Title: Scattering Vision Transformer: Spectral Mixing Matters
Venue: NeurIPS 2023
Accepted: yes
Keywords: Vision Transformer

Scatter Network

Spectral Transformer

Token Mixing

Channel Mixing

and Einstein Blending Method
Abstract: Vision transformers have gained significant attention and achieved state-of-the-art performance in various computer vision tasks, including image classification, instance segmentation, and object detection. However, challenges remain in addressing attention complexity and effectively capturing fine-grained information within images. Existing solutions often resort to down-sampling operations, such as pooling, to reduce computational cost. Unfortunately, such operations are non-invertible and can result in information loss. In this paper, we present a novel approach called Scattering Vision Transformer (SVT) to tackle these challenges. SVT incorporates a spectrally scattering network that enables the capture of intricate image details. SVT overcomes the invertibility issue associated with down-sampling operations by separating low-frequency and high-frequency components. Furthermore, SVT introduces a unique spectral gating network utilizing Einstein multiplication for token and channel mixing, effectively reducing complexity. We show that SVT achieves state-of-the-art performance on the ImageNet dataset with a significant reduction in a number of parameters and FLOPS. SVT shows 2\% improvement over LiTv2 and iFormer. SVT-H-S reaches 84.2\% top-1 accuracy, while SVT-H-B reaches 85.2\% (state-of-art for base versions) and SVT-H-L reaches 85.7\% (again state-of-art for large versions). SVT also shows comparable results in other vision tasks such as instance segmentation. SVT also outperforms other transformers in transfer learning on standard datasets such as CIFAR10, CIFAR100, Oxford Flower, and Stanford Car datasets.  The project page is available on this webpage.\url{https://badripatro.github.io/svt/}.

Scattering Vision Transformer: Spectral Mixing Matters

Abstract

Vision transformers have gained significant attention and achieved state-of-theart performance in various computer vision tasks, including image classification, instance segmentation, and object detection. However, challenges remain in addressing attention complexity and effectively capturing fine-grained information within images. Existing solutions often resort to down-sampling operations, such as pooling, to reduce computational cost. Unfortunately, such operations are non-invertible and can result in information loss. In this paper, we present a novel approach called Scattering Vision Transformer (SVT) to tackle these challenges. SVT incorporates a spectrally scattering network that enables the capture of intricate image details. SVT overcomes the invertibility issue associated with down-sampling operations by separating low-frequency and high-frequency components. Furthermore, SVT introduces a unique spectral gating network utilizing Einstein multiplication for token and channel mixing, effectively reducing complexity. We show that SVT achieves state-of-the-art performance on the ImageNet dataset with a significant reduction in a number of parameters and FLOPS. SVT shows 2% improvement over LiTv2 and iFormer. SVT-H-S reaches 84.2% top-1 accuracy, while SVT-H-B reaches 85.2% (state-of-art for base versions) and SVT-H-L reaches 85.7% (again state-of-art for large versions). SVT also shows comparable results in other vision tasks such as instance segmentation. SVT also outperforms other transformers in transfer learning on standard datasets such as CIFAR10, CIFAR100, Oxford Flower, and Stanford Car datasets. The project page is available on this webpage ( https://badripatro.github.io/svt/ ).

1 Introduction

Several adaptations of transformers have been introduced in the field of computer vision for various tasks. For image classification, notable vision transformers include ViT [13], DeIT [56], PVT [61], Swin [39], Twin [9], and CSWin transformers [12]. The different vision transformers improved the performance of image classification tasks significantly compared to Convolutional Neural Networks (CNNs) such as ResNets and RegNets, as discussed in efficient vision transformer research work [43]. This breakthrough in computer vision has led to state-of-the-art results in various vision tasks, including image segmentation such as SegFormer [66], TopFormer [77] and SegViT [74] and object

In recent years, there has been a remarkable surge in the interest and adoption of Large Language Models (LLMs), driven by the release and success of prominent models such as GPT-3, ChatGPT [1], and Palm [8]. These LLMs have achieved significant breakthroughs in the field of Natural Language Processing (NLP). Building upon their successes, subsequent research endeavors have extended the language transformer paradigm to diverse domains including computer vision, speech recognition, video processing, and even climate and weather prediction. In this paper, we specifically focus on exploring the potential of LLMs for vision-related tasks. By leveraging the power of these language models, we aim to push the boundaries of vision applications and investigate their capabilities in addressing complex vision challenges.

detection, models like DETR [4] and Yolo[15]. However, one challenge faced by vision transformers is the increasing computational complexity of the self-attention module as the sequence length or image resolution grows. Additionally, model size and the number of floating-point operations per second (FLOPS) also increase with image resolution or sequence length. These factors need to be carefully considered and addressed to ensure efficient and scalable deployment of vision transformers in real-world applications.

SVT uses the scattering network as the initial layer of the transformer, which captures fine-grained information (lines and edges, for instance) along with the energy component. SVT also uses attention nets in the deeper layers to capture long-range dependencies. The fine-grained information is captured by the high-frequency component of the scattering network by using the DTCWT transform while the low-frequency component is the energy component. SVT uses a Spectral Gating Network (SGN) to capture the effective features in both frequency components. Generally, the high-frequency component has extra directional information which makes it computationally complex while performing the gating operation. SVT addresses this complexity with a novel token and channel mixing technique using the Einstein Blending Method (EBM) in high-frequency component. SVT also uses a Tensor Blending Method (TBM) in the low-frequency component. We also observe that the DTCWT is more invertible compared to other spectral transformations in the literature which are based on Fourier transforms and discrete wavelet transforms [2, 28]. We quantify the invertibility in terms of reconstruction loss in the performance study section of this paper. The use of TBM for low-frequency components and EBM for high-frequency components is our contribution. It must be noted that low-frequency components contain the energy component of the signal which requires all the frequency components to provide energy compaction, while high-frequency components can be represented by only a few components, which can be achieved using Einstein multiplication. SVT is a generic recipe for componentizing the transformer architecture and efficiently implementing transformers with lesser parameters and computational complexity with the help of Einstein multiplication. So, this can be viewed as a simple and efficient learning transformer architecture with minimal inductive bias.

One way to address the computational complexity of attention-based Transformers is to replace the attention mechanism with a Multi-Layer Perceptron (MLP) based mixer layer [54, 55, 18]. However, it is difficult to capture spatial information in the MLP mixers. This was addressed by the paper MetaFormer [71]. MetaFormer uses a pooling operation to replace the attention layer. This however has the disadvantage that the pooling operation is not invertible and could possibly lose information. Fourier based Transformers such as FourierFormer[53], FNet[31], GFNet [47] and AFNO [16] minimizes the loss of information by using Fourier Transform. But it has an inherent problem of separating the low and high-frequency components. The ability to separate low-frequency and highfrequency components of an image is important. Recently, transformers such as LiTv2 [42] and iFormer [51] have been proposed to address this problem. However, both LiTv2 and iFormer have the same 𝑂 ( 𝑛 2 ) complexity as they use full-fledged self-attention networks, similar to ViT [13] and DeIT [56], which are weak in capturing fine-grained information of images. Vit, DeIT, LiTv2 and iFormer also have a limitation with respect to network size or number of parameters. We propose a Scattering Vision Transformer (SVT) which uses a spectral scattering network to address the attention complexity and Dual-Tree Complex Wavelet Transform (DTCWT) to capture the fine-grained information using spectral decomposition into low-frequency and high-frequency components of an image.

