S$^2$NN: Sub-bit Spiking Neural Networks
Authors: Wenjie Wei
Malu Zhang
Jieyuan Zhang
Ammar Belatreche
Shuai Wang
Yimeng Shan
Hanwen Liu
Honglin Cao
Guoqing Wang
Yang Yang
Haizhou Li
Venue: NeurIPS 2025 poster ()
Keywords: Spiking Neural Networks
Spiking Quantization
Neuromorphic Datasets
Primary area: neuroscience_and_cognitive_science
TL;DR: To further explore the compression and acceleration potential of SNNs, we propose Sub-bit Spiking Neural Networks (S$^2$NNs) that represent weights with less than one bit.

Abstract: Spiking Neural Networks (SNNs) offer an energy-efficient paradigm for machine intelligence, but their continued scaling poses challenges for resource-limited deployment. Despite recent advances in binary SNNs, the storage and computational demands remain substantial for large-scale networks. To further explore the compression and acceleration potential of SNNs, we propose Sub-bit Spiking Neural Networks (S$^2$NNs) that represent weights with less than one bit. Specifically, we first establish an S$^2$NN baseline by leveraging the clustering patterns of kernels in well-trained binary SNNs. This baseline is highly efficient but suffers from \textit{outlier-induced codeword selection bias} during training. To mitigate this issue, we propose an \textit{outlier-aware sub-bit weight quantization} (OS-Quant) method, which optimizes codeword selection by identifying and adaptively scaling outliers. Furthermore, we propose a \textit{membrane potential-based feature distillation} (MPFD) method, improving the performance of highly compressed S$^2$NN via more precise guidance from a teacher model. Extensive results on vision reveal that S$^2$NN outperforms existing quantized SNNs in both performance and efficiency, making it promising for edge computing applications.

SºNN: Sub-bit Spiking Neural Networks
Wenjie Wei', Malu Zhang Jieyuan Zhang', Ammar Belatreche", Shuai Wang', Yimeng Shant.4, Hanwen Liu', Honglin Cao', Guoqing Wang', Yang Yang', Haizhou Li 'University of Electronic Science and Technology of China,
Shenzhen Loop Area Institute, NNotthubrria University, *Liaoning Technical University The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen)
Abstract
Spiking Neural Networks (SNNs) offer energy-efficient paradigm for machine intelligence, but their continued scaling poses challenges for resource- limited de- ployment. Despite recent advances in binary SNNs, the storage and computational demands remain substantial for large-scale networks. To further explore the com- pression and acceleration potential of SNNs, we propose Sub-bit Spiking Neural Networks NS* that represent weights with less than one bit. Specifically, we first establish an N baseline by leveraging the clustering patterns of kernels in well-trained binary SNNs. This baseline is highly efficient but suffers from oulier-induced codeword selection bias during training. To mitigate this issue, we propose outlier-awee sub-bit weight quantization (OS-Qui (OS-Quant) method, which optimizes codeword selection by identifying and adaptively scaling outliers. Fur- thermore, we propose membrane potential-bant feature distillation (MPFD) method, improving the performance of highly compressed S2 via more precise guidance from a teacher model. Extensive results on vision tasks reveal that NN outperforms existing quantized SNNs in both performance and efficiency, making it promising for edge computing applications.
Introduction
Spiking Neural Networks (SNNs), with their unique event-driven paradigm, seen promising energy-efficient solution for realizing the next generation of machine intelligence 2 Specifically, SNNs employ binary spikes for information transmission and process them in a sparse event-driven manner. This computational paradigm transforms convolution operations in traditional artificial neural networks (ANNs) from computationally intensive multiply-accumulaa (MAC) efficient accumulate (AC), thereby significantly improving computational efficiency [3] Moreover, the event- driven nature of SNNs has spurred the development of neuromorphic hardware, such as SpiNNaker [4. TrueNorth [5. Loihi [6], and Tianjic 77 further harnessing their potential for energy efficiency. However, as large language models (LLMs) exhibit superior performance, the SNN community has
While this scaling has enhanced performance, it has sacrificed SNNs' inherent efficiency advantages, posing storage and computational challenges for their deployment
begun scaling up SNN models to improve their performance on complex tasks g 10
constrceconttrained edge devices.
In recent years, researchers have increasingly investigated compression techniques for SNNs, such as pruning [2], neural architecture search [33 14], quantization [1 16), and others [17 [8] [9].
As an extreme quantization technique, binarization restricts parameters to only two values. i.e.. -1 and +1 (20]. By applying binarization, researchers have developed lightweight binary SNNs (BSNNs). BSNNs not only inherit the sparse event-driven paradigm of SNNs, but also further convert
"Corresponding author: maaluzhng@uesstc.omm
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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convolution operations from AC to cost-effective bitwise operations. This greatly reduces the resource overhead of SNNs, especially on edge devices. However, as neural networks are scaled to greater depths to meet practical demands, the computational burden remains a significant challenge, even 22]. This raises an important question: "Can the compression and acceleration potential of SNNs be further exploited?"
for binary-weighted versions [21
Studies on Binary Neural Networks (BNNs) have shown that binarized convolutional kernels in well- trained BNNs exhibit clustering patterns within each layer. This phenomenon becomes increasingly pronounced as the network depth increases 23 24 In the case of 3x3 kernel, an analysis of the distribution of all possible 3x3 kernel values (2xx3 representing the full codebook) reveals that only a small subset of binary kernels (codewords) is frequently activated. Based on this observation. kernels in each layer can be restricted to a subset of binary convolution kernels (a compact codebook) during training. enabling sub-bit model compression. This method achieves higher compression ratios and faster inference speeds compared to BNNs. However, despite these promising efficiency gains, sub-bit techniques remain unexplored in the context of SNNs.
In this paper, we introduce sub-bit spiking neural networks (S NNs) to further harness the compression and acceleration potential of SNNs. We first construct an N baseline that encodes weights using less than I bit. However, we observe that this baseline is prone to outliers when mapping 32-bit kernels to binary kernel subset, resulting in suboptimal binary kernel selection. To address this issue, we propose an outlier-aware sub- bit weight quantization (OS- (us-uuatt) method that improves binary kernel selection by identifying and scaling outliers. Furthermore, to enhance the baseline performance, we introduce a membrane notential-based feature distillation (MPFD) method, which utilizes teacher model to guide the training of the highly compressed baseline. The main contributions are as follows:
We introduce N bascline that achieves extreme model compression by encoding weights with less than bit. This approach achieves higher compression ratios than BSNNs, further unlocking the potential of SNNs in terms of both compression and acceleration.
We identify that the baseline suffers from mutlier-induced codeword selection bias, negatively impacting performance. To address it, we propose an outlier-aware sub-bit weight quantiza- tion (OS-Quant). OS-Quant effectively eliminates the influence of outliers on quantization while preserving the spatial features of kernels, ensuring optimal codeword selection.
We propose a membrane potential-based based feature distillation (MPFD) framework which employs a teacher model to guide the training of the highly compressed baseline. By applying distillation at the membrane potential level, MPFD achieves more accurate knowledge transfer, improving performance without compromising compression benefits.
Extensive experiments demonstrate that integrating OS- OS-Quant and MPFD into the baseline enables N' to achieve state-of-the-ee (SOTA) performance and efficiency. Furthermore, tests across diverse tasks and architectures validate the scalability of our method.
2 Related Works
Binary Neural Network Binarization is traditionally considered the most extreme quantization method, which helps reduce computational overheads but compromises model accuracy. Therefore, most early BNN research focuses on narrowing the gap between BNNs and full precision models.
For example, [25] propose floating-point scaling factors for BNNs to fit real-value weights. [26)
approximate real-value weights by linearly combining multiple binary weight bases. [27 propose adding shortcuts similar to ResNet to reduce information loss during the binarization process. [28
retains information in BNNs by maximizing information entropy and minimizing gradient errors. [29 adopt generalized activation function to capture the distribution reshape and shift, achieving excellent accuracy on ImageNet-
As the performance gap between BNNs and full- pile-ireeiioon models continues to narrow, few recent studies have begun further compress BNNs, successfully reducing parameter bitwidths less than 1 bit. 330 propose a flexible encryption algorithm that encrypts subvectors of flattened weights into low-dimensional binary codes. (23 observe the kernel clustering distribution characteristic of BNNs and then constrain kernels within prescribed binary kernel subset during training. [II applies the concept of stacked low-dimensional filters and product quantization to achieve sub-bit model compression.
[3) propose minimum spanning tree compression, which uses the fact that output
2

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channels in binary convolution can be calculated using another output channel and XNOR operations. Recently, [33] has been learning sequences of binary tiles to populate the layers of DNNs, achieving sub-bit storage of neural network parameters.
Binary Spiking Neural Network Given SNNs' binary spike activations, researchers have devel- oped BSNNs with 1-bit synaptic weights to reduce storage and accelerate computation. Early works explore ANN-SNI conversion to obtain BSNNs. For instance, [34] first train binary convolutional neural network and then convert it to BSNN. [35] introduces weight-threshole balanced conversion approach to minimize conversion errors and enhance BSNN performance. However, these conversion- based methods inevitably suffer from accuracy degradation and fail to process sequential datasets. This leads researchers explore direct BSNN training methods. [36] directly train BSNNs using surrogate gradient (SG) methods. (37) propose novel Bayesian-based BSNNs learning algorithm that outperforms SG methods in accuracy. [38 presents a time-encoded BSNN where neurons emit at most one spike and learn in event-driven manner, thereby offering substantial energy benefits. By combining BNN and SNN advantages, [16] propose BitSNN, which enhances energy efficiency through binary weights, single-step inference, and sparse activations. Recently, [15] draw from information theory and introduce a weight-spike dual regulation method, aimed at achieving the performance of full-precision SNNs (FP SNNs) by improving BSNN's information capacity. Despite these advances, current methods still face key limitations. Firstly, these studies mainly focus on narrowing the performance gap between BSNNS and FP-SNNs. However, as networks scale up to meet practical application demands, the computational burden remains a challenge even with binary versions. Secondly, these studies are limited simple image classification tasks, leaving their scalability to complex tasks and diverse architectures unexplored.
To address these limitations, we propose novel method to further explore SNNs' potential in compression and acceleration. Moreover, our method emphasizes scalability across complex tasks and diverse architectures, making it practical for broad applications. These efforts will facilitate the efficient deployment of large-scale SNNs on edge devices.
3 Sub-bit Spiking Neural Networks Baseline
In this section, we construct sub-bit SNN baseline by leveraging existing knowledge, primarily including the employed spiking neuron models and the sub-bit weight quantization.
Spiking Neuron Model Various neuron models proposed replicate the information processing capabilities of biological neurons, such as Madgkin-Huxles [39], Izhikevich (2) and Leaky Integrate- and-Fire (LIF) 40) models. Due to its computational efficiency, the LIF model is widely used. Therefore, we also employ the classic LIF model in our work; its membrane potential is described as,
ū' [t] = ru' [t 1] + f(w/, s (()),,,
s [t] O 0),
u' [ ū ú' (1-)((1))
where T is the leaky factor, wy is the 32-bit weight matrix of layer E, f(-) is the convolution or linear operation followed by batch normalization (BN), and O(-) is the Heaviside step function. As described above, neurons integrate inputs and emit a spike when the membrane potential ñ exceeds the threshold 0. After each spike emission, the hard reset mechanism is invoked, where u is reset to zero upon emitting spike and remains unchanged in the absence of a spike.
Sub-Bit Weight Quantization The NN baseline is constructed based on the sub-bit weight quantization technique, as shown in Fig. [(b] This technique is applied to the binary convolutional kernels of the network. Therefore, we first construct BSNN by quantizing the weight matrix w/ in Eq. (T) 1-bit representation, described as
w = sign(w'), sign(w)) = (-, if if <0,
+1,, otherwise,
where a the channel-wise scaling factor that calculated as the average of the absolute value of weights in each output channel [25). and w is the binary weight matrix. By combining this
3

