.jpg)  
+Figure 2: Pipeline of our approach. Given a video, we represent its intricate 3D content using video Gaussians in the camera coordinate space. By associating them with motion parameters, we enable video Gaussians to capture the video dynamics. These video Gaussians are supervised by RGB image frames and 2D priors such as optical flow, depth, and label masks. This representation makes it convenient for users to perform various editing tasks on the video.
+
+Video Editing Decomposing videos into layered representations facilitates advanced video editing techniques. Kasten et al. [14] introduced layered neural atlases, enabling efficient video propagation and editing. Further advancements include deformable sprites [58], bi-directional warping fields [9], and innovations in rendering lighting and color details [4]. CoDeF [33] and GenDeF [49] focus on multi-resolution hash grids and shallow MLPs for frame-by-frame deformations. Latent diffusion models [38] and methodologies like ControlVideo [61], MaskINT [28], and VidToMe [20] have also been employed for data-driven video editing.
+
+Video Tracking Video tracking captures physical motion within video sequences. PIPs [8] and TAPIR [6] offer foundational approaches, while CoTracker [13] uses a sliding-window transformer for tracking. OminiMotion [48] and MFT [30] employ neural radiance fields and optical flow fields for dense tracking. State-of-the-art methods like RAFT [44] and FlowFormer [11] provide accurate flow estimations but struggle with long-term correspondences.
+
+Dynamic Gaussian Splatting Gaussian Splatting [16] enhances rendering in radiance fields and has been extended to dynamic scenes [26, 57, 51]. Methods like SC-GS [10] and 3DGStream [43] offer novel approaches for scene dynamics. Our method targets monocular video representation, eliminating the need for camera pose estimations and facilitating robust long-term tracking and editing in dynamic scenes.
+
+# 3 3D Gaussian Splitting
+
+Gaussian splatting [16] models 3D scenes using Gaussians learned from multiview images. Each Gaussian,  $G$ , is defined by a center  $\mu$  and a covariance matrix  $\Sigma$ :  $G(x) = \exp \left(-\frac{1}{2}(x - \mu)^T\Sigma^{-1}(x - \mu)\right)$ . Here,  $\Sigma$  is decomposed into  $RSS^T R^T$  for optimization, with  $R$  as a rotation matrix parameterized by a quaternion  $q$  and  $S$  as a scaling matrix parameterized by a vector  $s$ . Each Gaussian also has an opacity  $\alpha$  and spherical harmonic (SH) coefficients  $sh$ . Then 3D Gaussians can be formulated as:  $\mathcal{G} = \{G_j : \mu_j, q_j, s_j, \alpha_j, sh_j\}$ . Rendering is done via:
+
+$$
+C (u) = \sum_ {i \in N} T _ {i} \sigma_ {i} \mathcal {S H} \left(s h _ {i}, v _ {i}\right), T _ {i} = \Pi_ {j = 1} ^ {i - 1} \left(1 - \sigma_ {j}\right), \tag {1}
+$$
+
+where  $\sigma_{i}$  is calculated by projecting Gaussian  $G_{i}$  at the rendering pixel and  $v$  is the direction from view point to the Gaussian. Optimizing parameters  $\{G_{j}:\mu_{j},q_{j},s_{j},\alpha_{j},sh_{j}\}$  and adjusting densities allows for high-quality, real-time image synthesis. For a more detailed introduction to Gaussian Splatting, please refer to Sec. A.8. We extend 3D Gaussians to represent a video by adding attributes to Gaussians for versatile processing.
+
+# 4 Method
+
+Given a video, our goal is to use 3D Gaussians in a canonical space to represent its appearance and associate Gaussians with 3D motions for video dynamics. To facilitate this mapping, we incorporate 2D priors extracted from existing 2D models and apply 3D motion regularization. This representation allows us to efficiently perform various downstream applications. The pipeline of our method is depicted in Fig. 2. In the following, we elaborate on the video Gaussian representation in Sec.4.1. Then, we discuss the learning objectives and optimization details in Sec.4.2 and Sec. 4.3, respectively.
+
+# 4.1 Video Gaussian Representation
+
+Camera Coordinate Space Instead of utilizing an absolute 3D world coordinate system, we opt for the orthographic camera coordinate system to model a video's 3D structure, as demonstrated in Omnimotion [48]. In this space, the video's width, height, and depth correspond to the  $X$ ,  $Y$ , and  $Z$  axes, respectively. This enables us to circumvent the challenges associated with estimating camera poses or disentangling camera motion from scene dynamics, which can be not only time-consuming [40, 41] but also prone to failure in casually captured monocular videos with dynamic objects [34, 59]. By modeling the scene as dynamic 3D Gaussians in the camera coordinate space, we intertwine camera motion with object motion and treat them as the same type of motion, eliminating the need for camera calibration. During the rendering process, the 3D Gaussians in the camera coordinate space are rasterized into images from an identity pose camera. This approach simplifies the representation of dynamics and avoids the challenges of estimating camera pose from monocular casual videos.
+
+Video Gaussians Given a video  $\mathcal{V} = \{I_1, I_2, \dots, I_n\}$  consisting of  $n$  frames, our video Gaussian representation transforms it into a set of dynamic 3D Gaussians, parameterized as  $\mathcal{G} = \{G_1, G_2, \dots, G_m\}$ , to simultaneously represent the appearance and motion dynamics of the video. Each Gaussian is characterized by its position  $\mu$ , rotation quaternion  $q$ , scale  $s$ , spherical harmonics  $(\mathcal{SH})$  coefficients of appearance  $sh$ , and opacity  $\alpha$ . In addition to these fundamental Gaussian properties for appearance, dynamic attributes  $p$ , segmentation labels  $m$ , and image features  $f$  from any 2D base models (e.g., DINOv2 [31] and SAM [17]) can also be associated with 3D Gaussians to depict the video's scene content. Consequently, a Gaussian can be expressed as  $G = (\mu, q, s, \alpha, sh, p, m, f)$ . To learn these properties from a video, we enhance the differentiable 3D Gaussian renderer to render additional attributes beyond simple color, which we denote as  $\mathcal{R}(\mu, q, s, \alpha, x)$ , where  $x$  represents the specific attribute to be rendered. The rendering function  $\mathcal{R}$  follows the same procedure as color rendering in the original Gaussian Splatting method [16].