Our contributions are as follows:

We introduce a novel invertible scattering network based on DTCWT transformation into vision transformers to decompose image features into low-frequency and high-frequency features.

Detailed performance analysis shows that SVT outperforms all transformers including LiT v2 and iFormer on ImageNet data, with a significantly lesser number of parameters. In addition, SVT also has comparable performance on other transfer learning datasets.

We proposed a novel SGN, which uses TBM to mix low-frequency representations and EBM to mix high-frequency representations. We use an efficient way of mixing high-frequency components using channel and token mixing with the help of Einstein multiplication.

We show that SVT is efficient not only performance-wise but also in terms of a number of parameters (memory size) as well as in terms of computational complexity (measured in Gigaflops). We also show that SVT is efficient for inferencing, by measuring its latency and comparing it with other state-of-art transformers.

Elephant

Elephant

Classifier Head

MLP

Attention Layer

SVT x L

L-α

Layer Norm

MLP

Attention Layer

Layer Norm

Attention Layer

Layers

InverseScatteringTransformation

↑

Stage-3

Attention Layer

Attention Layer

Attention Layer

MHSA

Attention Layer

7-

ScatterLayer

LFR

HFR

Layer Norm

Attention Layer

1

TensorBlendingMethod

Layers

TBM

EBM

ScatterLayer

X(H,W,C)

Y(H,W,C)

W

W

ScatterLayer

ScatterLayer

TBM

W-(H,W,C)

X

PositionTEmbedding

LF

HF

PositionTEmbedding

0□0□□0□0

00

EinsteinBlendingMethod

+

ScatteringTransformation

+

X(r,H,W,Ch

Y(r,H,W,Cb,Ca)

LinearProjection of Patches

↑

Layer Norm

↑

Linear Projection of Patches

↑

W(ChCafd)

C=C*Cd

EBM

ScatterLayer

Hierarchical SVT

2 Method

2.1 Background: Overview of DTCWT and Decoupling of Low & High Frequencies

Discrete Wavelet Transform (DWT) replaces the infinite oscillating sinusoidal functions with a set of locally oscillating basis functions, which are known as wavelets [50, 27]. Wavelet is a combination of low-pass scaling function 𝜙 ( 𝑡 ) and a shifted version of a band-pass wavelet function known as 𝜓 ( 𝑡 ) . It can be represented mathematically as given below:

where 𝑐 ( 𝑛 ) is the scaling coefficients and 𝑑 ( 𝑗, 𝑛 ) is the wavelet coefficients. These coefficients are computed by the inner product of the scaling function 𝜙 ( 𝑡 ) and wavelet function 𝜓 ( 𝑡 ) with input 𝑥 ( 𝑡 ) .

Let ℎ 0 ( 𝑛 ) , ℎ 1 ( 𝑛 ) denote the low-pass and high-pass filter pair in the upper band, while 𝑔 0 ( 𝑛 ) , 𝑔 1 ( 𝑛 ) denote the same for the lower band. Two real wavelets are associated with each of the two real wavelet transforms as 𝜓ℎ ( 𝑡 ) , and 𝜓𝑔 ( 𝑡 ) . The complex wavelet 𝜓ℎ ( 𝑡 ) ∶= 𝜓ℎ ( 𝑡 ) + 𝜓𝑔 ( 𝑡 ) can be approximated using Half-Sample Delay[49] condition,i.e. 𝜓ℎ ( 𝑡 ) is approximately the Hilbert transform of 𝜓𝑔 ( 𝑡 ) like

DWT suffers from the following issues oscillations, shift variance, aliasing, and lack of directionality. One of the solutions to solve the above problems is the Complex Wavelet Transform (CWT) with complex-valued scaling and wavelet function. The Dual-Tree Complex Wavelet Transform (DTCWT) addresses the issues of the CWT. The DT-CWT [28, 26, 27] comes very close to mirroring the attractive properties of the Fourier Transform, including a smooth, nonoscillating magnitude, a nearly shift-invariant magnitude with a simple near-linear phase encoding of signal shifts, substantially reduced aliasing; and better directional selectivity wavelets in higher dimensions. This makes it easier to detect edges and orientational features of images. The six orientations of the wavelet transformation are given by 15 ◦ , 45 ◦ , 75 ◦ , 105 ◦ , 135 ◦ , and 165 ◦ . The dual-tree CWT employs two real DWTs, the first DWT gives the real part of the transform while the second DWT gives the imaginary part. The two real DWTs use two different sets of filters, which are jointly designed to give an approximation of the overall complex wavelet transform and satisfy the perfect reconstruction (PR) conditions.

Similarly, we can define 𝜓𝑔 ( 𝑡 ) , 𝜙 𝑔 ( 𝑡 ) , and 𝑔 1 ( 𝑛 ) . Since the filters are real, no complex arithmetic is required to implement DTCWT. It is just two times more expansive in 1D because the total output data rate is exactly twice the input data rate. It is also easy to invert, as the two separate DWTs can be inverted. Compare DTCWT with the Fourier Transform, which is difficult to obtain low pass and high pass components of an image and it is less invertible (Loss is high when we do Fourier and inverse Fourier transform) compared to DTCWT. Also, It cannot speak about time and frequency simultaneously.

2.2 Scattering Visual Transformer (SVT) Method

Given input image 𝐈 ∈ ℝ 3×224×224 , we split the image into the patch of size ℝ 16×16 and obtain embedding of each patch token using position encoder and token embedding network. 𝐗 =  𝑇 ( 𝐈 ) +  𝑃 ( 𝐈 ) , where  𝑇 ,  𝑃 refer to token and position encoding network. The detailed distinct components of the SVT architecture are illustrated in Figure 1. Scattering Visual Transformer consists of three components such as a) Scattering Transformation, b) Spectral Gating Network, c) Spectral Channel and Token Mixing.

A. Scattering Transformation:

The input image 𝐈 is firstly patchified into a feature tensor 𝐗 ∈ ℝ 𝐶 × 𝐻 × 𝑊 whose spatial resolution is 𝐻 × 𝑊 and the number of channels is 𝐶 . To extract the features of an image, we feed 𝐗 into a series of transformer layers. We use a novel spectral transform based on an invertibility scattering network instead of the standard self-attention network. This allows us to capture both the fine-grain and the global information in the image. The fine-grain information consists of texture, patterns, and small features that are encoded by the high-frequency components of the spectral transform. The global information consists of the overall brightness, contrast, edges, and contours that are encoded by the low-frequency components of the spectral transform. Given feature 𝐗 ∈ ℝ 𝐶 × 𝐻 × 𝑊 , we use scattering transform using DTCWT [50] as discussed in section-2.1 to obtain the corresponding frequency representations 𝐗 𝐹 by 𝐗 𝐹 =  scatter ( 𝐗 ) . The transformation in frequency domain 𝐗 𝐹 provides two components, one low-frequency component i.e. scaling component 𝐗 𝜙 , and one highfrequency component i.e. wavelet component 𝐗 𝜓 . The simplified formulation for the real component of    ( ⋅ ) is:

𝑀 refers to resolution/level of decomposition and 𝑘 refers to directional selectivity. Similarly, we compute transformation for the imaginary component of    ( ⋅ ) .