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Conspact Codebook
Redex
2
Beshiet
SKN
3
1+1
=112
4:136 a
H.
+
3tmm
+1++1
Dinary karnd Indes
Binery kernelidd
22g
B5MN
Kennel 10-544
Renod in SRUM (1222 21)
(+:1 les
Layer
Layer
bit
[B2*99 bits
Bits per welghtt
Nis
brt
o/libit
Figure I: (a) Convolutional kernels in well-trained BSNNs exhibit clustering patterns. This motivates us to achieve higher compression ratios than BSNNs by using a compact codebook P instead of the full codebook K. (b) The constructed NN baseline.
binarization with Eq. I. the convolution or linear operation can be transformed from w @ s to ar (sign(w") w @ (T)). where arithmetic operations are replaced with efficient bitwise operations @ For simplicity, we omit the scaling factor in subsequent analysis as is applied after the bitwise convolution operation.
Before introducing sub-bit weight quantization, we first present the clustering pattern of binary kernels. We formulate the weight matrix wi. as the channel-wise concatenation of each binary kernel, ie., w/. = =I[= Il of respectively. Each binary kernel w E IK is derived by w/.- sign(w f.e)) Here, K = {=1] ) denotes the set of all possible binary kernels (i.e., full codebook) with size ki x kin This full codebook contains [K] 2k unique binary kernels (i.e., codewords). Previous studies reveal these codewords exhibit layer-dependent clustering patterns in well-trained BNNs, especially in deep layers [23 24). We conduct a similar analysis in well-trained BSNNs and observe the same phenomenon, as illustrated in Fig [(a) Based on prior studies and the clustering patterns of BSNNs, compact codebook rather than the full codebook K. to construct the S baseline. We present Appendix the top-k codeword proportions for BSNN. The findings validate the clustering
wf Whe where & and cf, are the number of output and input channels in layer l,
and reveal increased clustering patterns in deeper layers.
The NN baseline is built by (1) sampling layer-specific codeword subsets Tp' (i.e., compact codebook), (2) mapping each 32-bit kernel to its nearest codeword in pe for inference, depicted in Fig. 1(b)
It defined as (23),
Forward propagation: wh.- = arg min |K
w
(5)
kEE'
OC
Oc
Backward propagation: dw f.c
1 1ww
(6)
Owh
where pº C K. |.||P = 2", and r7 < -k. Noteworthy, each binary codeword in p can be optimized during training, so sign(-) must be applied after optimization to preserve its binary representation. By integrating Eq. 5 with Eq. [1. the N' baseline is established.
The sub-bit quantization in Eq. computes the squared L2 distance between whe and each candidate codeword k in Fº and selects the nearest k replace w for forward propagation. This approach achieves below 1-bit compression by using index to represent each binary kernel. Specifically, for weights the L-th layer, i.e., w E {+1] ( kwu BSNN requires kiw k2 c, c. bits to store the whole parameters, while the S?NN baseline requires bits of 1 c E.ue for indicies and 2n h. kn for the storage of TP Since 17 is designed to be smaller than R. kh, the NN baseline achieves compression ratio of D.c.,, ee. +2" Rw R., -ff -c
T.. for each weight. Consider the commonly used 3 3x3 convolutional kernel, the N2 baseline represents each parameter with 0.44, 0.56, and 0.67 bit when y of 4, 5, and 6, respectively. A comprehensive analysis of the baseline's compression and acceleration advantages is presented in Appendix H

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Compact Codebook P'
4m-3
Dudien
ki
Kgn
36-3
2m-3
16-3
Weight
35.5
40.7
-1e.3
Binarization prrqucttion
0.900.0..7
ginie()
22:
0.900.0110 0.8.0.
-3.3
10
30
50
Ideal codeword: &
32-bit kernel w/.
Selected endewword:
Randomly sampled convolution kernnls
(a) Outlicr-induced codeword selection bias
(b) Outliers during the training of the N2 baseline
Figure 2: (a) The outliers dominate the distance calculations, diminishing the contributions of other elements and leading the baseline to select undesirable codeword for inference. (b) During the training process of the N' baseline, we randomly sample several kernels and analyze their weight distributions using box plot. Discrete points in the figure indicate outliers.
4
Method
In this section, we first analyze how the SANN baseline is adversely affected by outlier-induced code- word selection bias. Then, we propose an outlier-aware sub-bit weight quantization that eliminates outlier interference while preserving kernels' spatial features. Finally, we introduce a membrane potential- based distillation, improving the SºNN's performance via the guidance of teacher model.
4.1 Outlier-induced Codeword Selection Bias
key step the SaNN baseline is compute the squared L2 distance between the 32-bit kernel wf.- and each candidate codeword k in IP, then select the nearest codeword to replace w'.c for inference. While achieves sub-bit weight compression, this process is susceptible to outliers, leading to biased codeword selection. This bias can be regarded as quantization errors in parameter compression.
To illustrate this issue, we consider an example of 3x3 kernel, as shown in Fig. ((a) Given w/.c and some candidate codewords. From a binarization perspective, ky is the optimal choice to replace wf.e better maintains the sign patterns of the majority elements in w However, the presence of outlier 5 causes the baseline to choose kg instead. We define this inconsistency the outleerinduced codeword selection bias. This issue occurs since the employed squared 1.2 distance is sensitive large values, causing outliers to dominate the distance computation and overshadow the contribution of other elements. Unfortunately. the chosen non-optimal codeword cannot capture the true sign pattern of w',.. adversely affecting the baseline learning. In Fig. 2(b) we show that many codeword
outliers exist in the learning process, indicating that the baseline suffers from selection bias, This motivates us to address the bias for stable convergence and improved performance. In Appendix B we count the percentage of kernels containing outliers in each layer demonstrate that this bias a common phenomenon.
4.2 Outlier-Aware Sub-Bit Weight Quantization
We introduce the OS-Quant to address the codeword selection bias caused by outliers. The OS- OS-Quant comprises two steps: (1) interquartile range (IQR)-based outlier detection, and (2) spatially- outlier scaling. This approach effectively mitigates the negative impact of outliers on the quantization process, while simultaneously preserving the spatial information inherent in float-point kernels.
(QR-based Outlier Detection During the sub-bit quantization, we use quartile statistics to deter- mine the boundaries for normal weight values in w/,e and consider values outside these boundaries as outliers. Specifically, we first calculate the interquartile range of w wf. as defined in
IQR = Q3 - Q1
(7)
5

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Compact Codchook of
Teachee model
09
+1
+1
-1
G0
ExPr
Shudent model
0.9 -0.5 2u, -.. 8.8 -0.
-1
Rease
Q, Seate 0.9 orgnnien
Lawer Bounds
Upeer Bound Se.t 155 FOR 225
0.9000.009
IQR based outleer detection
Adjusted kemee of
Selected: k2
(a) Outlier-aware sub-bit weight quantization
(b) Membrane potential-based feature distillation
Figure 3: Schematic diagram of the proposed OS DS-Quant and MPFD method.
where Q1 and Q3 denote the first and third quartiles of wf, respectively. Then, threshold coefficient y is applied define the normal weight range as follows,
R e [Q: 7 JQ Q3 + IQR
(8)
where 7 is a coefficient that controls the sensitivity of outlier detection. A larger 7 yields fewer outliers, while smaller 7 leads to more. Following the well-established "Tukey's fences' 442 is
typically set to 1.5. Detail analysis of this hyperparameter is provided in Appendix [ Accordingly, any weights outside this range are classified as outliers. We define the set of outlier coordinates in wf. f.o as,
Of,c- = ((.,) (t..)) & Rf.)),
where (i.j) is the coordinates in the kernel, and w/, (1.j) corresponds to the weight value at the specified position. As shown Eq. 7 the interquartile range metric focuses on the central 50% of all weights in the kernel, thus being less affected by outliers. This makes our outlier detection method exhibit greater robustness than methods relying on mean and variance, thereby providing a more reliable kernel adjustment to resolve the codeword selection bias.
Spatially-Aware Outlier Scaling After detecting outliers, we propose a spatially-aware scaling approach to eliminate their impact on distance computation. This method leverages the relationships between outliers and their spatial neighbors, preserving the spatial feature of 32-bit kernels.
Given the outlier set O of the kernel w/.c- we determine spatial neighbors for each outlier within this set. For an outlier located at (i.j) Of flot its neighbor set is defined as,
N (L.) [ii + 1,j), (i,j + 1)} n [1, kw] x [1, ks), where hiw and kn denote the kernel's width and height, respectively. This set effectively represents the local spatial relationships of the outlier within the kernel. Based on this spatial information, we introduce a regularization term to adaptively scale the outliers, and it is computed as,
(10)
Σ (P.4)EN(.))
(w..o(i.j) - (p.)l) -V).(..))-///((.)))).
(11)
No
where |N (a.)) is the number of spatial neighbors. Pig is calculated as the mean of the absolute differences between outlier and its neighbors. Then, we derive an adjusted 32-bit kernel w
(i. (1/24.) (..ii,), if (i,j) E Of otherwise.
(12)
As a result, the OS-Quant method is described as follows,
Backward: OC
Oc Owe
Forward: w = arg min ||k 1o---f.lll w/.lll;
tw 1wg. Ow, Ow
(13)
kEM
f,c
In summary, OS-Quant effectively addresses the codeword selection sub-bit quantization bias by detecting and scaling outliers. Furthermore, in the process of outlier scaling, OS- preserves the spatial features of full-precision kernels, enhancing the training stability and performance of NO We also compare OS- OS-Quant with several alternative outlier- outlier-handling methods in Appendix D

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4.3 Membrane Potential-basee Feature Distillation
The NN baseline yields great efficiency gains but suffers from performance degradation. To overcome this, introduce distillation technique to preserve the compressed model's performance [3]. Distillation is categorized into logit-based knowledge distillation (LGKD) and bastee
distillation (FD) 44 FD typically distills better-performin student since mimicking the teacher's
intermediate features provides the student with more precise optimization directions [45 46].
In the SNN domain, existing FD methods regard the firing rate of neurons network intermediate features and aim at aligning the firing rates between teacher and student networks. These firing rate-based FD (FRFD) methods achieve this alignment by adjusting membrane potentials and further
controlling the spike generation in backpropagation [47]. Mathematically, the gradient of the distillation loss to the membrane potential is expressed as:
OLdistal FLFFFD 0s 8g(-) 0
(14)
[s([
dg(-) du [
where @CERER can be directly calculated from the distillation loss function. Notably, this distillation- [x*(8]
related gradient computation involves an extra surrogate gradient function g(-), which causes the gradient induced by distillation the membrane potential to be imprecise, thereby compromising the distillation optimization process. As a result, we propose a direct MPFD method at the membrane potential level to achieve more precise optimization directions. Mathematically, within MPFD, the gradient of the distillation loss to the membrane potential @Criva can be derived directly from the ou'10]
distillation loss function LMPFD. The MPFD is formulated as,
LMPED
Σ Σ (E,))ET
teal!] G [1]
G 1
(15)
Q(ū 10 ((7
Q(MMED
GM
(16)
where {e", {) P denotes layer pairs between teacher and student, M E {tea, stu} is the network type, and Q R.oohw + Rh-cchw is operation transforming tensor dimensions from [b, c, h, w] to [b. c x w). In Eq. (6). we introduce metric using a normalized Gram matrix of membrane potentials, which can effectively represent the network's 's semantic information 48).
We summarize the advantages of MPFD in two aspects. First, by directly imposing distillation on membrane potentials, it achieves more precise knowledge transfer from teacher to student networks. Second, the inner product-based formulation of G facilitates arosstarchtecctuee distillation, without requiring matched network layers or identical layer dimensions between teacher and student networks. As result, the MPFD significantly enhances the effectiveness and flexibility of knowledge distillation. Further analysis of MPFD is available in Appendix E
4.4 Workflow and Supplementary Details for S
We develop the N' by integrating the OS US-Quant and MPFD into the baseline, with its workflow described in Algorithm We provide more details of SºNN in the Appendix. In Appendix H we provide an in-depth discussion of how SNNN achieves below-1-bit compression and acceleration advantages. In Appendix | we present comparison between NN and BSNN on an FPGA, highlighting the advantages of N' in terms of both compression and acceleration.
The proposed NNNN aims to optimize SNN deployment efficiency. Notably, the involved distance calculations, outlier detection, outlier scaling, and distillation in our methods introduce no additional overhead during inference. After training, only the compact codebook IP and the weight-to-codeww indices are stored. During inference, the model reconstructs binary kernels directly from the indices and I., without performing distance calculations, OS- OS-Quant, or membrane potential-based distillation. Consequently, our N'N achieves below 1-bit model compression, which further explores the potential of SNNs for both compression and acceleration while maintaining high performance, making NN particularly suitable for applications in resource-limited scenarios that require reliable and efficient processing.
7

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Algorithm 1 One training iteration process of the SºNN.
1: Input: Initial SNN model: M = {w),"" wf); Size of fi A well-trained teacher model:
MMeai Input;
2: Initialize: Randomly sample layer-specific Pe, where [IP] 2" and 17 < kw Kus Initialize an
empty list F::
3: for + 1 to L do
4: for c + 1 to cout c/ do
5:
Calculltee IQR confine normal weight range: R,,. =QQ -7-111/// Q3 Q3+7-IQR
6:
> Get the coordinates of outliers in the kernel: (ii.j) w (i.(i. & R,c):
f.c 0
7:
> Calculate regularization term for each outlier:
a4j= o. E(p-g)eNn.. (i.ji w ().c(P,)))
8:
> Spatially- aware scale each outlier:
(1/52g)) wh, w (i.i),
(f.j) E O S.e
w (t.i)
otherwise;
9:
> Apply OS-Quam to achieve sub-bit quantization: wh. = arg mink lk f.ol:
10:
end for
11:
Concatenate and reshape: w + concat&reshape
b.e of
12:
[f ru'it
(N (wf, @ of (t)),
Record for distillation: F + F. F.append (T(() (t);
14:
> Calculate ['[ and u'[(] according to Eq. 2
15:
& Perform the inference on the model Meea based on Eq.
and record its membrane
potential:
end for
17: CMPFD = =E E. Gleald Go 1
Compute the loss: = Loe ACMPFD:
19: Backpropagation and update model parameters;
Experiment
In this section, we evaluate the performance of SºNN on various tasks, including classification, object detection, and semantic segmentation. Then, conduct ablation studies verify the effect of the OS-Quant and MPFD. Details on experimental setups are provided in Appendix K
5.1 Performance Comparison
Image Classification We assess NN various architectures like MS-ResNet
49 VGGSNN
50 and spike-driven Transformer v3 (SDT3) (1), comparing it with advanced compression methods
[51]. and BESTformer [52] Results in Tab. 1 reveal three conclusions. First, with y = (W is 0.67 bit). S NN achieves SOTA results on all
SNNs, like BitSNN (16), Q-SNN [15]. Q-Spikformer
datasets, rww.ueduunngggge size and OPs by 1.4x 6.8x. Second, despite significant reductions in size and OPs when 4, N2NN outperforms the base on CIFAR-10 and ImageNet-1KK with only small performance drop on other datasets. Third, SNNN performs well FP SNN on simple datasets with fewer resources. Despite a gap on ImageNet- IK, it outperforms the advanced work [52] by we supplement our comparison with related BNN methods.
3.5%.4.6%. Appendix F
Object Detection use the COCO dataset evaluate the efficacy S2NN detection tasks.
Following previous studies 53 21
we use the mmdetection codebase, with S as the backbone
for feature extraction and Mask R-CNN 54 for detection. The backbone initialized using network pre-trained on ImageNet- IK, with 7 6. Visualization and results are shown in Fig. 4())