+
+Gaussian Dynamics When parameterizing motion, there is a trade-off between incorporating more regularization from motion priors and achieving high fitting capability [46]. In line with recent popular methods [21, 18], we employ a flexible set of hybrid bases comprising polynomials [22] and Fourier series [1] to model smooth 3D trajectories. Specifically, we assign learnable polynomial and Fourier coefficients to each Gaussian, denoted as  $p = \{p_p^n\} \cup \{p_{\sin}^l,p_{\cos}^l\}$ , respectively. Here,  $n$  and  $l$  represent the order of coefficients. The position of a Gaussian at time  $t$  can then be determined as follows:
+
+$$
+\mu (t) = \mu_ {0} + \sum_ {n = 0} ^ {N} p _ {p} ^ {n} t ^ {n} + \sum_ {l = 0} ^ {L} \left(p _ {\sin} ^ {l} \cos (l t) + p _ {\cos} ^ {l} \sin (l t)\right). \tag {2}
+$$
+
+Polynomial bases  $\{t^n\}$  are effective in modeling overall trends and local non-periodic variations in motion trajectories and are widely used in curve representation, such as in Bezier and B-spline curves [32, 7]. Fourier bases  $\{\cos (lt),\sin (lt)\}$  offer a frequency domain parameterization of curves, making them suitable for fitting smooth movements [1], and excel in capturing periodic motion components. The combination of these two bases leverages the strengths of both, providing comprehensive modeling, enhanced flexibility and accuracy, reduced overfitting, and robustness to noise. This equips Gaussians with the adaptability to fit various types of trajectories by adjusting the corresponding learnable coefficients. It is important to note that for each Gaussian, the associated parameters  $p = \{p_p^n\} \cup \{p_{\mathrm{sin}}^l,p_{\mathrm{cos}}^l\}$  are learned from the video by optimizing the learning objective as described in Sec. 4.3.
+
+# 4.2 2D Monocular Priors and 3D Motion Regularization
+
+Learning video Gaussians in the camera coordinate space to achieve consistency with real-world content using photometric loss is challenging and often ill-posed. There are multiple solutions for video Gaussians to fit the observed 2D projections. For instance, relative depth orders among scene objects can be ambiguous without occlusion cues. Moreover, different Gaussians may sequentially represent the same object, and their motion may not precisely match the object's actual motion. Therefore, regularization is required during the training process.
+
+Thanks to advancements in 2D visual understanding methods, monocular 2D priors such as optical flow [44, 11] and depth estimation [54, 15, 53] are now accessible. Although not perfect, these priors can provide crucial cues to regularize learning. To stabilize our method's training and ensure a real-world consistent solution, we supervise the video Gaussians using priors from the estimated flow obtained from RAFT [44] and the estimated depth derived from Marigold [15].
+
+Flow Distillation Optical flow represents the 2D projection of 3D motion. Flow distillation serves to regularize the 2D projections of 3D Gaussian motions. To guarantee that the motion of video Gaussians aligns with the estimated optical flow, we project the 3D motion of Gaussians  $(\mu(t_2) - \mu(t_1))$  between frames  $t_1$  and  $t_2$  onto the 2D image plane and regularize it using the estimated optical flow:
+
+$$
+\mathcal {L} _ {\text {f l o w}} = \mathbb {E} _ {(t _ {1}, t _ {2})} \left(\| \mathcal {R} (\mu (t _ {1}), q, s, \alpha , \pi (\mu (t _ {2})) - \pi (\mu (t _ {1}))\right) - \operatorname {f l o w} _ {t _ {1} \rightarrow t _ {2}} \| _ {1}). \tag {3}
+$$
+
+Here,  $\pi$  denotes the projection function that maps camera coordinates to image coordinates after projection, and flow $_{t_1 \to t_2}$  represents the optical flow estimated by RAFT [44] from  $t_1$  to  $t_2$ . This prior aids video Gaussians in learning the scene flow by ensuring that the 2D projection of their 3D motion on the XY-plane is consistent with the optical flow instead of relying on relative depth changes along the Z-axis to fit frame colors.
+
+Depth Distillation Monocular depth estimation provides the per-frame depth of a video. Although these estimates may be inconsistent across long-range frames, they offer valuable cues for regularizing the scene geometry. As a result, we utilize depth maps estimated by Marigold [15] to ensure a reasonable geometry for our video Gaussians. We employ the scale- and shift-trimmed loss proposed in MiDaS [37]:
+
+$$
+\mathcal {L} _ {\text {d e p t h}} = \mathbb {E} _ {t} \left(\left|\left| \tau \left(D ^ {t}\right) - \tau \left(\hat {D} ^ {t}\right)\right|\right| ^ {2}\right), \tau \left(D ^ {t}\right) = \left. \right.\left(D ^ {t} - t \left(D ^ {t}\right)\right) / \overline {{\left| D ^ {t} - t \left(D ^ {t}\right)\right|}}, t \left(D ^ {t}\right) = \operatorname {m e d i a n} \left(D ^ {t}\right), \quad (4)
+$$
+
+where  $D^{t}$  is the rendered depth of 3D Gaussians at time  $t$ , and  $\hat{D}^t$  is the corresponding predicted depth. It is worth noting that, thanks to our 3D representation, our approach can, in turn, refine the inconsistent monocular depth estimations and yield consistent depth predictions for a video.
+
+In sum, flow distillation regularizes the projected 3D Gaussian motion on the 2D image plane, corresponding to the X-Y axes in the camera coordinate space. Meanwhile, depth distillation regularizes the relative video Gaussian positions corresponding to the Z-axis in the camera coordinate space. Together, they offer comprehensive 3D supervision and complement each other, effectively regularizing the learning of 3D motion for video Gaussians.