B. Spectral Gating Network:

We introduce a novel method, Spectral Gating Network (SGN), to extract spectral features from both low and high-frequency components of the scattering transform. Figure-1 shows the architecture of our method. We use learnable weight parameters to blend each frequency component, but we use different blending methods for low and high frequencies. For the low-frequency component 𝐗 𝜙 ∈  𝐶 × 𝐻 × 𝑊 , we use the Tensor Blending Method (TBM), which is a new technique. TBM blends 𝑋𝜙 with 𝑊𝜙 using elementwise tensor multiplication, also known as Hadamard tensor product.

 𝜙 having same dimension as in  𝜙 .  𝜙 is the low-frequency representation of the image and it captures global information of the image. One of the biggest challenges to getting effective features in the high-frequency components 𝐗 𝜓 ∈  𝑘 × 𝐶 × 𝐻 × 𝑊 ×2 , which are complex-valued and have 'k' times more dimensions than the low-frequency components ( 𝐗 𝜙 ). Therefore, using the same Tensor Blending Method for the high-frequency components 𝑋𝜓 would increase the number of parameters by 2k times and also the computational cost (GFLOPS), where 𝑘 refers to directional selectivity, a factor of '2' indicating complex value comprising real and imaginary. To address this issue, we propose a new technique, the Einstein Blending Method (EBM), to blend the high-frequency components 𝑋𝜓 with the learnable weight parameters 𝑊𝜓 efficiently and effectively in the Spectral Gating Network that we propose in this paper. By using EBM, we can capture the fine-grain information in the image, such as texture, patterns, and small features.

To perform EBM, we first reshape a tensor 𝐀 from ℝ 𝐻 × 𝑊 × 𝐶 to ℝ 𝐻 × 𝑊 × 𝐶𝑏 × 𝐶𝑑 , where 𝐶 = 𝐶𝑏 × 𝐶𝑑 , and 𝑏 >> 𝑑 . We then define a weight matrix of size 𝑊 ∈ ℝ 𝐶𝑏 × 𝐶𝑑 × 𝐶𝑑 . We then perform Einstein multiplication between 𝐀 and 𝑊 along the last two dimensions, resulting in a blended feature tensor 𝑌 ∈ ℝ 𝐻 × 𝑊 × 𝐶𝑏 × 𝐶𝑑 as shown in the Figure-2. The formula for EBM is:

C. Spectral Channel and Token Mixing:

Weperform EBM in the channel dimension of the high-frequency component we call Spectral Channel Mixing and following that we perform EBM in the token dimension of the high-frequency component, which we call Spectral Token Mixing. To perform EBM in the channel dimension, we first reshape the high-frequency component 𝑋𝜓 from ℝ 2× 𝑘 × 𝐻 × 𝑊 × 𝐶 to ℝ 2× 𝑘 × 𝐻 × 𝑊 × 𝐶𝑏 × 𝐶𝑑 , where 𝐶 = 𝐶𝑏 × 𝐶𝑑 , and 𝑏 >> 𝑑 . We then define a weight matrix of size 𝑊𝜓𝑐 ∈ ℝ 𝐶𝑏 × 𝐶𝑑 × 𝐶𝑑 . We then perform Einstein multiplication between 𝑋𝜓 and 𝑊𝜓𝑐 along the last two dimensions, resulting in a blended feature tensor 𝐒 𝜓𝑐 ∈ ℝ 2× 𝑘 × 𝐻 × 𝑊 × 𝐶𝑏 × 𝐶𝑑 . The formula for EBM in Channel mixing is:

To perform EBM in the Token dimension, we first reshape the high-frequency component 𝑆𝜓𝑐 from ℝ 2× 𝑘 × 𝐻 × 𝑊 × 𝐶 to ℝ 2× 𝑘 × 𝐶 × 𝑊 × 𝐻 , where 𝐻𝑒𝑖𝑔ℎ𝑡 ( 𝐻 ) = 𝑊𝑖𝑑𝑡ℎ ( 𝑊 ) . We then define a weight matrix of size 𝑊𝜓𝑡 ∈ ℝ 𝑊 × 𝐻 × 𝐻 . We then perform Einstein multiplication between 𝑋𝜓 and 𝑊𝜓𝑡 along the last two dimensions, resulting in a blended feature tensor 𝐒 𝜓𝑡 ∈ ℝ 2× 𝑘 × 𝐶 × 𝑊 × 𝐻 . The formula for EBM in Token mixing is:

Where ⧆ represents an Einstein multiplication, the bias terms 𝑏 𝜓 𝑐 ∈ ℝ 𝐶𝑏 × 𝐶𝑑 , 𝑏 𝜓𝑡 ∈ ℝ 𝐻 × 𝐻 . Now the total number of weight parameters in the high-frequency gating network is ( 𝐶𝑏 × 𝐶𝑑 × 𝐶𝑑 )+( 𝑊 × 𝐻 × 𝐻 ) instead of ( 𝐶 × 𝐻 × 𝑊 × 𝑘 × 2) where 𝐶 >> 𝐻 and bias is ( 𝐶𝑏 × 𝐶𝑑 ) + ( 𝐻 × 𝑊 ) . This reduces the number of parameters and multiplication while performing high-frequency gating operations in an image. We use a standard torch package [48] to perform Einstin multiplication. Finally, we perform inverse scattering transform using low-frequency representation( 4) and high-frequency representation( 6) to bring back the spectral domain to the physical domain. Our SVT architecture consists of L layers, comprising 𝛼 scatter layers and ( 𝐿 -𝛼 ) attention layers [59], where L denotes the network's depth. The scatter layers, being invertible, adeptly capture both the global and the fine-grain information in the image effectively via low-pass and high-pass filters, while attention layers focus on extracting semantic features and addressing long-range dependencies present in the image.

3 Experiment and Performance Studies

We evaluated SVT through various mainstream computer vision tasks including image recognition, object detection, and instance segmentation. To compare the quality of SVT transformer features, we conducted the following evaluations on standard datasets: a) We trained and evaluated ImageNet1K [10] from scratch for image recognition task. b) We performed transfer learning on CIFAR10 [30], CIFAR-100 [30], Stanford Car [29], and Oxford Flower-102 [41] for Image recognition task. c) We conducted ablation studies to analyze variants of SVT transformers using scatter net with the help of various spectral mixing techniques. We also compare our results with transformers having similar decomposition architecture. d) We fine-tune SVT for downstream instance segmentation tasks. e) We also perform an in-depth analysis of the SVT, by conducting layer-wise analysis as well as invertibility analysis as well as latency analysis, and comparison.