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Table Performance of image classification. The colored values in brackets are the reduction factor of NN relative to the baseline.
BIT (W/A)
SIZE (MBIT)
OPs
ACC.
DATASET
METHOD
ARCITECTURE
(G)
(%)
FP SNN
RESNET-19
32/1
400.07
15.70
96.68
BITSNN 16
RESNET-18
1/1
11.34
94.37
Q-SNN 15
RESNET-19
1/1
13.04 BASE
6.40 BASE
95.54 BASE
CIFAR-10
NN
RESNET-19
0.67/1
8.92 (1.5x)
4.29 (1.5%)
96.43 (+0.9)
S'
RESNET-19
0.56/1
7.55 (1.7X)
3.58 (1.8x)
96.36 (10.8)
NN
RESNET-19
0.44/1
6.05 (2.2x)
2.82 (2.3x)
95.99 (+0.5)
FP SNN
RESNET-19
32/1
401.55
18.89
80.42
Q-SNN 15
RESNET-19
14.52 BASH
6.50 nAmh
78.77 DASH
CIFAR-100
SN
RESNET-19
0.67/1
10.40 (1.4x)
4.39 (1.4%)
78.77 (+0.0)
SºNN
RESNET-1S 19
0.56/1
9.03 (1.6))
3.67 (1.8%)
78.43 (-0.3)
N
RESNET-IE 19
0.44/1
7.53 (19x)
2.88 (2.3x)
77.40
FP SNN
SDTv3- 19M
32/1
607.57
16.03
79.80
QSPIKF
SPIKFORMER-& --5112
1/1
36.80
2.12
54.54
IMAGENET- IK
BESTF 25
SPIKFORMER-8
44.56
BASE
5.67 BASE
63.46
BASE
N'
SDTv3-19M
0.67/1
17.32 (2.6%)
0.84 (6.8x)
68.02 (14.6)
S
SDTv3-19M
0.56/1
15.88
(2.8%)
0.78 (7.3x)
67.43 (+4.0)
SONN
SDTv3-19M
0.44/1
14.31
(3.1%)
0.73 (7.8x)
67.00 (+3.5)
FP SNN
VGGSNN
32/1
296.60
1.97
82.3
Q-SNN [5
VGGSNN
1/1
10.91
BASE
0.31 BASE
81.6 BASE
DVSCIFAR-10
Nº
VGGSNN
0.67/1
7.86 (1.4x)
0.20 (1.6x)
82.0 (40.4)
Nº
VGGSNN
0.56/1
6.85 (1.6))
0.17 (1.8))
8 81.6
NN
VGGSNN
0.44/1
5.74 (1.9))
0.13 (2.4xc)
81.3 (-0.3)
and Tab. We compare N2 with advanced SNN and BNN models, and results reveal two conclusions. First, the N achieves comparable results to FP SNNs while remarkably reducing resource cost. For instance, our mAP@0.5 is comparable to that of (2] (i.e., 41.8%), but saves 2.08 in model size and 3.27x in power consumption. Second, compared to BNN methods, S exhibits SOTA results, surpassing the advanced work (55) by 7.4%.
Semantic Segmentation We use the ADE20K dataset to evaluate the efficacy of the SNNN in segmentation tasks. Following prior studies [53 21). we use the mmsegmentation, with N the backbone for feature extraction and semantic FPN for segmentation. The initialization mirrors that of the detection task. Visualization and results are shown in Fig. 4(b)aa and Tab. We compare NN with advanced SNNs and BNNs, and results reveal two conclusions. First, S"NN is far superior to methods in BNN, e.g., it surpasses the advanced work (56) by 17.6%% Second, S2NN yields comparable performance to FP SNNs with fewer resources, sufficiently validating the efficacy of our model in segmentation tasks. For example, method outperforms (53] by 1.1% in the MIoU metric and reduces the model size and power consumption by 34.6x and 19.8x, respectively.
SAN
GT
SAN
SAN
GT
SAN
(a) Detection visualization of SºNN.
Segmentation visualization of $ NN.
Figure 4: Detection and segmentation visualization of NN on COCO 2017 and ADE20K.

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Table 2: Object detection results on COCO 2017.
Table 3: Segmentation results ADE20K.
METHOD
BIT (W/A)
TIME STEP
(MBIT) (G)
SIZE
OPs
(@ (0%)
MАP
METHOD (T/A)STEP(MBITT (MJ) ACC (6)
BIT TIME SIZE POWER
PIX
MIoU
(%)
57
32/1
64
547.2
33.1
29
1/1
62.8
9.22
38
32/1
7
803.2
41.9
1/1
59.5
7.16
32/1
1
2400
51.2
63
1/1
59.5
9.74
33
32/1
1
1117
44.0
1/1
67.3
18.8
32/1
8
1238
58.8
32/1
528
22.1
32.3
32/1
2
1238
41.8
32/1
1914
183.6
35.3
1/1
10.99
0.55
31.0
32/1
352
27.2
40.1
60
1/1
175.0
3.22
32.9
32/1
352
33.6
41.4
1/1
173.3
3.21
37.2
S'NN
0.67/1
55.39
9.27
77.3
36.4
55
1/1
15.52
0.6
32.6
SºNN 0.56/1
55.36
9.23
77.5
36.2
NN
0.67/1
8
595.5
0.84
40.0
SONN 0.44/1
55.32
9.19
77.4
35.8
5.2
Ablation Study
We conduct ablation studies the proposed OS-Quant and MPFD meth- ods to demonstrate their effectiveness. Experiments are conducted on CIFAR- 100 with 1 (weight set 0.67 bit). The results are summarized in Tab. 4 First, replace the sub-bit weight quantization in the baseline with the QS-Quant, resulting 0.48% per- formance improvement. This result underscores the effectiveness of OS- Quant. In addition, compare three distillation methods mentioned in Sec. to validate the effectiveness of MPFD. Specifically, the performance for LGKD, FRFD, and MPFD are 75.92%, 76.71 %, and 78.77%, respectively. These re- sults indicate that feature-based distillation outperform logit-based methods, and our MPFD achieves higher performance than firing rate-based feature distillation by providing more precise optimization directions. conclusin, integrating the OS-Quant and MPFD into the baseline improves NºN performance by 3.66%, underscoring their effectiveness.
Table 4: Ablation study of our SºNN.
BASELINE OS-QUANT LGKD FRED MPFD Acc. (%)
75.1 BASE
75.59 (40.48)
75.92 (+0.81)
/
76.71 (+1.60)
78.77 (43.66)
Conclusion
SNNs have emerged as promising paradigm for energy-efficient machine intelligence. However, SNNs scale up to meet practical demands, their storage and computational requirements pose deployment. This work introduces SºNN, novel sub-bit
challenges for sesource-consrrainee compression framework for SNNs that represents weights with less than one bit. Through the introduction of OS-QOQQee OS-Quant and MPFD, SºNN effectively addresses quantization bias and preserves performance. Experiments show that Nº achieves SOTA performance while significantly reducing model size and computational costs, making it particularly suitable for edge computing applications.
Acknowledgments
This work is supported in part by the National Natural Science Foundation of China (No. 62220106008 and 02222122322 (432),, in part by the Shenzhen Science and Technology Program (Shenzhen Key Laboratory, Grant No. ZDSYS20230626091 (300006)) and in part by the Program for Guang- dong Introducing Innovative and Entrepreneurial Teams, Grant No. 2023ZT10X044, in part by the State Key Laboratory of Brain Cognition and Brain-inspired Intelligence Technology, Grant No. SKLBI-K2025010. This work was partially supported by UESTC Kunpeng& Ascend Center of Cultivation.
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A Ratio of Top-k codewords in Clustering Distribution
We analyze the top-k ratio BSNN kernels for ResNet and VGG, with results shown Tab. Both models show clustered distributions, with the clustering becoming more noticeable in deeper layers.
Table 5: The top-k ratio in BSNN kernels for ResNet and VGG structures.
CIFAR-100 100, Res19
Top2
Top4
Top8
Top16
Top32
Top64
Top128
layer1.2 .2
20.7%
24.4%
29.6%
37.2%
47.5%
60.8%
74.7%
layer2.2
21.8%
24.8%
30.0%
38.6%
50.2%
65.0%
80.0%
layer3.
46.8%
50.2%
54.5%
60.1%
67.9%
75.7%
83.9%
DVSCIFAR10, VGGSNN
Top2
Top4
Top8
Top16
Top32
Top64
Top128
conv3
15.1% 1
18.1% %
23.6%
31.8%
43.2%
59.1%
74.7%
conv5
14.9%
18.6%
23.8%
31.6%
43.5%
59.9%
76. 1%
conv7
14.2%
18.0%
24.6%
34.1%
47.7%
65.8%
82.3%
B Analysis of Outlier Occurrence about Models, Datasets, and Augmentation
Fig 2(b) depicts a small example of the first 60 kernels from ResNet- 19's second layer on CIFAR-100
To demonstrate that the emergence of outliers a common phenomenon, we count the percentage of
convolutional kernels containing outliers in each layer. Results are shown in Tab. 6 indicating that while the number of outlier-containnng kernels varies by models, datasets, and augmentation, outlier occurrence remains universal and severe issue.
Table 6: The percentage of convolutional kernels containing outliers.
Configuration
layerI.1
layer2.1
layer3.1
CIFAR 10, Res19
21.15%
17.62%
22.33%
CIFAR-100, Res19
33.19%
18.99%
20.19%
CIFAR-I00 100, Res19, only RandomCrop
30.10%
26.24%
17.61%
CIFAR-100, Res19, only RandomHoorizontalFii
29.16%
21.61% %
29.90%
conv3
conv5
conv7
DVSCIFARIO, VGGSNN
33.73%
15.99%
27.39%
C Analysis of Hyperparameter y in OS-Quant
We test the performance of different y values on CIFAR-100 with ResNet-19 (1=6). Results are summarized as in Tab. 7 and the following conclusions are obtained:
0.5 (too low) treats too many values as outliers, causing excessive outlier scaling and destroying the kernel's spatial information.
=3300 (too high) fails to detect outliers effectively, degrading N to baseline performance.
>=1.5 achieves optimal balance between outlier detection and spatial preservation.
1=1.0 and 7=2.0 perform well but slightly below 7=1.5.
Table 7: Accuracy under different values.
0.5
1.0
1.5
2.0
3.0
Accuracy 74.48%
75.34%
75.59%
75.19%
75.08%
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D Comparison of OS-Quant with alternative outlier-handling methods
We test several outler-handling methods against OS-Quant on CIFAR-I00 using ResNet-19 (7=6);
Hamming distance: Binarizes 32-bit kernels before calculating distance to codewords. This is the simplest method solve codeword selection bias but fails to preserve spatial information.
Clip methods: Instead of our IQR-based outlier detection and spatially-aware outlier scaling methods, we select three clipping variants.
(1) Layer uses 1st999th percentile of layer weights as clipping bounds; (2) IQResspi uses IQR for outlier detection and clipping;
(3) Z-score.iip- uses Z-score for outlier detection and clipping.
Smooth operation: Applied weight decay (Ie-3 and le-4) during training.
Z-scoreecalo: Combines Z-score outlier detection with our spatially-awar outlier scaling.
Table 8: Accuracy comparison of different outlier- methods.
OS-Quant Hamming Layerc11p IQRer1 Z-scorec1sp 1e-3wd 1e-4wd Z-8corescale
Acc. 75.59
70.82
37.23
75.01
74.82
73.68
72.24
75.34
Experimental results are summarized in Tab. g which confirms the effectiveness of OS-Quant. Furthermore, we analyze the potential reasons for the poor performance of other methods.
Hamming distance: Removes outliers via binarization but creates selection ambiguity. For example, given a full precision kernel f = [0.8. 0.7, 5, -0.9, -0.8, -0.7, -0.9, -0.8, -0.9], and two codewords k, = [1, -1, 1, -1. -1, -1, -1, -1, -1] and k2 [1. 1, 1, -1]. f has equal Hamming distance, i.e., 1, to kj and k2, causing selection ambiguity. In contrast, OS OS-Quant clearly differentiates distances (0.33 vs 3.93), identifying k, as better match. This shows the importance of preserving the spatial information full-precision kernels when handling outliers.
-1, -1, 1, -1,
Clip methods: Fig [ shows significant variation in kernel distributions, meaning the same value may be an outlier in some kernels but not others. Thus, using neiwork-wide layer-wide parameters determine clipping thresholds for each kernel is inappropriate, as confirmed by Layerclsp- Other clipping methods perform better but still lag behind OS-Quant and Z- eccaaee due to their failure preserve outliers' spatial information.
Smooth operation: Based on our analysis of outlier distribution, this method can slightly mitigate outlier occurrence but slows convergence, thereby yielding lower performance with equal training epochs.
Z-scoregcaie: Achieves top-2 results, but its effectiveness is limited because mean and variance calculations are influenced by the outliers themselves. Its performance gap with OS DS-Quant would widen on larger complex datasets.
E Detail Analysis of MPFD
In this section, we conduct a more detailed analysis of MPFD. Our main contribution to MPFD performing distillation at the membrane potential level, providing more precise gradient guidance for highly compressed models. That is, directly using the 2-norm to calculate membrane potential errors between teachers and students also provide more accurate gradient guidance compared to traditional FRED and LGKD. Therefore, if your goal is to avoid introducing excessive computation during training, you can choose to use simpler error calculation methods rather than the Gram matrix. In the following, we will mainly discuss the impact of the Gram matrix in the MPFD method on the performance, well the advantages and disadvantages of using and not using the Gram matrix.
We first evaluate membrane potential- distillation performance with and without the Gram matrix and other approaches. LGKD makes the student mimic the teacher network's final logits (the raw output values before the softmax activation function applied) to transfer knowledge [43]. FRFD is classic feature distillation scheme in the SNN domain [47), which uses the firing rate as
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the intermediate features of the network and align the firing rates between teacher and student models. Therefore, neither LGKD nor FRFD uses Gram matrix. Experiments are conducted on CIFAR- 100 with ResNet-19, 19. As shown in Tab. 9 the 2-norm membrane potential distillation achieves 78.32% aceuracy, which demonstrates that direct membrane potential distillation can also offer more precise gradient guidance than FRFD, thus improving accuracy.
Table 9: Performance comparison of different knowledge distillation methods in sub-bit SNN.
LGKD
FRKD
MPFD(w/o Gram)
MPFD Gram)
Accuracy 75.92%
76.71%
78.32%
78.77%
Both MPFD (w/o Gram) and MPFD (w/ Gram) perform distillation on membrane potentials, offer- ing more precise gradient guidance. Their respective pros and cons follows:
MPFD (w/ Gram). Gram matrix is typically regarded capturing semantic relationships, so MPFD usually correctly classified with higher confidence, leading to superior performance. This facilitates cross-architecture distillation without requiring matched network layers or identical dimensions. However, it incurs small additional computational costs.
MPFD (w/o Gram). It offers simpler implementation and lower computational costs. How- ever, it has slightly lower performance than MPFD (w/ Gram), and it doesn't 't capture the semantic relationships between features that the Gram matrix version does.
Supplementary Comparison with BNNs on Image Classification task
We supplement the comparison with related methods in the BNN domain the image classification task. The experimental results are summarized in the Table 10 These results demonstrate that NN performs competitively against existing BNN methods on static datasets. Specifically, when compared to the sub-bit neural network [23 that also operates with weights below 1-bit, SN achieves notable accuracy improvements of 5%-6% CIFAR-10 and 6%-9% ImageNet- IK. Notably, NN outperforms the sub-bit neural network, even when the latter employs 32-bit activations. Furthermore, when compared conventional BNNs with 1-bit weights, NN shows superior performance on CIFAR-10 10 using sub-l-bit weights, while maintaining competitive accuracy with state-of-the-ar methods on ImageNet-IK, These comprehensive results, along with those presented in Table decisively validate the effectiveness of NNN
G Performance Improvement Analysis about OS-Quant and MPFD
OS Quant improves performance in three ways:
Observing the codeword selection bias, which is the quantization error that typically
causes performance degradation. Addressing it will yield accuracy improvements.
Using IQR for outliers detection, which remains reliable even with limited data and isn't 't influenced by extreme values.
Implementing spatially-aware scaling, which effectively eliminates outlier interference while preserving crucial kernel spatial features for proper codeword selection.
MPFD improves performance by applying distillation at the membrane potential level, enabling more precise optimization than firing rate-based distillation.
H Model Compression & Acceleration
Compression We explain in detail how NN achieves sub-1-bit model compression. For simplicity, we analyze the parameter storage of single layer in the model. Consider a layer with Cont XCin x kn x kn parameters. As shown in the 'kernel storage' on the left side of Figure 5] standard BSNN requires I bit to store each weight parameter, resulting in total storage requirement of Cout Cin. key X Kr.
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Table 10: Supplementary comparison with related BNN methods on the image classification task.
Dataset
Method
Arcitecture Sub-bit Weight Bit Activity Bit Accuracy (%)
IR-Net
28
[CVPR20]
Res18
32
92.9
Res18
0.67
32
92.7
SNN
2
RCCV2021
Res18
0.56
32
92.3
Res18
0.44
32
91.9
RR-Net 23
CVPR201
Res18
X
91.5
CIFAR-10
Res1 88
0.67
91.0
SNN 23
JCCV21
Res18
0.56
90.6
Res18
0.44
90.1
ProxConnect++
64
NNoarPPS23
Res20
90.2
A&B 65
[CVPR24]
ReAct18
92.3
SC
Res19
0.67
1
96.43
SºNN
Res19
0.56
1
96.36
SºNN
Res19
0.44
1
95.99
IR-Net
28
[CVP220]
Res34
32
70.4
Res34
0.67
32
68.0
SNN X
JCCV21J
Res34
0.56
32
66.9
Res34
0.44
32
65.1
Bi-Real 27
JCC220
Res34
X
62.2
IR-Net
28
(CVPR20)
Res34
1
62.9
Res34
0.67
61.4
ImageNet-I1 K
SNN 2
ACCV21
Res34
0.56
60.2
Res34
0.44
58.6
BiBert 66
[CCR22)
Swin-T
1
68.3
Binary VIT
67
[CVPR23W)
VIT
X
67.7
ProxConnect++
[NarlPS223)
ViT-B
1
66.3
A&B
65
[CVPR26
ReActA
66.9
SºNN
SDT3
0.67
1
68.02
NºNN
SDT3
0.56
1
67.43
SºNN
SDT3
0.44 1
67.00
bits per layer. In contrast, S2NN requires fewer than Cout X Cin kw X k2, bits achieve more efficient storage. Specifically, as described Section 3 SºNN performs forward propagation using a compact codebook P rather than full codebook. This allows S to achieve compression below 1-bit by storing two components: (1) the indices of each kernel parameter in P. and (2) the mapping relationship between indices and weights. For the first component, NN needs store the indices of each kernel parameter in the compact codebook, with a total number Cout X Cin kernels (also indices). Since the compact codebook contains 2 binary codewords, the indices range from I to 2", requiring 1j bits per index. Therefore, representing the indices for this layer's parameters requires Cin Cout X 1) bits. For the second component, SNNN involves storing the compact codebook IP, which contains 2n elements, and each element is a kw X k2, binary kernel. Thus, storing P requires 29 kw x ky, bits. Therefore, the total storage requirement for S is C; Co X +7 2" kw X ky, bits. Compared to BSNN, N2 achieves compression ratio of Cour XC x1+22 x Cont X X Kew xka
Given that
7 < kw x kn, this ratio approximates to k Xk,
Acceleration In addition to model compression, we also discuss the hardware-friendly charac- teristics of SºNN. Benefiting from the advantages of model compression above, the hardware implementation of NN is theoretically more efficient than that of the standard BSNN. Specifically,
25