+
+3D Motion Regularization In addition to depth and flow distillation, we employ local rigidity regularization to prevent Gaussians from overfitting the rendering targets through non-rigid motions [10, 26]. This approach encourages the 3D motion of individual Gaussians to be as locally rigid as possible [42]. As a result, Gaussians form locally rigid structures, aligning with real-world dynamics. To constrain the local rigidity of a Gaussian  $G_{i}$  from time  $t_1$  to  $t_2$ , we first identify the  $K$  nearest neighboring Gaussians  $G_{k}(k\in \mathcal{N}_{i})$  using its 3D position at  $t_1$ . Then, we apply the rigid loss to ensure that the edges between them  $(\mu_i(t_1) - \mu_k(t_1))$  adhere to a rigid transformation:
+
+$$
+\mathcal {L} _ {\mathrm {a r a p}} = \mathbb {E} _ {(i, t _ {1}, t _ {2})} \left(\sum_ {k \in \mathcal {N} _ {i}} | | (\mu_ {i} (t _ {1}) - \mu_ {k} (t _ {1})) - \hat {R} _ {i} (\mu_ {i} (t _ {2}) - \mu_ {k} (t _ {2})) | | ^ {2}\right), \tag {5}
+$$
+
+where  $R$  is the estimated rigid rotation transformation given by
+
+$$
+\hat {R} _ {i} = \underset {R \in \mathbf {S O} (3)} {\arg \min } \sum_ {k \in \mathcal {N} _ {i}} | | \mu_ {i} (t _ {1}) - \mu_ {k} (t _ {1})) - R (\mu_ {i} (t _ {2}) - \mu_ {k} (t _ {2})) | | ^ {2}. \tag {6}
+$$
+
+# 4.3 Optimization
+
+In addition to 2D priors and 3D regularization for learning 3D motion and geometry, we also incorporate a color rendering loss for appearance learning. Furthermore, we introduce an optional mask loss to facilitate the separation of background and foreground, which is particularly useful for editing applications.
+
+Color Rendering Loss Video Gaussian representation also learns to fit the color of video frames  $\{I_{gt}^{t}\}$  as in novel view synthesis methods [16, 29] with the rendering loss:
+
+$$
+\mathcal {L} _ {\text {r e n d e r}} = \mathbb {E} _ {t} \left(\left| \left| \mathcal {R} (\mu (t), q, s, \alpha , \mathcal {S H} (s h, v)) - I _ {g t} ^ {t} \right| \right|\right). \tag {7}
+$$
+
+Mask Loss Segmentation labels serve as a crucial attribute for pixels, enabling the identification of groups of pixels belonging to foreground objects. In our experiments, we separate pixels into foreground and background components by segmenting each frame and extracting the foreground mask  $\mathcal{M}^t$ . This mask is subsequently lifted to Gaussian, where it is associated with a learnable label attribute  $m \in \{0,1\}$ . The label attributes of Gaussians are supervised by the image segmentation results:
+
+$$
+\mathcal {L} _ {\text {l a b e l}} = \mathbb {E} _ {t} \left(\left| \left| \mathcal {R} (\mu (t), q, s, \alpha , m) - \mathcal {M} ^ {t} \right| \right| _ {2} ^ {2}\right). \tag {8}
+$$
+
+With the segmentation label, we can divide Gaussians into different parts and constrain their motion respectively, as shown in Eq. 5. Our approach can also manipulate (remove/duplicate) and edit specific objects in a video, as shown in Fig 7.
+
+Total Learning Objective The total learning objective is the weighted sum of all the losses:
+
+$$
+\mathcal {L} = \lambda_ {\text {r e n d e r}} \mathcal {L} _ {\text {r e n d e r}} + \lambda_ {\text {d e p t h}} \mathcal {L} _ {\text {d e p t h}} + \lambda_ {\text {f l o w}} \mathcal {L} _ {\text {f l o w}} + \lambda_ {\text {a r a p}} \mathcal {L} _ {\text {a r a p}} + \lambda_ {\text {l a b e l}} \mathcal {L} _ {\text {l a b e l}}. \tag {9}
+$$
+
+Adaptive Density Control We initialize the video Gaussians by uniformly sampling points in the camera coordinate space of the first frame, and apply a similar density control strategy as in vanilla Gaussian Splatting [16]. For more details, please refer to Sec. A.1.
+
+# 5 Video Processing Applications
+
+With our video Gaussian representation, we can perform various video processing tasks, including 1) dense tracking, 2) consistent depth/feature prediction, 3) geometry editing, 4) appearance editing, 5) frame interpolation, 6) novel view synthesis, and 7) stereoscopic video creation. In this section, we detail these applications, highlighting the versatility of video Gaussians.
+
+Dense Tracking Since the scene motion is captured by the dynamics of video Gaussians, we can project these dynamics onto the image plane as UV flow and rasterize the attributes as flow maps. This method handles both short and long-frame gaps effectively. The pixel flow map  $dU_{t_1 \to t_2}$  from  $t_1$  to  $t_2$  is calculated as:
+
+$$
+d U _ {t _ {1} \rightarrow t _ {2}} = \mathcal {R} (\mu (t _ {1}), q, s, \alpha , \pi (\mu (t _ {2})) - \pi (\mu (t _ {1}))). \tag {10}
+$$
+
+The rendered dense flow map provides pixel correspondences, facilitating tracking across frames.