3.1 Comparison with Similar architectures

We compare SVT with LiTv2 (Hilo) [42] which decomposes attention to find low and high-frequency components. We show that LiTv2 has a top-1 accuracy of 83.3%, while SVT has a top-1 accuracy of

85.2% with a fewer number of parameters. We also compare SVT with iFormer [51] which captures low and high-frequency information from visual data, whereas SVT uses an invertible spectral method, namely the scattering network, to get the low-frequency and high-frequency components and uses tensor and Einstein mixing respectively to capture effective spectral features from visual data. SVT top1 accuracy is 85.2, which is better than iFormer-B, which is at 84.6 with a lesser number of parameters and FLOPS. We compare SVT with WaveMLP [53] which is an MLP mixer-based technique that uses amplitude and phase information to represent the semantic content of an image. SVT uses a low-frequency component as an amplitude of the original feature, while a high-frequency component captures complex semantic changes in the input image. Our studies have shown, as depicted in Table- 1, that SVT outperforms WaveMLP by about 1.8%.

3.2 Comparison with State of the art methods

We divide the transformer architecture into three parts based on the computation requirements (FLOP counts) - small (less than 6 GFLOPS), base (6-10 GFLOPS), and large (10-30 GFLOPS). We use a similar categorization as WaveViT [70]. Notable recent works falling into the small category include C-Swin Transformers [12], LiTv2[42], MaxVIT[58], iFormer[51], CMT transformer, PVTv2[62], and WaveViT[70]. It's worth mentioning that WaveViT relies on extra annotations to achieve its best results. In this context, SVT-H-S stands out as the state-of-the-art model in the small category,

Model

Params(M)

FLOPS(G)

Top-1(%)

Top-5(%)

Params

FLOPS

Top-1

Top-5

Invertible

achieving a top-1 accuracy of 84.2%. Similarly, SVT-H-B surpasses all the transformers in the base category, boasting a top-1 accuracy of 85.2%. Lastly, SVT-H-L outperforms other large transformers with a top-1 accuracy of 85.7% when tested on the ImageNet dataset with an image size of 224x224.

When comparing different architectural approaches, such as Convolutional Neural Networks (CNNs), Transformer architectures (attention-based models), MLP Mixers, and Spectral architectures, SVT consistently outperforms its counterparts. For instance, SVT achieves better top-1 accuracy and parameter efficiency compared to CNN architectures like ResNet 152 [21], ResNeXt [67], and ResNeSt in terms of top-1 accuracy and number of parameters. Among attention-based architectures, MaxViT [58] has been recognized as the best performer, surpassing models like DeiT [56], CrossViT [6], DeepViT [78], T2T [72] etc. with a top-1 accuracy of 85.0. However, SVT achieves an even higher top-1 accuracy of 85.7 with less than half the number of parameters. In the realm of MLP Mixer-based architectures, DynaMixer [63] emerges as the top-performing model, surpassing MLPmixer[54], gMLP [37], CycleMLP [7], Hire-MLP[18], AS-MLP [36], WaveMLP[53], PoolFormer[71] and DynaMixer-L [63] with a top-1 accuracy of 84.3%. In comparison, SVT-H-L outperforms DynaMixer with a top-1 accuracy of 85.7% while requiring fewer parameters and computations. Hierarchical architectures, which include models like PVT [61], Swin [39] transformer, CSwin [12] transformer, Twin [9] transformer, and VOLO [73] are also considered. Among this category, VOLO achieves the highest top-1 accuracy of 85.4%. However, it's important to note that SVT outperforms VOLO with a top-1 accuracy of 85.7% for SVT-H-L. Lastly, in the spectral architecture category, models like GFNet[47], iFormer [51], LiTv2 [42], HorNet [46], Wave-ViT [70], etc. are examined. Wave-ViT was previously the state-of-the-art method with a top-1 accuracy of 85.5%. Nevertheless, SVT-H-L surpasses Wave-ViT in terms of top-1 accuracy, network size (number of parameters), and computational complexity (FLOPS), as indicated in Table 1.

3.3 What Matters: Does Initial Spectral or Initial Attention or Initial Convolution Layers?

The ablation study was conducted to show that initial scatter layers followed by attention in deeper layers are more beneficial than having later scatter and initial attention layers ( SVT-H-S-Inverse). We also compare transformer models based on an alternative to the attention and scatter layer (SVT-H-SAlternate) as shown in Table- 2. From all these combinations we observe that initial scatter layers followed by attention in deeper layers are more beneficial than others. We compare the performance of SVT when the architecture changes from all attention (PVTv2[62] ) to all spectral layers (GFNet[47])

Params

FLOPS

Top-1

Top-5

as well as a few spectral and remaining attention layers(SVT ours). We observe that combining spectral and attention boost the performance compared to all attention and all spectral layer-based transformer as shown in Table- 2. We have conducted an experiment where the initial layers of a ViT are convolutional networks and later layers are attention layers to compare the performance of SVT. The results are captured in Table- 1, where we compare SVT with transformers having initial convolutional layers such as CVT [64], CMT [17], and HorNet [46]. Initial convolutional layers in a transformer are not performing well compared to the initial scatter layer. Initial scatter layer-based transformers have better performance and less computation cost compared to initial convolutional layer-based transformers which is shown in Table- 2.

3.4 Ablation analysis

SVT uses a scattering network to decompose the signal into low-frequency and high-frequency components. We use a gating operator to get effective learnable features for spectral decomposition. The gating operator is a multiplication of the weight parameter in both high and low frequencies. We have conducted experiments that use tensor and Einstein mixing. Tensor mixing is a simple multiplication operator, while Einstein mixing uses an Einstein matrix multiplication operator [48]. Weobserve that in low-frequency components, Tensor mixing performs better as compared to Einstein mixing. As shown in Table- 6, we start with 𝑆𝑉 𝑇 𝑇𝑇𝑇𝑇 , which uses tensor mixing in both high and low-frequency components. We see that it may not perform optimally. Then we reverse it and use Einstein mixing in both low and high-frequency components - this also does not perform optimally. Then, we came up with the alternative method 𝑆𝑉 𝑇 𝑇𝑇𝐸𝐸 , which uses tensor mixing in low frequency and Einstein mixing in high frequency. The high-frequency further decomposes into token and channel mixing, whereas in low-frequency we simply tensor multiplication as it is an energy or amplitude component.

In the second ablation analysis, we compare various spectral architectures, including the Fourier Network (FN), Fourier Neural Operator (FNO), Wavelet Gating Network (WGN), and Fast Fourier Convolution (FFC). When we contrast SVT with WGN, it becomes evident that SVT exhibits superior directional selectivity and a more adept ability to manage complex transformations. Furthermore, in comparison to FN and FNO, SVT excels in decomposing frequencies into low and high-frequency components. It's worth noting that SVT surpasses other spectral architectures primarily due to its utilization of the Directional Dual-Tree Complex Wavelet Transform (DTCWT), which offers directional orientation and enhanced invertibility, as demonstrated in Table 3. For a more comprehensive analysis, please refer to the Supplementary section.