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BSNN Convolution
Output splie features
Ispet
Total parameters: Sin X Eaut Ky X
Imput sprke features
Index: e
O O
Indo; add. 1adeo aco. Eobo
SCNN Convolution
Outgust spslon
Inpuc
Bit Numberof Storages
conrolon
St 6
W,
Cadiorords
Input spike
Kernel Storage
Convolution Process
Figure 5: Comparison compression and acceleration between standard BSNN and our S2NN during the convolution process.
the weight transmission volume in each layer of S2 are significantly lower than those of BSNN, resulting in substantial reduction in data movement between on-chip and off-chip. This reduction yields three key improvements in hardware efficiency: first, significantly lowers data transmission latency between on-chip and off-chip. thereby accelerating the inference process; second, it reduces the demand for on-chip storage, minimizing memory overhead. When is set to 4, the on-chip storage requirement can be saved by approximately _ compared the standard BSNN; third, it decreases the number of off-chip memory accesses, which can lower energy consumption. Between on-chip and
off-chip data movement is typically one of the most power-hunge operations (68] By maximizing the reuse of weights in P. the movement of data between on-chip and off-chip is minimized, leading more energy-efficien design. The outstanding features demonstrated by Sº may inspire and drive algorithn-driven chip design, while simultaneously reducing the algorithm exploration costs required prior to hardware design.
I Hardware Validation
We compare NN and BSNN on an FPGA to quantify S2'''s advantages. Experimental Settings:
Platform: Xilinx Vivado 2021.2
Simulation Platform: Modelsim
Clock Frequency: 100MHz
AXI Bit Width: 32 bits
Model: SCNN:
T=8, 1F5
We measure SºNN and BSNN's per-layer data access and latency between on-chip and off-chip memory. As shown in Tab. II despite additional indices, NN achieves lower data access due to sub-bit compression (columns 2-3). and also reduces on/off-chip transmission latency (columns 4-5). Compared to FPSNN, NN' advantages are even more obvious. We also present the Modelsim simulation results for the 5th layer in Fig 6 showing data access and access latency for NN and BSNN. Notably, the benefits of sub-bit weight transmission are significant relative to the codebook overhead. As per (69, off-chip ARAM access requires 128 more energy than on-chip SRAM access and 6400x more than integer addition. Compared BSNN, SNNN reduces DRAM access by 3.6x (Columns 2-3), thereby leading to highly significant energy savings benefits.
26