+
+Consistent Depth/Feature Prediction Video Gaussians, supervised by monocular depth priors for each frame, conform to a reasonable geometry layout, providing consistent depth predictions across frames. Similarly, other image features can be distilled into video Gaussians; unifying per-frame features into a consistent 3D form. To distill image features (e.g., SAM [17] or DINOV2 [31]), we associate each video Gaussian with a feature attribute  $f$  and rasterize them to match the feature map  $\{\mathcal{F}_{gt}^{t}\}$  from 2D models:
+
+$$
+\mathcal {L} _ {\text {f e a t u r e}} = \mathbb {E} _ {t} \left(\| \mathcal {R} (\mu (t), q, s, \alpha , f) - \mathcal {F} _ {g t} ^ {t} \| _ {2} ^ {2}\right). \tag {11}
+$$
+
+Optimizing video Gaussians with  $\mathcal{L}_{\mathrm{feature}}$  unifies frame-wise 2D features in a 3D form, enabling the rendering of view-consistent feature maps  $\{\mathcal{F}_t\}$ :
+
+$$
+\mathcal {F} _ {t} = \mathcal {R} (\mu (t), q, s, \alpha , f). \tag {12}
+$$
+
+Consistent feature prediction is crucial for applications like video segmentation and re-identification.
+
+![](images/5ce3cdb0e3133700e78deb86a3c1280bc24070a50a58f0c05c53baa1a45848bb.jpg)  
+Figure 3: Qualitative comparison of video reconstruction using our method and SOTA methods.
+
+![](images/eed22a551b36e5af51cbfda4e6a332f8a3156c5e631d14d6ab8920786d55600d.jpg)  
+Figure 4: Dense tracking results on diverse complex motion patterns.
+
+Geometry Editing In the unified 3D space, geometry editing is straightforward. By distilling segmentation labels into video Gaussians, we can select Gaussians of the target identity and transform their positions  $\mu$ , quaternions  $q$ , and scales  $s$  for translation, resizing, and rotation. Adjusting their opacities changes the transparency of the edited objects. It also facilitates easy object removal within a video and supports object copying both between and within videos.
+
+Appearance Editing Appearance editing with video Gaussians can also be easily achieved. Users can select a specific frame  $t$  and perform painting, recoloring, or stylization. We fix all attributes except the  $\mathcal{SH}$  coefficients representing Gaussian appearance and optimize them to fit the edited image  $I_{\mathrm{edit}}^{t}$  using:
+
+$$
+\mathcal {L} _ {\mathrm {e d i t}} = \left\| \mathcal {R} (\mu (t), q, s, \alpha , \mathcal {S H} (s h, v)) - I _ {\mathrm {e d i t}} ^ {t} \right\| _ {2} ^ {2}. \tag {13}
+$$
+
+The edited results can propagate throughout the video, maintaining temporal consistency.
+
+Frame Interpolation The learned smooth trajectories of video Gaussians enable interpolation of scene dynamics at any up-sampling rate. Interpolated Gaussians' dynamic attributes can render interpolated video frames. By re-mapping the timestep values  $\{t\} \to \{t'\}$  with an arbitrary continuous function, we can freely adjust the video playback speed.
+
+Novel View Synthesis Applying a global rigid transformation  $\mathcal{T} \in \mathcal{S}\mathcal{E}(3)$  to video Gaussians allows for camera position adjustments. The rendering results of transformed Gaussians  $\mathcal{R}(\mathcal{T}(\mu(t)), \mathcal{T}(q), s, \alpha, \mathcal{SH}(sh, v))$  provide synthesized views from different perspectives.
+
+Stereoscopic Video Creation Similar to the novel view synthesis application, we can achieve stereoscopic frames by slightly translating video Gaussians horizontally by a fixed distance, representing the interocular distance. This application is crucial in filmmaking and gaming.
+
+# 6 Experiments
+
+Evaluation We conducted experiments on the DAVIS dataset [36] as well as some videos used by Omnimotion [48] and CoDeF [33]. Our approach is evaluated based on two criteria: 1) reconstructed video quality and 2) downstream video processing tasks. In addition to general video representation methods DeformableSprites [58], Omnimotion [48] and CoDeF [33], we also compare with dynamic
+
+Table 1: Comparison with existing methods on Tap-Vid benchmark (DAVIS).  
+
+<table><tr><td>Methods</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>AJ↑</td><td>δxavg↑</td><td>OA↑</td><td>TC↓</td><td>Training Time</td><td>GPU Memory</td><td>FPS↑</td></tr><tr><td>4DGS [51]</td><td>18.12</td><td>0.5735</td><td>0.5130</td><td>5.1</td><td>10.2</td><td>75.45</td><td>8.11</td><td>~40 mins</td><td>10G</td><td>145.8</td></tr><tr><td>RoDynRF [23]</td><td>24.79</td><td>0.723</td><td>0.394</td><td>/</td><td>/</td><td>/</td><td>/</td><td>&gt;24 hours</td><td>24G</td><td>&gt;1min</td></tr><tr><td>DeformableSprites [58]</td><td>22.83</td><td>0.6983</td><td>0.3014</td><td>20.6</td><td>32.9</td><td>69.7</td><td>2.07</td><td>~30 mins</td><td>24G</td><td>1.6</td></tr><tr><td>Omnimotion [48]</td><td>24.11</td><td>0.7145</td><td>0.3713</td><td>51.7</td><td>67.5</td><td>85.3</td><td>0.74</td><td>&gt;24 hours</td><td>24G</td><td>&gt;1min</td></tr><tr><td>CoDeF [33]</td><td>26.17</td><td>0.8160</td><td>0.2905</td><td>7.6</td><td>13.7</td><td>78.0</td><td>7.56</td><td>~30 mins</td><td>10G</td><td>8.8</td></tr><tr><td>Ours</td><td>28.63</td><td>0.8373</td><td>0.2283</td><td>41.9</td><td>57.7</td><td>79.2</td><td>1.82</td><td>~30 mins</td><td>10G</td><td>149</td></tr></table>
+
+![](images/3d96a40b74a0c2416484f489bc445470b23a17e0ba0d2c639d75c97a8009c4fe.jpg)  
+Figure 5: Qualitative comparison of video depth and features generated by our method and SOTA single-frame estimation methods. Our method yields more consistent estimations.