3.5 Transfer Learning and Task Learning

We train SVT on ImageNet1K data and fine-tune it on various datasets such as CIFAR10, CIFAR100, Oxford Flower, and Stanford Car for image recognition tasks. We compare SVT-H-B performance with various transformers such as Deit [56], ViT [13], and GFNet [47] as well as with CNN architectures such as ResNet50 and MLP mixer architectures such as ResMLP. This comparison is shown in Table- 4. It can be observed that SVT-H-B outperforms state-of-art on CIFAR10 with a top-1 accuracy of 99.1%, CIFAR100 with a top-1 accuracy of 91.3%, Flowers with a top-1 accuracy of 98.9% and Cars with top-1 accuracy of 93.7%. We observe that SVT has more representative features and has an inbuilt discriminative nature which helps in classifying images into various categories. We use a

Model

Type

Params

GMAC

Top-1

Latency

Model

MSE loss(

Table 8: Invertibility: This table shows the invertibility of SVT(DTCWT) compared with Fourier and DWT. We also compare different directional orientations and show the reconstruction loss (MSE) in an image.

)

PSNR (db)(

)

pre-trained SVT model for the downstream instance segmentation task and obtain good results on the MS-COCO dataset as shown in Table- 5.

3.6 Latency Analysis

It's important to highlight that Fourier Transforms, as mentioned in the GFNet[47], are not inherently capable of performing low-pass and high-pass separations. In contrast, GFNet consistently employs tensor multiplication, a method that, while effective, may be less efficient compared to Einstein multiplication. The latter approach is known for reducing the number of parameters and computational complexity. As a result, SVT does not lag behind in terms of performance or computational complexity; rather, it gains enhanced representational power. This is exemplified in Table 7, which provides a comparison of latency, FLOPS (Floating-Point Operations per Second), and the number of parameters. Table 7 specifically demonstrates the latency (measured in milliseconds) of SVT in relation to various network types, including convolution-based networks, Attention-based Transformer networks, Poolbased Transformer networks, MLP-based Transformer networks, and Spectral-based Transformer networks. The reported latency values are on a per-sample basis, measured on an A100 GPU.

3.7 Invertibility Versus Redundancy trade-off Analysis

We conducted an experiment to illustrate that invertibility not only enhances performance but also contributes to image comprehension. To do this, we passed an image through the raw DTC-WT and performed an inverse DTC-WT operation to calculate the reconstruction loss. The experiment was executed across various values of "M" corresponding to the level of decomposition and orientations in SVT. We observed that the reconstruction loss decreased as the value of "M" increased, indicating that SVT's ability to comprehend the image improved. These orientations effectively captured higherorder image properties, enhancing SVT's performance. We further compared different spectral transforms, including the Fourier Transform, Discrete Wavelet Transform (DWT), and DTC-WT. Our findings demonstrated that the reconstruction loss was lower for DTC-WT compared to other spectral transforms, as depicted in Table 8 below. In Table 8, we quantified the mean squared error (MSE) for FFT, DWT at stages 1, 2, and 3, and DTCWT at stages 1, 2, and 3. The MSE decreased as we increased the level of decomposition (M) and the degree of selectivity. DTCWT consistently exhibited lower MSE compared to DWT. Furthermore, the peak signal-to-noise ratio (PSNR) of DTCWT surpassed that of DWT and the Fourier Transform. PSNR gauges the quality of the reconstructed image, expressing the ratio between the maximum possible power of an image and the power of noise affecting its representation, measured in decibels (dB). A higher PSNR indicates superior image quality. A high-quality reconstructed image is characterized by low MSE and high PSNR values. For further details on redundancy, please consult the supplementary table. Additionally, we have visualized the filter coefficients for all six orientations in the supplementary materials.

3.8 Limitations

SVTcurrently uses six directional orientations to capture an image's fine-grained semantic information. It is possible to go for the second degree, which gives thirty-six orientations, while the third degree gives 216 orientations. The more orientations, the more semantic information could be captured, but

this leads to higher computational complexity. The decomposition parameter 'M' is currently set to 1 to get single low-pass and high-pass components. Higher values of 'M' give more components in both frequencies but lead to higher complexity.

4 Related Work

The Vision Transformer (ViT) [13] was the first transformer-based attempt to classify images into pre-defined categories and use NLP advances in vision. Following this, several transformer based approaches like DeiT[56], Tokens-to-token ViT [72], Transformer iN Transformer (TNT) [19], CrossViT [6], Class attention image Transformer(CaiT) [57] Uniformer [32], Beit. [3], SViT[45], RegionViT [5], MaxViT [58] etc. have all been proposed to improve the accuracy using multi-headed self-attention (MSA). PVT [61], SwinT [39], CSwin[12] and Twin [39] use hierarchical architecture to improve the performance of the vision transformer on various tasks. The complexity of MSA is O( 𝑛 2 ). For high-resolution images, the complexity increases quadratically with token length. PoolFormer [71] is a method that uses a pooling operation over a small patch which has to obtain a down-sampled version of the image to reduce computational complexity. The main problem with PoolFormer is that it uses a MaxPooling operation which is not invertible. Another approach to reducing the complexity is the spectral transformers such as FNet [31], GFNet [47], AFNO [16], WaveMix [24], WaveViT [70], SpectFormer [44], FourierFormer [40], etc. FNet [31] does not use inverse Fourier transforms, leading to an invertibility issue. GFNet [47] solves this by using inverse Fourier transforms with a gating network. AFNO [16] uses the adaptive nature of a Fourier neural operator similar to GFNet. SpectFormer [44] introduces a novel transformer architecture that combines both spectral and attention networks for vision tasks. GFNet, SpectFormer, and AFNO do not have proper separation of low-frequency and high-frequency components and may struggle to handle the semantic content of images. In contrast, SVT has a clear separation of frequency components and uses directional orientations to capture semantic information. FourierIntegral [40] is similar to GFNet and may have similar issues in separating frequency components.

One of the challenges in MSA is its inability to characterize different frequencies in the input image. Hilo attention (LiTv2) [42] helps to find high-frequency and low-frequency components by using a novel variant of MSA. But it does not solve the complexity issue of MSA. Another parallel effort named Inception Transformer came up [51], which uses an Inception mixer to capture high and low-frequency information in visual data. iFormer still has the same complexity as it uses attention as the low-frequency mixer. SVT in comparison, uses a spectral neural operator to capture low and high frequency components using the DTCWT. This removes the O ( 𝑛 2 ) complexity as it uses spectral mixing instead of attention. iFormer [51] uses a non-invertible max pooling and convolutional layer to capture high-frequency components, whereas, in contrast, SVT's mixer is completely invertible. SVT uses a scatter network to get a better directional orientation to capture fine-grained information such as lines and edges, compared to Hilo attention and iFormer.

WaveMLP [53] is a recent effort that dynamically aggregates tokens as a wave function with two parts, amplitude and phase to capture the original features and semantic content of the images respectively. SVT uses a scattered network to provide low-frequency and high-frequency components. The highfrequency component has six or more directional orientations to capture semantic information in images. We use Einstein multiplication in token and channel mixing of high-frequency components leading to lower computational complexity and network size. In Wave-ViT [70], the author has discussed the quadratic complexity of the self-attention network using a wavelet transform to perform lossless down-sampling using wavelet transform over keys and values. However, WaveViT still has the same complexity as it uses attention instead of spectral layers. SVT uses the scatter network which is more invertible compared to WaveViT.