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2--
Waight Movement sunditi 1930
Weight Movement 2
Figure 6: Modelsim simulation results for the 5th layer, including data access and access latency for
NCNN (top) and BSNN (bottom).
Table 11: Hardware validation of N2NN and BSNN.
Data Access
Access Lateney (us)
SºNN
BSNN
SºNN
BSNN
Layerl
5138
18432
61.32
194.14 14
Layer2
10258
36864
122.44
388.38
Layer3
20498
73728
244.68
776.86
Layer4
40978
147456
489.17
1553.82
Layer5
81938
294912
978.12
3107.74
Layer6
81938
294912
978.12
3107.74
J Energy Consumption Calculation
We use the standard SNN energy calculation 70 Etotal EMAC FLOPs cont EAC (E NO ΣM SOPs where both SOPs fr. FLOPs and FAC bit-width dependent. Most SNN research uses 45nm technology for energy calculations. After investigation, binary weights reduce SOPs 1/64 of fp32 values[71], but the literature lacks EAC for binary operations at 45nm. For fair comparison with existing work, we use fp32-based EAC at 45nm (0.9pJ). As result, our actual energy is lower than reported.
K
Experiment Details
K.1 Image Classification
Dataset CIFAR-10 [72) is a widely used computer vision dataset that contains 10 categories, with
6,000 x2x32 pixel color images per category, totaling 60,000 images. CIFAR-100 [72 maintains identical image dimensions and total count, containing 100 fine-grained categories grouped into 20 superclasses. ImageNet-IK (73 a large-scale visual database comprising over 1.2 million training images and 50,000 validation images across 1,000 object categories. Its extensive category coverage and image diversity have established ImageNet- IK as pivotal benchmark dataset in deep learning and computer vision research. DVS-CIFAR 10 77 consists of 10,000 event streams generated by converting the original CIFAR-IO images using an event-based sensor with a resolution of 128x 128 pixels. The dataset preserves the original 10 categorization structure. These datasets hold substantial importance within machine learning and neuromorphic computing, serving as standard benchmarks for evaluating diverse methodologies.
Setup We conduct three experiments across all datasets. For convolutional layers with kernel size greater than 1, we set the cardinality the compact codebook IP to 16. 32, and 64, corresponding to parameter values of 4, 5, and 6, respectively. Noteworthy, our S2NN method can also be extended to convolutions with a kernel size 1. Since x convolutions already have relatively low computational complexity and parameter count, we do not apply further compression to these kernels. In our experiments, we employ the Spike-driven Transformer v3 model with 19M parameters, which uses time step of 1 during training and 4 during inference. For the CIFAR- 10 and CIFAR- 100
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datasets, we adopt MS-ResNer with a time step of 6. following prior work [7 49 In contrast, Q-SNN and BitSNN use SpikingResNet, which typically uses fewer time steps, such employ MS-ResNet due to its advanced membrane potential-based residual connections. Additionally, we supplement experiments with SpikingResNet, achieving 95.56% accuracy on CIFAR-10 and 78.51% % on CIFAR-100 100 with bit-width of 0.44 and time step 2. In our experiments, we employ the full-precision counterpart as the teacher model. The detailed hyperparameter settings are provided in
4. We Table 12
Table 12: Hyper-parametee for image classification.
ImageNet-IK IK
CIFAR-10
CIFAR-100
DVS-CIFARI0
Timestep
1x4
Epochs
200
250
250
300
Resolution
224x224
32x32
32x32
48x48
Batch size
1024
128
128
32
Optimizer
Adam
Adam
Adam
Adam
Weight decay
0
0
0
Initial learning rate
6e-4
5e-4
5e-4
5e-4
Learning rate decay
Cosine
Cosine
Cosine
Cosine
Warmup epochs
10
None
None
None
Label smoothing
0.1
None
None
None
K.2
Object Detection
Dataset COCO 2017 (76 a large-scale computer vision dataset designed for multiple tasks, including object detection, segmentation, and image captioning. The dataset consists of 118,287 training images, 5,000 validation images, and 40,670 test images. It provides multiple types of annotations, including object detection annotations (covering 80 common object categories), instance segmentation masks (detailing the contours of each object), and natural language annotations with five descriptive sentences per image. COCO emphasizes contextual relationships between objects in everyday scenes, making it a crucial benchmark for evaluating computer vision algorithms in practical applications.
21 we first convert the mmdettcctio codebase to the spike version and then use it for our experiments. We employ our highly compressed S the backbone and use Mask R-CNN as the detector to obtain the final model. The backbone is initialized with the weights of the N' (1 = 6) pre-trained on ImageNet- while the additional layers are initialized using Xavier [77] initialization. We fine-tune the model for 30 epochs on the COCO dataset. During fine-tuning, we resize and crop both the training and test images to 1333 x800 Additionally, we apply random horizontal flipping and resize the training images with ratio of 0.5. The batch size is set to 16. We use the Adam optimizer with an initial learning rate of 1e-4, and the learning rate decays according
Setup In the COCO experiment, similar to previous work 553
polynomial schedule with exponent of 0.9. The results of our method on the COCO dataset are shown in Figure ((a)
K.3 Semantic Segmentation
Dataset ADE20K is a comprehensive semantic segmentation dataset widely used in computer vision research. It contains over 20,000 images spanning diverse range of indoor and outdoor scenes, with pixel-level annotations for 150 object and stuff categories, such as person, tree, and sky. ADE20K covers variety of challenging environments, including urban areas, streets, buildings, and natural landscapes, making it ideal for training models to recognize and segment complex scenes. The dataset is particularly valuable for evaluating semantic segmentation algorithms, it provides detailed ground truth annotations, including both object categories and background elements.
Setup Following our object detection experiments, we convert the mmsegmeentation 78 codebase to its spike version and use it for our experiments. Similar to the object detection task experiments,
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employ our highly compressed Nº as the backbone for feature extraction. integrated with Semantic FPN (79 for segmentation. In this task, we conduct three experiments using NN pre-trained on ImageNet-1K with y/ values of 4, 5, and 6 for the backbone initialization. The newly added layers are initialized using Xavier initialization [77] During training, we use the Adam W optimizer with an initial learning rate of 1 x 10 that follows polynomial decay schedule with an exponent of 0.9. We train for 160K iterations with batch size of 16, incorporating a linear warm-up period during the first 1500 iterations. The results of our method on the ADE20K dataset are shown in Figure 4(b)
L Novelty of SANN
N2 is an incremental innovation based on existing research, with contributions in three key aspects.
SNN domain. First, we pioneer introducing the sub-bit concept and realize below 1-bit SNN models. This provides a potential solution for the deployment of SNN at edge devices and the future development of neuromorphic hardware. Second, we provide a MPFD, which offers more accurate gradient guidance than existing feature distillation, improving highly compressed SNN performance.
Model compression domain. We first identify the 'Outlier-induced Codeword Selection Bias' and propose OS-Quant to address the quantization error caused by outliers, improving accuracy and convergence.
Comprehensive evaluation. We extensively evaluate NN on diverse architectures, vi- sion, and NLP tasks, establishing a new comparative benchmark lacking in previous SNN compression research.
29