+
+NeRF/3DGS methods, namely 4DGS [51] and RoDynRF [23]. Note that for 4DGS, we estimate camera poses using monocular depth estimation method Unidepth [35] and DROID-SLAM [45]. Despite these efforts, the performance remains unsatisfactory, further highlighting the challenges of accurate camera pose estimation in causal dynamic videos. In contrast, our approach demonstrates the capability to handle more complex motions and achieves significantly higher reconstruction quality. For downstream tasks, our method also shows comparable performance to those specifically designed for these tasks.
+
+Video Reconstruction To demonstrate our method's fitting ability for casual videos, we compare it with Omnimotion [48] and CoDeF [33]. Omnimotion tends to render blurred results due to the smooth bias of the MLP when modeling the canonical space, while CoDeF struggles with complex motions due to the limited representation ability of the 2D canonical image. We report the rendering quality metrics and visualizations on the Tap-Vid DAVIS dataset [36] in Table 1 and Figure 3. More comparison with RyDynRF [23] and 4DGS [51] are provided in the supplementary materials.
+
+# 6.1 Video Processing Applications
+
+Dense Tracking. Our approach enables dense tracking by projecting the dynamics of Gaussians onto 2D image planes to obtain correspondences. Tracking results are visualized in Fig. 4 and evaluated in Table 1. Despite Omnimotion's specialization in tracking, our approach supports a wider array of video processing tasks with higher computational and training efficiencies. It achieves comparable results with better reconstruction quality using fewer resources. Tracking performance comparisons with similar-cost methods (CoDeF / 4DGS) are shown in the supplementary materials 11, highlighting our superior outcomes.
+
+Consistent Depth / Feature Generation. We present the results of consistent video depth and features (using SAM [17]) in Fig. 5, compared to per-frame prediction. Due to the unified 3D representation of video frames, the predicted depth and features exhibit significantly better consistency than those obtained from monocular predictions. We also evaluate the effectiveness of consistent SAM feature in the segmentation task in the supplementary materials 15. We recommend that readers watch the supplemental videos for better illustrations.
+
+Geometry Editing By manipulating the Gaussians associated with specific labels, we can achieve geometric editing of target identities, as demonstrated in Fig. 7. By deleting foreground Gaussians, we can remove foreground elements and render a clean background. Our approach also supports geometric edits such as duplicating, resizing, and translating. Additionally, the motion of these elements can be adjusted by setting different motion attributes.
+
+Appearance Editing Users can edit the appearance in a specific frame by drawing, stylizing, or recoloring, and these edits will be propagated across the entire video with cross-frame consistency. In
+
+![](images/49dd4fbb9d13d2cb8926410309c06aa331eddaf5698d999c71b42ce12b56e491.jpg)  
+Frames  
+Figure 6: Appearance editing results using the 2D prompt editing method [60].
+
+![](images/c8ddce3c3ff706b4660dc7d9ec8eab6a0ac115ad46a4e61badc4231f650e1a6f.jpg)  
+Fig.6, we demonstrate appearance editing using ControlNet[60]. Appearance editing is user-friendly in our representation, as it only requires single-frame editing.  
+Figure 7: Geometry editing results including object deleting, resizing, copying, and translating.
+
+Novel View Synthesis & Stereoscopic Video Creation Benefitting from depth regularization, the 3D Gaussians maintain a meaningful 3D structure, even from a monocular video. This facilitates novel view synthesis tasks, with examples provided in the supplemental video. Stereoscopic videos can also be produced, as shown in Fig. 8.
+
+# 6.2 Ablation Study.
+
+We perform ablation studies to validate the importance of the proposed modules, including camera model (perspective/orthographic), flow loss, depth loss (L2/scale-shift invariant). The results are reported in Table 2.
+
+Table 2: Ablation of each module in our framework.  
+
+<table><tr><td>Methods</td><td>Ours</td><td>Perspective Camera</td><td>w/o Flow Loss</td><td>w/o Depth Loss</td><td>L2 Depth Loss</td></tr><tr><td>PSNR ↑</td><td>29.61</td><td>22.51</td><td>25.16</td><td>29.18</td><td>28.15</td></tr><tr><td>SSIM ↑</td><td>0.8624</td><td>0.6908</td><td>0.6937</td><td>0.8475</td><td>0.8214</td></tr><tr><td>LPIPS ↓</td><td>0.1845</td><td>0.3958</td><td>0.4724</td><td>0.2449</td><td>0.3328</td></tr></table>
+
+Using a predefined pinhole camera intrinsic led to unstable optimization, resulting in artifacts in both geometry and appearance. This instability likely stems from the rasterization process, where the denominators of Gaussians' screen coordinates UV include depth, complicating the gradients. Replacing the shift- and scale-invariant depth loss with absolute L2 loss degrades performance, as monocular depth cues are ambiguous regarding scale and shift.
+
+Moreover, the depth prior is crucial for maintaining the 3D structure of Gaussians. Without the depth prior, Gaussians collapse into a flat 2D plane, hindering novel view synthesis and resembling 2D-layer methods.
+
+We also demonstrate the significance of motion regularization (rigid loss) and the selection of motion coefficients  $n$  and  $l$ , which effectively suppress unorganized Gaussian motion. Please refer to Sec. A.3 for more details.
+
+![](images/858ecc2b4b6d939d0ed51e06c71c62a29548315c5bf72485173c6ee93c592bfc.jpg)  
+Figure 8: Stereo view synthesis. One original frame is visualized in the first column for comparison.
+
+# 6.3 Limitations
+
+Although achieving satisfying performance, there are still some limitations to be enhanced. First, our approach suffers from significant changes in the scene, since large deformation is hard to optimize. Initializing the scene with dynamic point clouds might alleviate this problem. In addition, our approach still relies on existing correspondence estimation methods (e.g., RAFT), which might fail when processing rapid and highly non-rigid motion. Extending this representation to more general scenarios is still worth exploring. We have illustrated two scenarios in Fig 9.