5 Conclusions and Future Research Directions

We have proposed SVT, which helps in separating low-frequency and high-frequency components of an image, while simultaneously reducing computational complexity by using Einstein multiplicationbased technique for efficient channel and token mixing. SVT has been evaluated on standard benchmarks and shown to achieve state-of-the-art performance on standard benchmark datasets on both image classification tasks and instance segmentation tasks. It also achieves comparable performance on object detection tasks. We shall experiment with SVT in other domains such as speech and NLP as we believe that it offers significant value in these domains as well.

References

https://openai.com/blog/chatgpt/ , 2022.

Hezam Albaqami, G Hassan, and Amitava Datta. Comparison of wpd, dwt and dtcwt for multi-class seizure type classification. In 2021 IEEE Signal Processing in Medicine and Biology Symposium (SPMB) , pages 1-7. IEEE, 2021.

Hangbo Bao, Li Dong, Songhao Piao, and Furu Wei. Beit: Bert pre-training of image transformers. In International Conference on Learning Representations , 2021.

Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to-end object detection with transformers. In Computer Vision-ECCV 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part I 16 , pages 213-229. Springer, 2020.

Chun-Fu Chen, Rameswar Panda, and Quanfu Fan. Regionvit: Regional-to-local attention for vision transformers. In International Conference on Learning Representations , 2022.

Chun-Fu Richard Chen, Quanfu Fan, and Rameswar Panda. Crossvit: Cross-attention multi-scale vision transformer for image classification. In Proceedings of the IEEE/CVF international conference on computer vision , pages 357-366, 2021.

Shoufa Chen, Enze Xie, GE Chongjian, Runjian Chen, Ding Liang, and Ping Luo. Cyclemlp: A mlp-like architecture for dense prediction. In International Conference on Learning Representations , 2022.

Aakanksha Chowdhery, Sharan Narang, Jacob Devlin, Maarten Bosma, Gaurav Mishra, Adam Roberts, Paul Barham, Hyung Won Chung, Charles Sutton, Sebastian Gehrmann, et al. Palm: Scaling language modeling with pathways. arXiv preprint arXiv:2204.02311 , 2022.

Xiangxiang Chu, Zhi Tian, Yuqing Wang, Bo Zhang, Haibing Ren, Xiaolin Wei, Huaxia Xia, and Chunhua Shen. Twins: Revisiting the design of spatial attention in vision transformers. Advances in Neural Information Processing Systems , 34:9355-9366, 2021.

Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition , pages 248-255. Ieee, 2009.

Mingyu Ding, Bin Xiao, Noel Codella, Ping Luo, Jingdong Wang, and Lu Yuan. Davit: Dual attention vision transformers. In Computer Vision-ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23-27, 2022, Proceedings, Part XXIV , pages 74-92. Springer, 2022.

Xiaoyi Dong, Jianmin Bao, Dongdong Chen, Weiming Zhang, Nenghai Yu, Lu Yuan, Dong Chen, and Baining Guo. Cswin transformer: A general vision transformer backbone with cross-shaped windows. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 1212412134, 2022.

Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representations , 2020.

Stéphane d'Ascoli, Hugo Touvron, Matthew L Leavitt, Ari S Morcos, Giulio Biroli, and Levent Sagun. Convit: Improving vision transformers with soft convolutional inductive biases. In International Conference on Machine Learning , pages 2286-2296. PMLR, 2021.

Yuxin Fang, Bencheng Liao, Xinggang Wang, Jiemin Fang, Jiyang Qi, Rui Wu, Jianwei Niu, and Wenyu Liu. You only look at one sequence: Rethinking transformer in vision through object detection. Advances in Neural Information Processing Systems , 34:26183-26197, 2021.

John Guibas, Morteza Mardani, Zongyi Li, Andrew Tao, Anima Anandkumar, and Bryan Catanzaro. Efficient token mixing for transformers via adaptive fourier neural operators. In International Conference on Learning Representations , 2022.

Jianyuan Guo, Kai Han, Han Wu, Yehui Tang, Xinghao Chen, Yunhe Wang, and Chang Xu. Cmt: Convolutional neural networks meet vision transformers. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 12175-12185, 2022.

Jianyuan Guo, Yehui Tang, Kai Han, Xinghao Chen, Han Wu, Chao Xu, Chang Xu, and Yunhe Wang. Hire-mlp: Vision mlp via hierarchical rearrangement. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) , pages 826-836, June 2022.

Kaiming He, Georgia Gkioxari, Piotr Dollár, and Ross Girshick. Mask r-cnn. In Proceedings of the IEEE international conference on computer vision , pages 2961-2969, 2017.

Kai Han, An Xiao, Enhua Wu, Jianyuan Guo, Chunjing Xu, and Yunhe Wang. Transformer in transformer. Advances in Neural Information Processing Systems , 34:15908-15919, 2021.

Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition , pages 770-778, 2016.

Jie Hu, Li Shen, and Gang Sun. Squeeze-and-excitation networks. In Proceedings of the IEEE conference on computer vision and pattern recognition , pages 7132-7141, 2018.

Qibin Hou, Zihang Jiang, Li Yuan, Ming-Ming Cheng, Shuicheng Yan, and Jiashi Feng. Vision permutator: A permutable mlp-like architecture for visual recognition. IEEE Transactions on Pattern Analysis & Machine Intelligence , (01):1-1, 2022.

Pranav Jeevan and Amit Sethi. Wavemix: Resource-efficient token mixing for images. arXiv preprint arXiv:2203.03689 , 2022.

Nick Kingsbury. Image processing with complex wavelets. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences , 357(1760):2543-2560, 1999.

Zi-Hang Jiang, Qibin Hou, Li Yuan, Daquan Zhou, Yujun Shi, Xiaojie Jin, Anran Wang, and Jiashi Feng. All tokens matter: Token labeling for training better vision transformers. Advances in Neural Information Processing Systems , 34:18590-18602, 2021.

Nick Kingsbury. Complex wavelets for shift invariant analysis and filtering of signals. Applied and computational harmonic analysis , 10(3):234-253, 2001.

Jonathan Krause, Michael Stark, Jia Deng, and Li Fei-Fei. 3d object representations for fine-grained categorization. In Proceedings of the IEEE international conference on computer vision workshops , pages 554-561, 2013.

Nick G Kingsbury. The dual-tree complex wavelet transform: a new technique for shift invariance and directional filters. In IEEE digital signal processing workshop , volume 86, pages 120-131. Citeseer, 1998.

Alex Krizhevsky et al. Learning multiple layers of features from tiny images. 2009.

Kunchang Li, Yali Wang, Junhao Zhang, Peng Gao, Guanglu Song, Yu Liu, Hongsheng Li, and Yu Qiao. Uniformer: Unifying convolution and self-attention for visual recognition. arXiv preprint arXiv:2201.09450 , 2022.