Reviews:
[{"review_id": "coFA6IE8Zx", "reviewer": "Reviewer_d7VG", "summary": "This work aims at addressing the storage and computational demands for large-scale spiking neural networks (SNNs). Its key idea is to employ sub-bit SNNs that represent weights with less than one bit (S2NNs). To do this, the work employs three main steps. First, it establishes an S2NN baseline that encodes weights using less than 1 bit. Second, it proposes an outlier-aware sub-bit weight quantization (OS-Quant) for improving binary kernel selection. Then third, it employs membrane potential-based feature distillation (MPFD) that aims to preserve the model’s performance.", "questions": "1)\tProvide some evaluations for real-world deployment/scenarios, and if possible for bigger datasets as well.\n\n2)\tIt mentions that “In the case of a 3×3 kernel, an analysis of the distribution of all possible 3×3 kernel values (23×3, representing the full codebook) reveals that only a small subset of binary kernels (i.e., codewords) is frequently activated”. This should be supported with some empirical data and/or illustrations.\n\n3)\tIt also mentions that “… as networks scale up to meet practical application demands, the computational burden remains a challenge even with binary versions”. This should be supported with some empirical data to show that the existing binary versions are not enough.\n\n4)\tHow to perform the sampling process to establish the compact codebook from the full codebook? It is important to clearly describe one of the main steps of the proposed techniques.\n\n5)\tThe codeword selection process is somewhat unclear. Specifically, how the kernel of S2NN is obtained from its FP-SNN domain as shown in Figures 1-3. \n\n6)\tWhat variable “d” represents in Figures 1-3 is also not clear. It seems to be a critical part in understanding the proposed technique, especially for codeword selection. \n\n7)\tHow to decide the suitable number of bits-per-kernel (η)? \n\n8)\tThe sub-bit implementation is somewhat not clear. It is suggested to refine Figure 1(b) to illustrate the sub-bit implementation step-by-step while relating to examples in the description text. The discussion on the role of “d” may be critical here to help explaining the key steps. For this, information from appendix A.11 can be used in the main paper.  \n\n9)\tIt is also suggested to refine Figure 2 to show the steps in showing how actual outlier values from Figure 2(b) lead to outlier-induced codeword selection bias in Figure 2(a). Currently, Figure 2(a) and Figure 2(b) do not look connected to each other. \n\n10)\tWhat is the rationale of using the squared L2 criteria for identifying the outliers?\n\n11)\tThis paper does not completely provide power and energy consumption results for all evaluated tasks (i.e., image classification, object detection, and semantic segmentation). Hence, it is recommended to complete these studies as well.\n\n12)\tThere are several typos which need correction, such as “Nnormalized” in Figure 3(b) and “conclusin” in Ablation Study section.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\n•\tIts idea in applying sub-bit concept in SNNs is interesting and holds promising efficiency benefits.  \n\n•\tThe challenges of performance degradation in realizing sub-bit SNNs are addressed systematically. \n\n•\tIt performs evaluation considering different datasets: image classification (i.e., CIFAR-10, CIFAR-100, ImageNet-1K, and DVSCIFAR-10), object detection (COCO), and semantic segmentation (ADE20K); and the results show promising improvements on model sizes, number of operations, and accuracy.\n\n\nWeaknesses\n\n•\tThe sampling process to establish the compact codebook from the full codebook is not sufficiently discussed.   \n\n•\tThe codeword selection process is somewhat still vague.\n\n•\tThe study does not completely provide power and energy consumption results for all evaluated tasks (i.e., image classification, object detection, and semantic segmentation).\n\n•\tEvaluation for bigger datasets and real-world deployment/scenarios is missing\n\n•\tThere are several issues with the technical descriptions and writing quality as highlighted in the question section below.", "was_revised": true, "final_justification": "Authors have addressed most of my comments, thought real-world evaluations can be improved. Anyway I think paper can be accepted, and authors have done a very good job in further improving this paper, and have sufficient feedback for future work as well.", "rebuttal": "Dear d7VG, thank you for your detailed feedback on our work. Below, we will address each of the weaknesses and questions you raised one by one. We hope this response can resolve your concerns.\n# Q1 & W4: Larger datasets and real-world evaluation.\n\n**The utilized datasets in our paper, ImageNet-1K, COCO, and ADE20K, are recognized as large-scale and challenging datasets in the fields of image classification, object detection, and semantic segmentation.** These datasets widely serve as standard benchmarks for evaluating various methods.\n\n**As per your suggestion, we have now included validations for real-world tasks, i.e., event-based object tracking tasks.** Experiments are conducted on three standard benchmarks, i.e., FE108, FELT, and VisEvent. We use a well-trained 0.56-bit S²NN on ImageNet-1k as the backbone for feature extraction, and other settings follow the advanced spike-based tracker, i.e., SDTrack[1]. As shown in the table below, with only 0.56-bit, our S²NN tracker surpasses most RGB-based trackers and is comparable to the state-of-the-art spike-based tracker SDTrack (all of which are full-precision). These results demonstrate the practical applicability of our approach in dynamic real-world environments.\n\n|Methods|Bit|FE108|FE108|VisEvent|VisEvent|\n|-|-|-|-|-|-|\n|||AUC(%)|PR(%)|AUC(%)|PR(%)|\n|STARK (ICCV 2021)|32|57.4|89.2|34.1|46.8|\n|SimTrack (ECCV 2022)|32|56.7|88.3|34.6|47.6|\n|OSTrack256 (ECCV 2022)|32|54.6|87.1|32.7|46.4|\n|ARTrack256 (CVPR 2023)|32|56.6|88.5|33.0|43.8|\n|SeqTrack-B (CVPR 2023)|32|53.5|85.5|28.6|43.3|\n|HIT-B (ICCV 2023)|32|55.9|88.5|34.6|47.6|\n|HIPTrack (CVPR 2024)|32|50.8|81.0|32.1|45.2|\n|ODTrack (AAAI 2024)|32|43.2|69.7|24.7|34.7|\n|STNet (CVPR 2022)|32|-| -|35.0|50.3|\n|SNNTrack (TIP 2025)|32|-| -| 35.4|50.4|\n|SDTrack$_{Tiny}$ (2025)|32| 55.3|88.1| 35.4| 49.5|\n|**S²NN**|0.56|54.3|85.5|35.4|49.5|\n>[1] SDTrack: A Baseline for Event-based Tracking via Spiking Neural Networks. 2025\n\n# Q2: Empirical data support for the clustering distribution of the binary kernel.\n\nWe analyze **the total percentage of the top-k most frequent binary kernels** in binary ResNet and binary VGG, with results shown below. Clearly, by only using the most frequent 128 (from the total 512) binary kernels, the total percentage can surpass 74%. Besides, these clustering distributions become more noticeable in deeper layers.\n\n|Dataset|Top2|Top4|Top8|Top16|Top32|Top64|Top128|\n|-|-|-|-|-|-|-|-|\n|CIFAR100,Res19| 20.7% | 24.4% | 29.6% | 37.2% | 47.5% | 60.8% | 74.7%  |\n|layer1.2| 20.7% | 24.4% | 29.6% | 37.2% | 47.5% | 60.8% | 74.7% |\n|layer2.2| 21.8% | 24.8% | 30.0% | 38.6% | 50.2% | 65.0% | 80.0% |\n|layer3.1| 46.8% | 50.2% | 54.5% | 60.1% | 67.9% | 75.7% | 83.9% |\n|DVSCIFAR10,VGGSNN ||||||||\n| conv3| 15.1% | 18.1% | 23.6% | 31.8% | 43.2% | 59.1% | 74.7%|\n| conv5| 14.9% | 18.6% | 23.8% | 31.6% | 43.5% | 59.9% | 76.1%|\n| conv7| 14.2% | 18.0% | 24.6% | 34.1% | 47.7% | 65.8% | 82.3%|\n\n# Q3: Data support for 1-bit computational burden.\n\nThanks for your valuable comments. Consider the spike-driven Transformer v3 (SDT-V3) [1]: even with 1-bit weight quantization, its best model requires ~21.63 MB storage and ~0.68 G operations. These resource demands may exceed the limits of low-resource hardware. For instance, the Loihi 1 neuromorphic chip has only 16 MB of neuron core memory[2,3], making it unable to deploy even the 1-bit compressed SDT-V3 model. In contrast, our method offers greater compression. With η = 4, the model size is reduced to about 9.5 MB and operations to 0.3 G, making deployment on resource-constrained hardware much more feasible.\n\n[1] Scaling spike-driven transformer with efficient spike firing approximation training. TPAMI 2025.\n\n[2] Loihi: A Neuromorphic Manycore Processor with On-Chip Learning. IEEE Micro 2018.\n\n[3] Loihi asynchronous neuromorphic research chip. Energy 2018.\n\n# Q4 & W1: Construction of compact codebook.\n\nWe build the compact codebook in two steps: initialization and learning.\n-  **Initialization.** We randomly select a compact codebook $|ℙ| = 2^\\eta$ from the full codebook to create the initial compact codebook. \n- **Learning.** Since random sampling may not be optimal, we refine ℙ through learning, following prior studies.* Specifically, $ℙ$ is stored as a tensor $\\mathbf{p} \\in \\mathbb{R}^{k \\times k \\times |ℙ|}$ formed by concatenating initially sampled codewords. During training, $\\mathbf{p}$ is updated continuously. Since $\\mathbf{p}$ is no longer limited to {±1} during optimization, so sign(·) is applied on it after optimization to preserve the binary representation of the compact codebook $ℙ$.\n\n# Q5 & W2: Detailed codeword selection. \n\nWe apologize for not clearly explaining the codeword selection and sub-bit implementation. Below, we outline the process.\n\n(1) **How is the kernel in S²NN baseline derived from the FP-SNN?**\n\nIn the S²NN baseline, each FP kernel in the FP-SNN is replaced by the closest codeword from a compact codebook $\\mathbb{P}$, following common sub-bit quantization approaches. Specifically, for a given FP kernel $w_f$, the codeword $k ∈ ℙ$ with the smallest L2 distance to $w_f$ is selected and used (namely *d* in  Fig 1-3)  for forward inference (Eq.5). A toy example is provided to illustrate this process:\n\n- FP kernel: $w_f$ = [0.8, 0.7, 5, -0.9, -0.8, -0.7, -0.9, -0.8, -0.9].\n- Two candidate codewords: $k_2$ = [1,1,-1,-1,-1,-1,-1,-1,-1], $k_3$ = [1,-1,1,1,1,1,1,1,-1].\n- Distances using L2 distance: *d*($w_f$, $k_2$) = 36.3, *d*($w_f$, $k_3$) = 35.5.\n- Codeword selection: Since *d*($w_f$, $k_2$) > *d*($w_f$, $k_3$), the S²NN baseline will use $k_3$ instead of $w_f$ for forward inference (Fig2a).\n\n(2) **Why does the baseline suffer from selection bias?**\n\n- **Ideal codeword**: Ideally, from a binarization view, $k_2$ best matches $w_f$, as it retains the sign patterns of the majority elements in $w_f$, except for an outlier value \"5\", while $k_3$ retains fewer correct signs.\n- **Selection bias**: However, the outlier \"5\" skews the distance metric, leading to incorrect selection of $k_3$. We define this inconsistency as the outlier-induced codeword selection bias.\n- **Prevalent outliers**: In Fig. 2b, we show the first 60 kernels of the second layer of ResNet19 on CIFAR100, showing prevalent outliers in FP kernels, which leads to severe sub-bit quantization error.\n\n(3) **How S²NN solve this problem?** (Fig.3)\n\nWe introduce the OS-Quant to address the above bias, detailed as:\n\n- **IQR-based outlier detection** (Eq.7-8): For $w_f$=[0.8, 0.7, 5, -0.9, -0.8, -0.7, -0.9, -0.8, -0.9], so Q₁ = -0.9 and Q₃ = 0.75, thus the normal range of $w_f$ is [-3.375, 3.225]. This indicates that \"5\" in $w_f$ is an outlier.\n- **Spatially-aware scaling** (Eq.11-12): The outlier \"5\" is scaled to 1 using neighbor differences, yielding adjusted kernel: *ŵ* = [0.8, 0.7, 1, -0.9, -0.8, -0.7, -0.9, -0.8, -0.9].\n- **Outlier-Aware Sub-Bit Weight Quantization** (Eq.13): *d*(ŵ, $k_2$) = 4.3, *d*(ŵ, $k_3$) = 19.5. Therefore, S²NN correctly selects $k_2$, preserving more spatial sign patterns (Fig. 3a).\n\n# Q6: Clarification of *d*.\n\nWe apologize for this missing. The variable *d* in Fig 1–3 is the squared L2 distance.\n\n# Q7: The suitable value of η.\n\nη is the bit-width used in quantization. **Once we determine η, it applies to the entire model, not just individual kernels.** The selection of η should depend on the specific needs of the deployment. Larger η is optimal for accuracy-sensitive scenarios, preserving model performance with moderate compression. Lower η is suitable for resource-limited devices, maximizing compression and speed while accepting manageable accuracy loss.\n\n# Q8: Step-by-step clarification of sub-bit implementation.\n\nGiven a FP kernel $w_f$ and the compact codebook $ℙ={k_1,k_2,...,k_{2^\\eta}}$, the vanilla sub-bit comprises three key steps:\n\n- **Step 1: Distance Computation**. For each codeword $k_i\\in ℙ$, compute the distance between $w_f$ and $k_i$, i.e., *d*($w_f$,$k_i$)=$||k_i − w_f||^2_2$.\n- **Step 2: Nearest Codeword Selection.** This is done by comparing the distances calculated in Step 1. The selected codeword is $k_{\\text{select}}=\\arg\\min_{k_i\\in ℙ}d(w_f, k_i)$\n- **Step 3: Inference Execution.** Replace $w_f$ with $k_{\\text{select}}$ for forward inference to complete the sub-bit quantization.\n\n# Q9: Relation between Fig 2(a) and Fig 2(b).\n\nFig 2(a) is a specific example of Fig 2(b). Fig 2(a) aims to illustrate what the \"outlier-induced codeword selection bias\" is and how this issue affects the quantization process. Fig 2(b) presents the problem of prevalent outliers within the real FP-SNN.\n\n# Q10: Rationale for squared L2 criteria for identifying outliers.\n\nWe draw on the advanced sub-bit quantization within the model compression domain and follow this calculation in our baseline. Notably, the novelty of OS-Quant lies in the IQR-based outlier identification and spatially-aware outlier scaling, rather than squared L2 criteria.\n\n# Q11 & W3: Lack of energy consumption results.\n\nWe **adopt OPs instead of energy consumption** for efficiency evaluation **due to limitations in the existing energy calculation method.** Almost all SNN studies compute energy using 45nm technology with $E\\_{MAC}$ = 4.6 pJ and $E\\_{AC}$ = 0.9 pJ for FP32. However, our S²NN uses sub-1-bit beyond the first and final layer, making the FP32 $E\\_{AC}$ value in the SNN inapplicable. After investigation, we find **no $E_{AC}$ value exists for binary operations under 45nm technology**. To ensure fair comparisons, we adopt OPs from model compression research [1,2]:\n$$\nOPs = FLOPs^{1} _ {32-bit} + \\sum _ {l=2}^{L} SOPs _ {1-bit}^l, \\text{ with } SOPs _ {1-bit} = fr \\cdot T \\cdot \\frac{1}{64} FLOPs _ {32-bit}\n$$\n\n[1] BinaryBERT: Pushing the Limit of BERT Quantization. ACL 2021.\n\n[2] PokeBNN: A Binary Pursuit of Lightweight Accuracy. CVPR 2022.\n\n# Q12 & W5: Typos correction.\nThank you for helping us improve the quality of our work. We will ensure that these corrections are made in our revised version."}, {"review_id": "sqCU36m10a", "reviewer": "Reviewer_TU7S", "summary": "This paper proposes a sub-bit spiking neural network (S²NN) aimed at further exploring the compression and acceleration potential of spiking neural networks (SNNs). S²NN utilizes a compact codebook to encode weights into representations smaller than 1 bit, significantly reducing model storage and computational overhead. The authors introduce outlier-aware sub-bit weight quantization (OS-Quant), which optimizes codeword selection by identifying and adaptively scaling outliers.", "questions": "Does knowledge distillation in the method increase additional energy consumption?", "limitations": "None.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths: The paper features a comprehensive experimental design and a clear structure. By introducing sub-bit compression into SNNs, with weight representations below 1 bit (0.44 to 0.67 bit), it significantly enhances the potential for edge deployment. The paper features thorough experimentation, covering both vision and non-vision tasks, which validates the universal applicability of the proposed method.\nWeaknesses: The paper lacks an in-depth theoretical analysis of OS-Quant and MPFD (e.g., convergence of gradient propagation, and why membrane potential-based distillation provides more precise optimization directions). In the classification task, more methods could be added for comparison.", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer TU7S, thank you for your review. Your concerns regarding our work primarily focus on four aspects: **(1) theoretical analysis of OS-Quant, (2) how MPFD provides a more accurate optimization direction, (3) more method comparison in the classification task, and (4) additional energy consumption for distillation**. We have provided detailed explanations for each of these concerns and hope to help resolve them.