+
+![](images/1466678cc18f63033f3e828e55c505d026e10746690557931f5f081eb4437087.jpg)  
+Figure 9: Our method may underperform in two scenarios. (1) Without camera estimation, it has to learn large camera rotations as scene motions, causing distant background blur (left). (2) It fails to track fast-moving transients, as photometric loss is insufficient for motion fitting (right).
+
+# 7 Conclusion
+
+In this paper, we introduced a novel explicit video Gaussian representation (VGR) based on 3D Gaussians to address the challenges of video processing. By modeling video appearance in a canonical 3D space and associating each Gaussian with time-dependent 3D motion attributes, our approach effectively handles complex motions and occlusions. Leveraging recent advancements in monocular priors, such as optical flow and depth, we lift 2D information into a compact 3D representation, facilitating a wide range of video-processing tasks. Our VGR method demonstrates efficacy in dense tracking, improving monocular 2D priors, video editing, interpolation, novel view synthesis, and stereoscopic video creation, providing a robust and versatile framework for sophisticated video processing applications.
+
+# Acknowledgement
+
+This work has been supported by Hong Kong Research Grant Council - Early Career Scheme (Grant No. 27209621), General Research Fund Scheme (Grant No. 17202422), and RGC Matching Fund Scheme (RMGS). Part of the described research work is conducted in the JC STEM Lab of Robotics for Soft Materials funded by The Hong Kong Jockey Club Charities Trust. We would like to thank Ziyi Yang for the insightful discussion and generous help.
+
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+
+# A Appendix / Supplemental Material
+
+# A.1 Implementation Details
+
+Typically, we use a video clip of about 50-100 frames and train the system iteratively for 20,000 steps. The training duration is approximately 15-20 minutes on an NVIDIA 3090 GPU. The Gaussians are initialized as 10,0000 points randomly sampled in a  $[-1,1] \times [-1,1] \times [0,1]$  box. We use an orthographic camera for rendering for simplicity, which is fixed at the origin. We also modify the rasterization pipeline of 3DGS to support the orthographic projection by replacing the  $J$  in EWA projection with  $\left[ \begin{array}{ccc}W / 2 & 0 & 0\\ 0 & H / 2 & 0 \end{array} \right]$ , where  $W$  and  $H$  are the resolution of the image. For each attribute attached to Gaussians, we set different learning parameters and annealing strategies, list in Tab 3. Note that the dynamics of Gaussians' rotation is also modelled in the same way as position.
+
+During training, the number of video Gaussians is adaptively adjusted as in vanilla Gaussian Splating [16]. Every 100 steps, Gaussians with an accumulated gradient scale of positions above a threshold will be densified. Based on their projected size, they will be either split or cloned. Concurrently, Gaussians with opacities below a threshold will be pruned. To avoid floaters, the opacity of Gaussians is reset to 0.01 every 3000 steps. After optimization, there are around  $10^{5} - 10^{6}$  3D Gaussian for a video containing  $10^{7} - 10^{8}$  pixels (resolution  $\times$  frame number).
+
+The loss weights for render, depth, flow, motion regularization, and label are set to  $\lambda_{\text{render}} = 5.0$ ,  $\lambda_{\text{depth}} = 1.0$ ,  $\lambda_{\text{flow}} = 2.0$ ,  $\lambda_{\text{arap}} = 0.1$ , and  $\lambda_{\text{label}} = 1.0$ .
+
+Table 3: Gaussian attributes table  
+
+<table><tr><td>Attributes</td><td>Position</td><td>Rotation</td><td>Scaling</td><td>SH0</td><td>SH1,2,3</td><td>Polynomial</td><td>Fourier</td><td>Seg Label</td><td>SAM Feature</td></tr><tr><td>lr</td><td>6e-5</td><td>1e-3</td><td>5e-3</td><td>2.5e-3</td><td>1.25e-4</td><td>1e-3</td><td>1e-3</td><td>1e-3</td><td>1e-3</td></tr><tr><td>Annealed lr</td><td>1.6e-6</td><td>/</td><td>/</td><td>/</td><td>/</td><td>1e-5</td><td>1e-5</td><td>/</td><td>/</td></tr></table>
+
+# A.2 More Visual Comparison
+
+We have visualized the comparison with 4DGS [51] and RoDynRF [23] in Fig 10. We also visualized the tracking comparison in Fig 11. Note that due to the inaccurate camera pose for the wild scene, the performance of 4DGS is very limited.
+
+![](images/c17af7b46aed151af28784b056fe5f5033950f5835ef4713da9a78af5aa7bdb6.jpg)  
+Figure 10: Video reconstruction compared with RoDynRF [23] and 4DGS [51]. Our method has higher PSNR and better visual quality.
+
+![](images/34cdb7d0ce096a4d7aa06eb501692c9d042a6ea6045f3514d5a77c2d4588ed41.jpg)  
+Figure 11: Tracking results visualization shows our method outperforms CoDeF [? ] and 4DGS [? ], especially in handling large-scale view changes and scene motions.
+
+# A.3 Ablation Study
+
+Depth Regularization. Without the depth prior, Gaussians collapse into a 2D plane. Although overfitting ability remains largely unaffected, the 3D structure is lost, and novel view synthesis is no longer possible, as shown in the right part of Fig. 12. Our approach then resembles 2D layer-based methods.
+
+Rigid Loss. Without the rigid motion constraint, undesirable floaters appear, degrading rendering quality and reducing reconstruction PSNR by 1.51 dB, as illustrated in the left part of Fig. 12.
+
+![](images/221ce1aa4409a77b2413994f75ed9960696e9febec3d23d88d1d42c178eb4978.jpg)  
+Figure 12: The depth prior (w/ depth) ensures video Gaussians conform to a realistic layout, while rigidity regularization (w/ rigid) eliminates floaters.
+
+We also conducted ablation studies on the selection of camera models and depth loss formats as shown in Fig 13.