James Lee-Thorp, Joshua Ainslie, Ilya Eckstein, and Santiago Ontanon. Fnet: Mixing tokens with fourier transforms. arXiv preprint arXiv:2105.03824 , 2021.

Yanghao Li, Chao-Yuan Wu, Haoqi Fan, Karttikeya Mangalam, Bo Xiong, Jitendra Malik, and Christoph Feichtenhofer. Mvitv2: Improved multiscale vision transformers for classification and detection. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 4804-4814, 2022.

Yehao Li, Ting Yao, Yingwei Pan, and Tao Mei. Contextual transformer networks for visual recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence , 2022.

Yanyu Li, Geng Yuan, Yang Wen, Ju Hu, Georgios Evangelidis, Sergey Tulyakov, Yanzhi Wang, and Jian Ren. Efficientformer: Vision transformers at mobilenet speed. Advances in Neural Information Processing Systems , 35:12934-12949, 2022.

Dongze Lian, Zehao Yu, Xing Sun, and Shenghua Gao. As-mlp: An axial shifted mlp architecture for vision. In International Conference on Learning Representations , 2022.

Ze Liu, Han Hu, Yutong Lin, Zhuliang Yao, Zhenda Xie, Yixuan Wei, Jia Ning, Yue Cao, Zheng Zhang, Li Dong, et al. Swin transformer v2: Scaling up capacity and resolution. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 12009-12019, 2022.

Hanxiao Liu, Zihang Dai, David So, and Quoc V Le. Pay attention to mlps. Advances in Neural Information Processing Systems , 34:9204-9215, 2021.

Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo. Swin transformer: Hierarchical vision transformer using shifted windows. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 10012-10022, 2021.

Tan Minh Nguyen, Minh Pham, Tam Minh Nguyen, Khai Nguyen, Stanley Osher, and Nhat Ho. Fourierformer: Transformer meets generalized fourier integral theorem. In Advances in Neural Information Processing Systems , 2022.

Maria-Elena Nilsback and Andrew Zisserman. Automated flower classification over a large number of classes. In 2008 Sixth Indian Conference on Computer Vision, Graphics & Image Processing , pages 722-729. IEEE, 2008.

Zizheng Pan, Jianfei Cai, and Bohan Zhuang. Fast vision transformers with hilo attention. In Advances in Neural Information Processing Systems , 2022.

Badri N Patro and Vijay Agneeswaran. Efficiency 360: Efficient vision transformers. arXiv preprint arXiv:2302.08374 , 2023.

Badri N Patro, Vinay P Namboodiri, and Vijay Srinivas Agneeswaran. Spectformer: Frequency and attention is what you need in a vision transformer. arXiv preprint arXiv:2304.06446 , 2023.

Tianming Qiu, Ming Gui, Cheng Yan, Ziqing Zhao, and Hao Shen. Svit: Hybrid vision transformer models with scattering transform. In 2022 IEEE 32nd International Workshop on Machine Learning for Signal Processing (MLSP) , pages 01-06. IEEE, 2022.

Yongming Rao, Wenliang Zhao, Yansong Tang, Jie Zhou, Ser Nam Lim, and Jiwen Lu. Hornet: Efficient high-order spatial interactions with recursive gated convolutions. Advances in Neural Information Processing Systems , 35:10353-10366, 2022.

Yongming Rao, Wenliang Zhao, Zheng Zhu, Jiwen Lu, and Jie Zhou. Global filter networks for image classification. Advances in Neural Information Processing Systems , 34:980-993, 2021.

Alex Rogozhnikov. Einops: Clear and reliable tensor manipulations with einstein-like notation. In International Conference on Learning Representations , 2022.

Ivan W Selesnick. Hilbert transform pairs of wavelet bases. IEEE Signal Processing Letters , 8(6):170-173, 2001.

Ivan W Selesnick, Richard G Baraniuk, and Nick C Kingsbury. The dual-tree complex wavelet transform. IEEE signal processing magazine , 22(6):123-151, 2005.

Chenyang Si, Weihao Yu, Pan Zhou, Yichen Zhou, Xinchao Wang, and Shuicheng YAN. Inception transformer. In Advances in Neural Information Processing Systems , 2022.

Aravind Srinivas, Tsung-Yi Lin, Niki Parmar, Jonathon Shlens, Pieter Abbeel, and Ashish Vaswani. Bottleneck transformers for visual recognition. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages 16519-16529, 2021.

Yehui Tang, Kai Han, Jianyuan Guo, Chang Xu, Yanxi Li, Chao Xu, and Yunhe Wang. An image patch is a wave: Phase-aware vision mlp. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 10935-10944, 2022.

Ilya O Tolstikhin, Neil Houlsby, Alexander Kolesnikov, Lucas Beyer, Xiaohua Zhai, Thomas Unterthiner, Jessica Yung, Andreas Steiner, Daniel Keysers, Jakob Uszkoreit, et al. Mlp-mixer: An all-mlp architecture for vision. Advances in Neural Information Processing Systems , 34:24261-24272, 2021.

Hugo Touvron, Piotr Bojanowski, Mathilde Caron, Matthieu Cord, Alaaeldin El-Nouby, Edouard Grave, Gautier Izacard, Armand Joulin, Gabriel Synnaeve, Jakob Verbeek, et al. Resmlp: Feedforward networks for image classification with data-efficient training. IEEE Transactions on Pattern Analysis and Machine Intelligence , 2022.

Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Hervé Jégou. Training data-efficient image transformers & distillation through attention. In International Conference on Machine Learning , pages 10347-10357. PMLR, 2021.

Hugo Touvron, Matthieu Cord, Alexandre Sablayrolles, Gabriel Synnaeve, and Hervé Jégou. Going deeper with image transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 32-42, 2021.

Zhengzhong Tu, Hossein Talebi, Han Zhang, Feng Yang, Peyman Milanfar, Alan Bovik, and Yinxiao Li. Maxvit: Multi-axis vision transformer. In Computer Vision-ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23-27, 2022, Proceedings, Part XXIV , pages 459-479. Springer, 2022.

Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems , 30, 2017.

Pichao Wang, Xue Wang, Hao Luo, Jingkai Zhou, Zhipeng Zhou, Fan Wang, Hao Li, and Rong Jin. Scaled relu matters for training vision transformers. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 36, pages 2495-2503, 2022.

Wenhai Wang, Enze Xie, Xiang Li, Deng-Ping Fan, Kaitao Song, Ding Liang, Tong Lu, Ping Luo, and Ling Shao. Pyramid vision transformer: A versatile backbone for dense prediction without convolutions. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 568-578, 2021.

Wenhai Wang, Enze Xie, Xiang Li, Deng-Ping Fan, Kaitao Song, Ding Liang, Tong Lu, Ping Luo, and Ling Shao. Pvt v2: Improved baselines with pyramid vision transformer. Computational Visual Media , 8(3):415-424, 2022.

Ziyu Wang, Wenhao Jiang, Yiming M Zhu, Li Yuan, Yibing Song, and Wei Liu. Dynamixer: a vision mlp architecture with dynamic mixing. In International Conference on Machine Learning , pages 22691-22701. PMLR, 2022.