\n\n\n# W1: Theoretical analysis of OS-Quant about convergence of gradient propagations.\n\n\n- **Convergence of Vanilla Sub-bit Quantization.** \n  The vanilla sub-bit quantization converges using a straight-through estimator (STE) during backpropagation. In the forward pass, full-precision weights $\\mathbf{w} _ {f,c}^{\\ell}$ are mapped to the nearest binary codeword $\\mathbf{k} \\in \\mathbb{P}^{\\ell}$ via $\\arg\\min$, which is non-differentiable. During backward propagation, gradients $\\partial\\mathcal{L}/\\partial\\mathbf{w} _ {b,c}^{\\ell}$ are approximated by bypassing the $\\arg\\min$ operation, treating it as an identity function within the STE. Formally, gradients are computed as $\\partial\\mathcal{L}/\\partial\\mathbf{w} _ {f,c}^{\\ell} \\approx \\mathbb{1} _ {|\\mathbf{w} _ {b,c}^{\\ell}| \\leq 1} \\cdot \\partial\\mathcal{L}/\\partial\\mathbf{w} _ {b,c}^{\\ell}$, ignoring quantization discontinuities. This allows gradient flow but introduces bias when outliers dominate the distance computation, leading to suboptimal codeword selection and unstable convergence.\n\n- **Convergence of OS-Quant.** OS-Quant ensures improved convergence by integrating outlier-aware adjustments into both forward and backward passes. During the forward pass, outliers in $\\mathbf{w} _ {f,c}^{\\ell}$ are detected via IQR and adaptively scaled using spatial neighbor relationships, yielding adjusted weights $\\hat{\\mathbf{w}} _ {f,c}^{\\ell}$. Quantization then operates on $\\hat{\\mathbf{w}} _ {f,c}^{\\ell}$. For backpropagation, gradients are computed via the chain rule: $\\partial\\mathcal{L}/\\partial\\mathbf{w} _ {f,c}^{\\ell} = (\\partial\\mathcal{L}/\\partial\\hat{\\mathbf{w}} _ {f,c}^{\\ell}) \\cdot (\\partial\\hat{\\mathbf{w}} _ {f,c}^{\\ell}/\\partial\\mathbf{w} _ {f,c}^{\\ell})$. Compared with the vanilla sub-bit quantization, the term $\\partial\\hat{\\mathbf{w}} _ {f,c}^{\\ell}/\\partial\\mathbf{w} _ {f,c}^{\\ell}$ explicitly accounts for outlier scaling, enabling gradients to reflect spatial corrections.  \n\n- **Theoretical Advantage of OS-Quant.** OS-Quant converges better than vanilla quantization due to its outlier-robust gradient formulation. Let $\\nabla _ {\\text{vanilla}} = \\partial\\mathcal{L}/\\partial\\mathbf{w} _ {b,c}^{\\ell}$ (simplified STE) and $\\nabla _ {\\text{OS-Quant}} = (\\partial\\mathcal{L}/\\partial\\hat{\\mathbf{w}} _ {f,c}^{\\ell}) \\cdot (\\partial\\hat{\\mathbf{w}} _ {f,c}^{\\ell}/\\partial\\mathbf{w} _ {f,c}^{\\ell})$. The scaling term $\\partial\\hat{\\mathbf{w}} _ {f,c}^{\\ell}/\\partial\\mathbf{w} _ {f,c}^{\\ell}$ acts as a preconditioner: for outlier weights $(i,j) \\in \\mathbf{O} _ {f,c}^{\\ell}, \\partial\\hat{\\mathbf{w}} _ {f,c}^{\\ell}(i,j)/\\partial\\mathbf{w} _ {f,c}^{\\ell}(i,j) = 1/\\Omega _ {i,j}$, where $\\Omega _ {i,j}$ scales inversely with local weight variance. This downweights outlier gradients and amplifies spatially consistent signals. Consequently, $\\nabla _ {\\text{OS-Quant}}$ reduces gradient noise from outliers, while $\\nabla _ {\\text{vanilla}}$ propagates distorted gradients from biased codeword selections. The $\\Omega _ {i,j}$-modulated gradients in OS-Quant thus promote stable descent toward minima that preserve kernel semantics, accelerating convergence and improving performance.\n\n# W2: Theoretical analysis of MPFD. (How does MPFD provide a more accurate optimization direction?)\n\nDear Reviewer TU7S, we can understand that MPFD provides a more accurate optimization direction by analyzing the gradient of the distillation loss with respect to the membrane potential.\n\nFiring rate-based feature distillation (FRFD) methods treat the firing rate of neurons as network intermediate features and aim at aligning the firing rates between teacher and student networks. To achieve this alignment, the FRFD needs to adjust the membrane potentials in backpropagation and further control spike generation. Mathematically, the gradient of the distillation loss to the membrane potential is expressed as:\n$$\n\\frac{\\partial \\mathcal{L _ {distill}}}{\\partial\\tilde{\\mathbf{u}}^{\\ell}[t]}=\\frac{\\partial\\mathcal{L} _ {FRFD}}{\\partial\\mathbf{s}^{\\ell}[t]}\\cdot \\frac{\\partial \\mathbf{s}^{\\ell}[t]}{\\partial g(\\cdot)}\\cdot\\frac{\\partial g(\\cdot )}{\\partial\\tilde{\\mathbf{u}}^{\\ell}[t]},\n$$\nwhere $\\frac{\\partial\\mathcal{L}_{FRFD}}{\\partial\\mathbf{s}^{\\ell}[t]}$ can be directly calculated from the distillation loss function. Notably, **this gradient computation involves a surrogate gradient function $g(\\cdot)$**, which causes the gradient induced by distillation on the membrane potential to be imprecise, thereby compromising the distillation optimization process.\n\nThe proposed membrane potential-based distillation feature method (MPFD) applies distillation directly at the membrane potential level, enabling more precise optimization. Mathematically, **within MPFD, the gradient of the distillation loss to the membrane potential $\\frac{\\partial \\mathcal{L_{distill}}}{\\partial\\tilde{\\mathbf{u}}^{\\ell}[t]}$ can be derived directly from our distillation loss function, resulting in less perturbation compared to FRFD method due to the absence of $g(\\cdot)$ at this step.**\n\n# W3: More method comparison in the classification task.\n\nThank you for your valuable suggestion. Due to space constraints, our manuscript compared only some advanced BSNN works. The revised version will add more comparisons of related works. Here, we **first present the added comparisons on the static ImageNet-1K and the neuromorphic DVS-CIFAR10.** \n\n## ImageNet-1K\nOn the ImageNet-1K, we compare related work in both ANN and SNN domains. As shown below, whether compared with the model compression methods in ANN or SNN, our S²NN consistently achieves close to SOTA performance under sub-1bit compression.\n\n> ||Method|Arc.|Bit (W/A)|Size (MBit)|OPs (G)|Acc. (%)|\n> |-|-|-|-|-|-|-|\n> | *Method in the ANN domain* | Bi-Real (IJCV 2020)| Res34            | 1/1        | -           | -        | 62.2      |\n> || IR-Net (CVPR 2020) | Res34  | 1/1        | -      | -        | 62.9      |\n> || SNN (ICCV 2021)  | Res34            | 0.67/1     | -           | -        | 61.4      |\n> || SNN (ICCV 2021)  | Res34            | 0.56/1     | -           | -        | 60.2      |\n> || SNN (ICCV 2021)           | Res34            | 0.44/1     | -           | -        | 58.6      |\n> || BiBert (ICLR 2022)        | Swin-T           | 1/1        | -           | -        | 68.3      |\n> || BinaryViT (CVPR 2023W)    | ViT              | 1/1        | -           | -        | 67.7      |\n> || ProxConnect++ (NIPS 2023) | ViT-B            | 1/1        | -           | -        | 66.3      |\n> || A&B (CVPR 2024)           | ReActA           | 1/1        | -           | -        | 66.9      |\n> | *Method in the SNN domain* | SDTv3 (TPAMI 2025)        | SDTv3-19M        | 32/1       | 607.57      | 16.03    | 79.80     |\n> || QSPIKF (CVPR 2024)        | SPIKFORMER-8-512 | 1/1        | 36.80       | 2.12     | 54.54     |\n> |  | AGMM (AAAI 2025)          | ResNet-18        | 1/1        | -           | -        | 64.67     |\n> || BESTF (IJCAI 2025)        | SPIKFORMER-8-512 | 1/1        | 44.56       | 5.67     | 63.46     |\n> || **S²NN**                  | **SDTv3-19M**    | **0.67/1** | **17.32**   | **0.84** | **68.02** |\n> | | **S²NN**                  | **SDTv3-19M**    | **0.56/1** | **15.88**   | **0.78** | **67.43** |\n> |  | **S²NN**                  | **SDTv3-19M**    | **0.44/1** | **14.31**   | **0.73** | **67.00** |\n\n## DVSCIFAR-10\nOn the DVSCIFAR-10, we add comparisons with more related BSNN works. As shown below, S²NN achieves near-identical accuracy to FP-SNNs (82.0% vs. 82.3% with TET), while reducing model size 38× (7.86 vs. 296.6 MBit) and operations 90% (0.2 vs. 1.97 OPs).\n\n> | Method                                  | Arc.       | Bit (W/A)  | Size (MBit) | OPs (G)  | Acc. (%) |\n> | --------------------------------------- | ---------- | ---------- | ----------- | -------- | -------- |\n> | TET (ICLR 2022)                         | VGGSNN     | 32/1       | 296.6       | 1.97     | 82.3     |\n> | ALBSNN (Frontiers in Neuroscience 2023) | 5Conv1FC   | 1/1        | -           | -        | 69.0     |\n> | CBP (IEEE JESTCS 2023)                  | 16Conv1FC  | 1/1        | -           | -        | 74.7     |\n> | Q-SNN (ACMMM 2024)                      | VGGSNN     | 1/1        | 10.91       | 0.31     | 81.6     |\n> | AGMM (AAAI 2025)                        | VGGSNN     | 1/1        | 10.91       | 0.31     | 82.4     |\n> | **S²NN**                                | **VGGSNN** | **0.67/1** | **7.86**    | **0.20** | **82.0** |\n> | **S²NN**                                | **VGGSNN** | **0.56/1** | **6.85**    | **0.17** | **81.6** |\n> | **S²NN**                                | **VGGSNN** | **0.44/1** | **5.74**    | **0.13** | **81.3** |\n\n# Q1: Additional energy consumption for distillation.\n\nWe fully understand your concern about the additional overhead from knowledge distillation, but we would like to emphasize that it incurs zero energy consumption during deployment. **S²NN is designed to optimize SNN deployment efficiency while maintaining high performance.** By quantizing the weight to sub-bit, S²NN achieves extreme model compression. **Without distillation, S²NN will suffer from severe performance degradation and even convergence failures in challenging tasks.** Thus, distillation is essential for preserving S²NN's accuracy. Notably, distillation operates exclusively during training and remains fully decoupled from inference. **This ensures no additional inference overhead, aligning with our design goal for S²NN.**"}, {"review_id": "x1OudsoiZR", "reviewer": "Reviewer_WU5Y", "summary": "This paper proposes a sub-bit Spiking Neural Network framework that compresses weights to below 1-bit. The authors build upon clustering patterns in binary kernels and introduce an OS-Quant method to reduce the codeword selection bias caused by outliers. Additionally, a MPFD approach is proposed to guide the highly compressed model more effectively. The paper provides extensive experimental evidence across classification, detection, and segmentation tasks and includes FPGA hardware validations, demonstrating significant improvements in both performance and efficiency compared to state-of-the-art methods.", "questions": "1. Could you provide more details about how the compact codebook is initialized? Is it random or based on some clustering of pretrained weights?\n\n2. Is the codebook generation method sensitive to initialization or data distribution across datasets?\n\n3. For practical deployment, what would be a good η value that balances accuracy and efficiency? Does this vary much across different tasks? Could you provide more details about how it is determined for different layers?\n\n4. Could you elaborate on how the outlier detection threshold (γ=1.5) is chosen? Are there any sensitivity analyses performed?\n\n5. Are the outliers detected by OS-Quant common across different tasks and structures? Can they be determined in advance?", "limitations": "Yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. For the SNN domain, this paper proposes a novel sub-bit SNN compression approach that significantly reduces model size and computational cost.\n\n2. Includes hardware-oriented results, confirming the efficiency and deployment benefits of S2NN on FPGA platforms.\n\nWeaknesses:\n\n1. The paper does not provide sufficient analysis of model behavior at extremely low bit-width settings (η < 4), leaving questions about the practical lower limits of the proposed compression approach.\n\n2. The paper lacks formal theoretical analysis explaining why sub-bit quantization is particularly effective, missing an opportunity to provide deeper insights into the underlying mechanisms.\n\n3. The paper lacks detailed description of the important component compact codebookd, such as how it is initialized.", "was_revised": true, "final_justification": "My questions have been resolved. Since my initial rating was positive, I will maintain my rating and recommend acceptance.", "rebuttal": "Dear reviewer WU5Y, thank you for your review. Your concerns regarding our work primarily focus on **seven** aspects: **(1) effectiveness of S²NN at η < 4, (2) why OS-Quant is effective, (3) initialization of compact codebook, (4) is the codebook generation sensitive to initialization or data distribution across datasets, (5) the suitable value of η, (6) analysis of the outlier detection threshold γ, and (7) generalizability of outlier occurrence.** We have provided detailed explanations for each of these concerns and hope to help resolve them.\n\n# W1: Effectiveness of S²NN at extremely low bit-widths (η < 4).\n\nWhen η=4, the convolution kernel (e.g., 3*3 kernel) can only take 2⁴ different values. **This configuration reduces the codebook size from 2⁹ (full codebook) to 2⁴ (compact codebook), achieving a reduction of approximately 97%. Therefore, the configuration with η = 4 can be considered an extremely low-bit width setting**. To explore S²NN's potential at η<4 regimes, we conduct a supplementary experiment on CIFAR100 using ResNet-19 with η=3 (8 codewords, 0.33 bits/weight). As shown below, with only 8 codewords, S²NN demonstrates a remarkable 98.44% compression rate in kernel space representation. While this ultra-compact configuration achieves significant efficiency, we observe there's a noticeable performance impact (from 77.4% to 70.31%) due to the severely limited convolution kernel representation. Thus, we recommend maintaining η above 4 to ensure adequate representational capacity.\n\n| η        | Kernel Space        | Compression level of kernel space | Bit per weight | Accuracy (CIFAR100, ResNet19) |\n| -------- | ------------------- | --------------------------------- | -------------- | ----------------------------- |\n| 9 (BSNN) | 512 (Full codebook) | 0%                                | 1.00           | 78.77%                        |\n| 6        | 64                  | 87.5%                             | 0.67           | 78.77%                        |\n| 5        | 32                  | 93.75%                            | 0.56           | 78.43%                        |\n| 4        | 16                  | 96.88%                            | 0.44           | 77.40%                        |\n| 3        | 8                   | 98.44%                            | 0.33           | 70.31%                           |\n\n#  W2: Why the proposed sub-bit quantization method, i.e., OS-Quant, is effective.\n\nThe effectiveness of our sub-bit quantization stems from two aspects.\n\n- **First is the data distribution-driven compression.** By analyzing weight distributions in well-trained BSNNs, we reveal obvious clustering patterns in convolutional kernels. This intrinsic characteristic enables compression of the full codebook for 3×3 kernels (512 combinations) into compact codebooks containing only high-frequency codewords (e.g., η=4 reduces to 16 codewords). This data distribution-driven compression preserves critical structural information during the quantization process.\n\n- **Second is the observation and resolution of the outlier-induced codeword selection bias in the vanilla sub-bit compression process.** Specifically, this sub-bit quantization method is highly sensitive to outliers. When a weight value in a kernel deviates significantly from the main distribution, it will lead to suboptimal codebook selection. Furthermore, our OS-Quant detects outlier boundaries through the interquartile range statistical method, and adaptively scales based on spatial neighborhood relationships. This not only eliminates the interference of outliers on the quantization process, but also fully preserves the spatial characteristics of the convolution kernel. \n\nThrough the above two aspects, S²NN achieves compression below 1-bit while maintaining high model performance.\n\n# Q1 & W3: Construction of compact codebook.\n\nThe compact codebook is established through initialization followed by learning. **During initialization**, a compact codebook $|ℙ| = 2^\\eta$ is **randomly sampled from the full codebook** $\\mathbb{K}$. *However, since random sampling cannot guarantee an optimal compact codebook, we refine the sampled subset through learning, following prior studies.