+
+![](images/e27031693082b7e231453aa756f13aac67b90594a85b76ec77655e37c35509a5.jpg)  
+Figure 13: Ablation for camera model / L2 depth loss
+
+We also evaluated the effect of different motion representation parameters (n/l in Eq 2), reported in Table 4 and Fig. 14.
+
+Table 4: Ablation of n/l choice.  
+
+<table><tr><td>Metric</td><td>n=8 / l=0</td><td>n=0 / l=8</td><td>n=l=4</td><td>w/o n=l=8</td><td>n=l=12</td></tr><tr><td>PSNR ↑</td><td>28.02</td><td>26.87</td><td>28.46</td><td>29.61</td><td>27.47</td></tr><tr><td>SSIM ↑</td><td>0.8392</td><td>0.7989</td><td>0.8512</td><td>0.8624</td><td>0.8357</td></tr><tr><td>LPIPS ↓</td><td>0.3099</td><td>0.3245</td><td>0.2271</td><td>0.1845</td><td>0.2532</td></tr></table>
+
+![](images/b8bf3b201d9671f52093ccc4f1f05cec3effbe95b2c83a95e9a6bd3eebd9b744.jpg)  
+$\mathrm{n = 0,l = 8}$  
+n=1=8
+
+![](images/3ffe85bdd01b98f39d4ad5a066b5899eb80b50e14f579349dd94b63d417dfe79.jpg)  
+$\mathrm{n = 8, l = 0}$  
+n=1=12
+
+![](images/09a1d1a4dc3527b2e088037fe443dda36426c214d11d05745a93be966bca9616.jpg)  
+n=1=4  
+GT
+
+![](images/574f43dd5f8a9a2b1d0644f5586e156056eb5ee6cb9801290fa8858b1bc3c109.jpg)  
+Figure 14: Ablation study on hyperparameters  $n$  and  $l$  reveals optimal quality at  $n = l = 8$ .
+
+![](images/dc0610b8377987d43ed6e8bae5da13ea62e6b43dc25745d29978cc6e0f699599.jpg)
+
+![](images/e10c170c653974d0ec6104784bd976ecf593d5df8d55cbd87335dd426aaf0333.jpg)
+
+# A.4 Consistent SAM Feature Evaluation
+
+By adding SAM features to Gaussian points, our method can obtain consistent SAM features. Compared with SAM extracted from per single frame, this feature has better consistency for moving objects and removes the requirements of passing through the SAM image encoder, hence obtaining more accurate video segmentation results at a faster speed, as reported in the Table 5 and Fig. 15.
+
+Table 5: Comparison of per-frame SAM feature and ours.  
+
+<table><tr><td>Metric</td><td>Per-frame SAM</td><td>Ours</td></tr><tr><td>IOU ↑</td><td>0.753</td><td>0.827</td></tr><tr><td>Times ↑</td><td>0.513</td><td>0.025</td></tr></table>
+
+# A.5 Video Interpolation
+
+Thanks to the continuous parameterization of dynamics, our approach can interpolate video frames over time. We present the interpolation results in Fig. 16. Our method supports any video interpolation using an arbitrary continuous time re-mapping function at any frame rate.
+
+# A.6 Multi-object Editing
+
+By adding a multi-channel mask attribute to each Gaussian point, our method can achieve separate editing of multiple objects. We visualize an example of multi-object geometry / appearance editing in Fig. 17.
+
+# A.7 Expanded Related Work
+
+Video Editing Decomposing videos into layered representations facilitates advanced video editing techniques. Kasten et al.[14] introduced layered neural atlases that decompose an image into textured layers and learn a corresponding deformation field, thereby enabling efficient video propagation and editing. Subsequent advancements have introduced more sophisticated models. Ye et al.[58] developed deformable sprites, segregating videos into distinct motion groups, each driven by an MLP-based representation. Huang et al.[9] proposed employing a bi-directional warping field to support extensive video tracking and editing capabilities over longer durations. Recent innovations have also focused on enhancing the rendering of lighting and color details. Chan et al.[4] extended this approach by incorporating additional layers and introducing residual color maps, enhancing the representation of illumination effects within the video. The most current development in this
+
+![](images/65ea13dcb83aaaf8bffb1b68cca0e9a796205143f3f41ab5c15aa6b455652ec9.jpg)  
+Ours t1  
+Ours t2  
+SAM t2  
+Figure 15: Segmentation comparison reveals that our lifted SAM features outperform per-frame SAM segmentation, delivering higher resolution feature maps and superior IoU.
+
+![](images/c23813528dec1fcc970b7348c1aad9ad5266fc3a7aa9a3c9656b2b021bc65f01.jpg)  
+t  
+$\mathrm{t} + 0.25$  
+t+0.5  
+$\mathrm{t} + 0.75$  
+Figure 16: Video interpolation results. Please refer to the supplementary video for better visualization.
+
+area is CoDeF [33], which leverages a multi-resolution hash grid and a shallow MLP to model frame-by-frame deformations relative to a canonical image. This approach allows for editing in the canonical space, with changes effectively propagated across the entire video. GenDeF [49] uses a similar representation to generate controllable videos.
+
+Several studies have exploited the generative capabilities of latent diffusion models[38] for data-driven video editing. ControlVideo[61] adopts the methodology of ControlNet [60], integrating control signals into the network during the video reconstruction process to guide editing. Employing a related technique to manage control signals, MaskINT[28] utilizes frame interpolation to generate edited videos from specifically edited keyframes. In contrast, VidToMe[20] implements a token merging approach to incorporate control signals into the editing process. Additionally, certain research efforts [19, 62] have explored using inversion solutions to achieve video editing.