* Specifically, $ℙ$ is represented as a tensor $\\mathbf{p} \\in \\mathbb{R}^{k \\times k \\times |ℙ|}$ formed by concatenating initially sampled codewords. **During training**, $\\mathbf{p}$ is **updated continuously**. Since $\\mathbf{p}$ is no longer limited to {±1} during optimization, so sign(·) is applied on it after optimization to preserve the binary representation of the compact codebook $ℙ$.\n\n# Q2: Is the codebook generation sensitive to initialization or data distribution across datasets?\n\nWe discuss the sensitivity of the codebook generation to initialization and data distribution.\n\n- Regarding the sensitivity of codebook generation to initialization. Although the codebook $ℙ$ is initialized through random sampling, gradient optimization during training ensures its final performance is insensitive to initial states. Stable convergence across multiple tasks corroborates this robustness.\n\n- Regarding the sensitivity of codebook generation to data distribution. For visual data, CNN and Transformer architectures exhibit strong clustering patterns, enabling consistent high compression efficacy. In contrast, non-visual data (e.g., SpikeLM for NLP) exhibit weaker clustering (Appendix A.8), resulting in about a 3% decrease in accuracy on GLUE tasks (73.7% vs. 76.5%, Table 10). Thus, the codebook generation is somewhat sensitive to data distribution.\n\nIn summary, the codebook generation demonstrates initialization robustness but exhibits performance dependency on data modality (visual > non-visual).\n\n# Q3: The suitable value of η.\n\nη is a model-wise parameter rather than layer-wise, it represents the quantization bit-width. **Once we determine η, it applies to the entire model.** The selection of η should depend on the specific needs of the deployment:\n\n- **Larger η**: Optimal for accuracy-sensitive scenarios like safety-critical systems, preserving model performance with moderate compression.\n- **Lower η**: Suitable for resource-limited devices (e.g., mobile/IoT), maximizing compression and speed while accepting manageable accuracy trade-offs.\n\n# Q4: Analysis of the outlier detection threshold γ.\n\nYes, we conduct an analysis by setting the outlier detection threshold γ to different values. Specifically, we test various γ values on the CIFAR100 dataset using the ResNet19 architecture, with η set to 6. The results are as follows:\n\n- γ=0.5 (too low) treats too many values as outliers, causing excessive outlier scaling and destroying the kernel's spatial information.\n- γ=3.0 (too high) fails to detect outliers effectively, degrading S²NN to baseline performance.\n- γ=1.5 achieves optimal balance between outlier detection and spatial preservation. This is the value we chose in the manuscript.\n- γ=1.0 and γ=2.0 perform well but slightly below γ=1.5.\n\n| γ    | 0.5   | 1.0   | 1.5   | 2.0   | 3.0   |\n| ---- | ----- | ----- | ----- | ----- | ----- |\n| Acc. | 74.48 | 75.34 | 75.59 | 75.19 | 75.08 |\n\n# Q5: Generalizability of outlier occurrence. Can the outlier be determined in advance?\n\nIn our original manuscript, Fig 2(b) shows the first 60 kernels from ResNet19's second layer on CIFAR100, indicating the **prevalent issue of outlier occurrence** in a well-trained BSNN. **To further demonstrate that the emergence of outliers is a universal phenomenon, we explore whether the emergence of outliers varies by model, dataset, or data augmentation**. Specifically, we count the percentage of convolutional kernels containing outliers in each layer. Results are shown below, indicating that **while the number of outlier-containing kernels varies by model/dataset/augmentation, outliers occurrence remains a universal and severe issue**. Furthermore, as the network updates, the weights are continuously changed, resulting in varying outliers across training iterations. Consequently, the outliers cannot be determined in advance.\n\n|                                            | layer1.1  | layer2.1  | layer3.1  |\n| ------------------------------------------ | --------- | --------- | --------- |\n| CIFAR10, Res19                             | 21.15%    | 17.62%    | 22.33%    |\n| CIFAR100, Res19                            | 33.19%    | 18.99%    | 20.19%    |\n| CIFAR100, Res19, only RandomCrop           | 30.10%    | 26.24%    | 17.61%    |\n| CIFAR100, Res19, only RandomHorizontalFlip | 29.16%    | 21.61%    | 29.90%    |\n|                                            | **conv3** | **conv5** | **conv7** |\n| DVSCIFAR10, VGGSNN                         | 33.73%    | 15.99%    | 27.39%    |"}, {"review_id": "LROMQaEJta", "reviewer": "Reviewer_LxTv", "summary": "The paper introduces Sub-bit Spiking Neural Networks (S²NNs), a framework for compressing and accelerating SNNs by representing weights with less than one bit. Extensive experiments on vision and non-vision tasks demonstrate that S²NN achieves state-of-the-art performance and efficiency, making it suitable for edge computing applications.", "questions": "1.The OS-Quant method detects and scales outliers based on spatial neighbors. How does this approach perform in cases where outliers are spatially isolated or when kernel sizes vary significantly (e.g., 1x1 vs. 3x3 or others)?\n\n\n2.The paper compares S²NN to BSNNs and FP-SNNs. How does it fare against state-of-the-art quantized ANNs in terms of performance and efficiency, especially for similar bit-widths?\n\n3.The results on NLP tasks (e.g., GLUE benchmark) show a performance drop compared to vision tasks. What are the key challenges in adapting S²NN to non-vision domains?", "limitations": "Yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\n1.The paper introduces the novel concept of sub-bit weight representation in SNNs, pushing the boundaries of model compression and efficiency. The proposed OS-Quant and MPFD methods are well-motivated and address critical challenges in SNN quantization and performance preservation.\n\n2.The paper validates S²NN across diverse tasks (classification, object detection, segmentation, and NLP) and architectures, demonstrating its scalability and robustness. This paper also performs additional hardware verification.\n\nWeaknesses:\n\n1.The reliance on clustering patterns in binary kernels may not hold universally across all architectures or tasks.\n\n2.The MPFD method, while effective, introduces additional computational overhead during training, which could be a limitation for resource-constrained scenarios.\n\n3.The comparison with other sub-bit quantization methods (e.g., in non-spiking networks) is limited.", "was_revised": true, "final_justification": "Thanks for response.", "rebuttal": "Dear reviewer LxTv, thank you for your review. Your concerns regarding our work primarily focus on **five** aspects: **(1) kernel clustering universality, (2) additional energy consumption for distillation, (3) how does OS-Quant perform in cases in informal scene, (4) the comparison with methods in ANNs, and (5) key challenges in adapting S²NN to non-vision domains.** We have provided detailed explanations for each of these concerns and hope to help resolve them.\n\n# W1: Limitation of kernel clustering universality.\n\nYes, you raise a valid point. Indeed, the clustering patterns in binary kernels are more pronounced in vision architectures (i.e., image data) due to the prior of spatial correlation in convolutional filters. This phenomenon is less evident in sequential or text data, where structural assumptions differ.\nBut luckily, our sub-bit compression method also delivers satisfactory performance when applied to non-image data. To demonstrate the effectiveness of our method in non-image tasks, we conduct additional experiments in the NLP task. \n\n*Experimental details*: We extend S²NN to SpikeLM[1] using the method in Appendix A.7 (η=5). We select a BERT architecture featuring a 12-layer encoder transformer within SpikeLM and conduct experiments on the GLUE benchmark. All other training parameters are maintained according to the original paper.\n\n*Experimental results*: The results are summarized in the table below. Clearly, our method maintains the satisfactory performance of 73.7%. **These results fully validate the effectiveness of S²NN for complex non-image tasks.**\n\n|         | SST-2 | MRPC | RTE  | MNLI$_m$ | QNLI | QQP$_F1$ | CoLA | STS-B | Avg. |\n| ------- | ----- | ---- | ---- | -------- | ---- | -------- | ---- | ----- | ---- |\n| SpikeLM | 87.0  | 85.7 | 69.0 | 77.1     | 85.3 | 83.9     | 38.8 | 84.9  | 76.5 |\n| S²NN    | 85.1  | 84.4 | 61.8 | 74.3     | 83.6 | 84.0     | 31.8 | 84.4  | 73.7 |\n\n*[1] SpikeLM: Towards General Spike-Driven Language Modeling via Elastic Bi-Spiking Mechanisms. ICML2024.*\n\n# W2. Additional energy consumption for distillation in edge scenarios.\n\nMPFD introduces additional computational overhead during training, but **zero overhead during deployment.** S²NN is designed to optimize SNN deployment efficiency while maintaining high performance. By quantizing the weight to sub-bit, S²NN achieves extreme model compression. Without distillation, S²NN will suffer from severe performance degradation and even convergence failures in challenging tasks. Thus, distillation is essential for preserving S²NN's accuracy. **Notably, distillation operates exclusively during training and remains fully decoupled from inference. This ensures no additional inference overhead, aligning with our design goal for S²NN.**\n\n# Q1. How does OS-Quant perform in cases where outliers are spatially isolated or when kernel sizes vary significantly?\n\nWe appreciate this insightful question regarding outlier handling. The concern about spatially isolated outliers can be addressed through two key considerations. \n\n- First, for conventional kernels (kernel size >1), the inherent spatial connectivity of the receptive field ensures that all weights have adjacent neighbors, making spatially isolated outliers impossible by definition. \n- Second, for 1×1 kernels, we use the method in Appendix A.7 to aggregate adjacent 1×1 kernels into a large kernel, and then perform sub-bit compression, thereby creating meaningful neighbor relationships for all values, including potential outliers.\n\nIn addition, the OS-Quant quantization method is designed to be architecture-agnostic, capable of handling any kernel, regardless of the size and change of the kernel size.\n\n\n# Q2 & W3. The comparison with related methods in the ANN domain is limited, especially for similar bit-widths.\n\nWe supplement the comparison with related methods in the BNN domain on the image classification task. The experimental results are summarized in the table below. These results demonstrate that S²NN performs competitively against existing BNN methods on static datasets. Specifically, **when compared to the sub-bit neural network that also operates with weights below 1-bit, S²NN achieves notable accuracy improvements of 5%-6% on CIFAR-10 and 6%-9% on ImageNet-1K.** Notably, S²NN outperforms the sub-bit neural network, even when the latter employs 32-bit activations. Furthermore, **when compared to conventional BNNs with 1-bit weights, S²NN shows superior performance on CIFAR-10 using sub-1-bit weights while maintaining competitive accuracy with state-of-the-art methods on ImageNet-1K.** These comprehensive results, along with those presented in Table1, decisively validate the effectiveness of S²NN.\n\n### CIFAR-10\n\n| Method                       | Arc.    | Weight Bit | Activity Bit | Acc. (%)  |\n| ---------------------------- | ------- | ---------- | ------------ | --------- |\n| IR-Net (CVPR 2020)           | Res18   | 1          | 32           | 92.9      |\n| SNN (ICCV 2021)              | Res18   | 0.67       | 32           | 92.7      |\n| SNN (ICCV 2021)              | Res18   | 0.56       | 32           | 92.3      |\n| SNN (ICCV 2021)              | Res18   | 0.44       | 32           | 91.9      |\n| IR-Net (CVPR 2020)           | Res18   | 1          | 1            | 91.5      |\n| SNN (ICCV 2021)              | Res18   | 0.67       | 1            | 91.0      |\n| SNN (ICCV 2021)              | Res18   | 0.56       | 1            | 90.6      |\n| SNN (ICCV 2021)              | Res18   | 0.44       | 1            | 90.1      |\n| ProxConnect++ (NeurIPS 2023) | Res20   | 1          | 1            | 90.2      |\n| A&B (CVPR 2024)              | ReAct18 | 1          | 1            | 92.3      |\n| **S²NN**                     | Res19   | 0.67       | 1            | **96.43** |\n| **S²NN**                     | Res19   | 0.56       | 1            | **96.36** |\n| **S²NN**                     | Res19   | 0.44       | 1            | **95.99** |\n\n### ImageNet-1K\n\n| Method                       | Arc.   | Weight Bit | Activity Bit | Acc. (%)  |\n| ---------------------------- | ------ | ---------- | ------------ | --------- |\n| IR-Net (CVPR 2020)           | Res34  | 1          | 32           | 70.4      |\n| SNN (ICCV 2021)              | Res34  | 0.67       | 32           | 68.0      |\n| SNN (ICCV 2021)              | Res34  | 0.56       | 32           | 66.9      |\n| SNN (ICCV 2021)              | Res34  | 0.44       | 32           | 65.1      |\n| Bi-Real (IJCV 2020)          | Res34  | 1          | 1            | 62.2      |\n| IR-Net (CVPR 2020)           | Res34  | 1          | 1            | 62.9      |\n| SNN (ICCV 2021)              | Res34  | 0.67       | 1            | 61.4      |\n| SNN (ICCV 2021)              | Res34  | 0.56       | 1            | 60.2      |\n| SNN (ICCV 2021)              | Res34  | 0.44       | 1            | 58.6      |\n| BiBert (ICLR 2022)           | Swin-T | 1          | 1            | 68.3      |\n| Binary ViT (CVPR 2023W)      | ViT    | 1          | 1            | 67.7      |\n| ProxConnect++ (NeurIPS 2023) | ViT-B  | 1          | 1            | 66.3      |\n| A&B (CVPR 2024)              | ReActA | 1          | 1            | 66.9      |\n| **S²NN**                     | SDT3   | 0.67       | 1            | **68.02** |\n| **S²NN**                     | SDT3   | 0.56       | 1            | **67.43** |\n| **S²NN**                     | SDT3   | 0.44       | 1            | **67.00** |\n\n# Q3. What are the key challenges in adapting S²NN to non-vision domains?\n\nThe key challenge in applying S²NN to non-vision domains primarily lies in the **weaker kernel clustering patterns in these architectures compared to vision models**. Due to local receptive fields and translation invariance, vision models naturally exhibit strong spatial clustering in their weight distributions. However, non-vision models lack such an inherent prior assumption, resulting in less pronounced clustering patterns. Nevertheless, our experiments on NLP tasks (Response to W1/Appendix A.8) demonstrate that even with these weaker clustering patterns, S²NN can still achieve an accuracy of 73.7% through its outlier-aware sub-bit quantization and membrane potential-based distillation techniques. This suggests that S²NN retains some cross-domain adaptability despite this architectural limitation."}]

Author rebuttal:

Meta-review: The main idea of the paper is to use sub-bit SNNs that represent weights with less that one bit. First an encoding is presented based on clustering patterns of kernels. The authors mitigate an associated outlier induced codeword selection bias during training, by optimizing codeword selection. A membrane potential feature distillation procedure is also proposed.


Strengths identified by the reviewers include:

1. Interesting idea of using sub-bit concepts to SNNs, including the addressing of challenges associated with performance degradation. This should be useful for edge deployment.

2. Good evaluation across a diverse set of tasks (classification, object detection, segmentation, and NLP)

Weaknesses identified, which the authors should try to address, include:

1. The codebook generation process is a bit vague

2. Some writing quality issues

3. Lack of in-depth theoretical analysis of the algorithm

Overall the authors made a good effort addressing the concerns of the reviewers making some things in the exposition clearer vis-a-vis typos and evaluation, convincing some reviewers that the paper scores should be upgraded post rebuttal. 

The paper received two Accepts, one Borderline accept and one Borderline reject. Its average ranking was above the typical acceptance threshold for NeurIPS, and in conjunction with the authors' addressing many of the reviewer concerns in the rebuttal leads me to recommend its acceptance.
Decision: Accept (poster)