+
+Video Tracking. Video tracking is essential for capturing the physical motion of each point within a video sequence[39]. PIPs[8] track motion within fixed-size windows and include an occlusion branch, though they lack the ability to re-detect targets following prolonged occlusions. Building on the temporal processing concepts from PIPs, TAPIR [6] introduces TAP-Net[5], which precisely locates per-frame points. CoTracker[13] advances this by tracking individual query points using a sliding-window transformer approach. OminiMotion[48] pioneers the use of neural radiance fields[29] to model scene flow in NDC space. Its bijection network, which represents scene flow, is optimized for photometric consistency across frames, thereby enabling dense tracking. MFT[30] employs a sequential and dense point tracking methodology using optical flow fields computed across varying time spans. SpatialTracker[52] transforms each frame into a triplane and estimates trajectories by iteratively predicting movements with a transformer, facilitating 2D tracking within a 3D space. While state-of-the-art optical flow methods such as RAFT[44] and FlowFormer[11] provide accurate flow estimations for consecutive frames, they struggle with maintaining long-term frame correspondences.
+
+Gaussian Splatting Gaussian Splatting [16] has emerged as a potent method for enhancing rendering quality and speed in radiance fields. Following that, Lu et al. [25] further organize the Gaussians distribution by introducing anchor points, and Yang et al. [56] enrich its fitting ability in the specular setting with anisotropic sphere gaussian. These approaches have been extended to dynamic scenes in various recent studies. Luiten et al.[26] utilize frame-by-frame training, making it well-suited for
+
+![](images/d507159d954b3b26faed189289f747b011bbf7eca2694cefdd3617243d582b7e.jpg)  
+Figure 17: Segmentation and editing results are displayed from left to right: multi-object segmentation and video editing across two frames. The first row illustrates geometry editing where two individuals are copied and repositioned to lie on the ground and stand aside. The second shows the appearance editing with "2024" and "NeurIPS" painted on two fish.
+
+multi-view scenes. Yang et al.[57] advance this by segmenting scenes into 3D Gaussians coupled with a deformation field, particularly for monocular scenes. Building upon this work, Wu et al.[51] have replaced the traditional MLP with multi-resolution hex-planes [3] and a shallow MLP. Additionally, Yang et al.[55] integrate time as an additional dimension in their 4D Gaussian model. SC-GS[10] introduces a novel approach using sparse control points to learn a spatially compact representation of scene dynamics. 3DGStream [43] offers a high-quality free viewpoint video (FVV) stream of dynamic scenes generated in real-time, though it necessitates multi-view video streams as input. Gaussian-Flow [22] hybrid the basis of polynomial and Fourier to represent the Gaussian motion. These methods typically rely on pre-estimated camera poses. Our approach specifically targets monocular video representation, obviating the need for camera pose estimations. This facilitates more robust long-term tracking and editing capabilities in dynamic scenes.
+
+# A.8 Detailed Introduction to 3D Gaussian Splitting
+
+Gaussian splatting [16] models 3D scenes using 3D Gaussians by learning from posed multiview images. Each Gaussian, denoted as  $G$ , is defined by a central point  $\mu$  and a covariance matrix  $\Sigma$ ,
+
+$$
+G (x) = \exp \left(- \frac {1}{2} (x - \mu) ^ {T} \Sigma^ {- 1} (x - \mu)\right). \tag {14}
+$$
+
+The covariance matrix  $\Sigma$  undergoes decomposition into  $RSS^T R^T$  for efficient optimization. Here,  $R$  represents a rotation matrix, parameterized by a quaternion  $q$  from  $\mathbf{SO}(3)$ , and  $S$  is a scaling matrix defined by a positive 3D vector  $s$ . Additionally, each Gaussian is assigned an opacity value  $\alpha$  to modulate its rendering impact and is equipped with spherical harmonic (SH) coefficients  $sh$  for capturing view-dependent effects. The collection of Gaussians is represented as  $\mathcal{G} = \{G_j : \mu_j, q_j, s_j, \alpha_j, sh_j\}$ . Rendering is achieved through the equation:
+
+$$
+C (u) = \sum_ {i \in N} T _ {i} \sigma_ {i} \mathcal {S H} \left(s h _ {i}, v _ {i}\right), \text {w h e r e} T _ {i} = \Pi_ {j = 1} ^ {i - 1} \left(1 - \sigma_ {j}\right). \tag {15}
+$$
+
+Here,  $\mathcal{SH}$  denotes the spherical harmonic function and  $v_{i}$  the viewing direction. The value of  $\sigma_{i}$  is determined by evaluating the corresponding projection of Gaussian  $G_{i}$  at pixel  $u$  as follows:
+
+$$
+\sigma_ {i} = \alpha_ {i} \exp \left(- \frac {1}{2} \left(u - \mu_ {i} ^ {\prime}\right) ^ {T} \Sigma_ {i} ^ {\prime} \left(u - \mu_ {i} ^ {\prime}\right)\right), \tag {16}
+$$
+
+where  $\mu_i'$  and  $\Sigma_i'$  represent the projected 2D center and covariance matrix of Gaussian  $G_i$ , respectively. By optimizing the Gaussian parameters  $\{G_j : \mu_j, q_j, s_j, \alpha_j, sh_j\}$  and dynamically adjusting Gaussian densities, high-quality and real-time image synthesis is facilitated. However, vanilla Gaussian splatting can only be used to represent a static scene. In this paper, we integrate this representation with video by assigning additional attributes to each Gaussian, enabling more versatile video processing.
+
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+
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+
+# 2. Limitations
+
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+
+# 3. Theory Assumptions and Proofs
+
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+
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+
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+
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+# 8. Experiments Compute Resources
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+# 10. Broader Impacts
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+# 11. Safeguards
+
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+# 12. Licenses for existing assets
+
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+# 13. New Assets
+
+Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
+
+Answer: [NA]
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+Justification: The paper does not release new assets.
+
+Guidelines:
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+# 14. Crowdsourcing and Research with Human Subjects
+
+Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
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+Answer: [NA]
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+Justification: The paper does not involve crowdsourcing nor research with human subjects.
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+# 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
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+Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
+
+Answer: [NA]
+
+Justification: The paper does not involve crowdsourcing or research with human subjects.
+
+Guidelines:
+
+- The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.  
+- Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.  
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