Haiping Wu, Bin Xiao, Noel Codella, Mengchen Liu, Xiyang Dai, Lu Yuan, and Lei Zhang. Cvt: Introducing convolutions to vision transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 22-31, 2021.

Zhuofan Xia, Xuran Pan, Shiji Song, Li Erran Li, and Gao Huang. Vision transformer with deformable attention. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages 4794-4803, 2022.

Enze Xie, Wenhai Wang, Zhiding Yu, Anima Anandkumar, Jose M Alvarez, and Ping Luo. Segformer: Simple and efficient design for semantic segmentation with transformers. Advances in Neural Information Processing Systems , 34:12077-12090, 2021.

Saining Xie, Ross Girshick, Piotr Dollár, Zhuowen Tu, and Kaiming He. Aggregated residual transformations for deep neural networks. In Proceedings of the IEEE conference on computer vision and pattern recognition , pages 1492-1500, 2017.

Weijian Xu, Yifan Xu, Tyler Chang, and Zhuowen Tu. Co-scale conv-attentional image transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 9981-9990, 2021.

Jianwei Yang, Chunyuan Li, Pengchuan Zhang, Xiyang Dai, Bin Xiao, Lu Yuan, and Jianfeng Gao. Focal self-attention for local-global interactions in vision transformers. arXiv preprint arXiv:2107.00641 , 2021.

Ting Yao, Yingwei Pan, Yehao Li, Chong-Wah Ngo, and Tao Mei. Wave-vit: Unifying wavelet and transformers for visual representation learning. In Computer Vision-ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23-27, 2022, Proceedings, Part XXV , pages 328-345. Springer, 2022.

Weihao Yu, Mi Luo, Pan Zhou, Chenyang Si, Yichen Zhou, Xinchao Wang, Jiashi Feng, and Shuicheng Yan. Metaformer is actually what you need for vision. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages 10819-10829, 2022.

Li Yuan, Yunpeng Chen, Tao Wang, Weihao Yu, Yujun Shi, Zi-Hang Jiang, Francis EH Tay, Jiashi Feng, and Shuicheng Yan. Tokens-to-token vit: Training vision transformers from scratch on imagenet. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 558-567, 2021.

Li Yuan, Qibin Hou, Zihang Jiang, Jiashi Feng, and Shuicheng Yan. Volo: Vision outlooker for visual recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence , 2022.

Bowen Zhang, Zhi Tian, Quan Tang, Xiangxiang Chu, Xiaolin Wei, Chunhua Shen, et al. Segvit: Semantic segmentation with plain vision transformers. Advances in Neural Information Processing Systems , 35:49714982, 2022.

Hang Zhang, Chongruo Wu, Zhongyue Zhang, Yi Zhu, Haibin Lin, Zhi Zhang, Yue Sun, Tong He, Jonas Mueller, R Manmatha, et al. Resnest: Split-attention networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 2736-2746, 2022.

Pengchuan Zhang, Xiyang Dai, Jianwei Yang, Bin Xiao, Lu Yuan, Lei Zhang, and Jianfeng Gao. Multi-scale vision longformer: A new vision transformer for high-resolution image encoding. In Proceedings of the IEEE/CVF International Conference on Computer Vision , pages 2998-3008, 2021.

Wenqiang Zhang, Zilong Huang, Guozhong Luo, Tao Chen, Xinggang Wang, Wenyu Liu, Gang Yu, and Chunhua Shen. Topformer: Token pyramid transformer for mobile semantic segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 12083-12093, 2022.

Daquan Zhou, Bingyi Kang, Xiaojie Jin, Linjie Yang, Xiaochen Lian, Zihang Jiang, Qibin Hou, and Jiashi Feng. Deepvit: Towards deeper vision transformer. arXiv preprint arXiv:2103.11886 , 2021.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nAssistant\nThe article is about the Scattering Vision Transformer, or SVT, which is a new adaptation of transformers for computer vision tasks. Its unique feature is the use of a spectrally scattering network, which captures fine-grained information about an image and addresses the issue of information loss caused by down-sampling operations. The SVT achieves state-of-the-art performance on the ImageNet dataset and outperforms other transformers in other vision tasks.\n\nSTRENGTHS:\n1. The use of DT-CWT, which is both shift-invariant and free from aliasing, is a robust and reliable approach.\n2. The combination of tensor multiplication in low-frequency components and Einstein multiplication in high-frequency components is a compelling and efficient technique.\n3. The experimental results demonstrate great promise for the proposed method.\n\nWEAKNESSES:\n1. To enhance comprehension, a brief illustration of DT-CWT through either descriptive language or illustrative figures should be provided in the method or background section.\n2. Not all layers of vision transformers are scatter layers. An explanation and study should be provided.\n3. The paper would benefit from the inclusion of mathematical theorem proofs. I found it difficult to understand the benefits of DT-CWT, particularly the decoupling of low and high frequencies. The paper's methods read more like a technical report than a compelling story of motivation, possibly due to the omission of some logistic or preliminary details.\n4. To further evaluate the performance, provide comparisions of speed and over-fitting tests on ImageNet V2. \n5. There are some typos in the paper. For example, on line 260, \"figure ??\" is not clear. Additionally, the official reference uses brackets [xx], whereas parentheses are used in the text. Please ensure consistency and accuracy in all references.\n\nQUESTIONS:\n1. Could you provide an explanation for why scatter layer is used in shallower layers but traditional attention layer is used in deeper layers?\n2. Is there any evidence, such as references, mathematical induction, or related research studies, to support the idea of using tensor multiplication for low pass and Einstein multiplication for high pass?\n\nLIMITATIONS:\nThere appear to be no potential negative social impacts.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-26"}

comments: SUMMARY:
This paper proposed to use DTCWT in the transformer model in early stage, with the motivation of saving computation cost without information loss. The proposed scattering module has a low-pass and a high-pass branch. Tensor multiplication & Einstein multiplication are applied to the LF & HF branch respectively. The proposed module replaces the first 2 stages in hierarchical ViT models. The proposed SVT model achieved superior performance on image classification on the ImageNet1K dataset, and the downstream transfer learning results.

STRENGTHS:
Replacing low level transformer modules with the proposed DTCWT based module achieves better accuracy and latency trade-off. Applying DTCWT to transformer models may not be explored yet by prior work.

WEAKNESSES:
DTCWT has been used in the previous CNN based deep learning method, e.g. Uses of Complex Wavelets in Deep Convolutional Neural Networks. 
Existing work already shows that replacing early transformer stage with convolutional layers could give better accuracy and latency trade-off.

QUESTIONS:
How do we compare SVT with combining ViT with CNN, i.e. an architecture with CNN in early stages while transformer in late stages?

LIMITATIONS:
The proposed module is a highly hand engineered component. Adding manually designed components to the deep learning models may contradict with the motivation of removing inductive bias to the model and building generic learning based architectures.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 2
DATE: 2023-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes to use Dual-time Complex Wavelet Transforms to decompose images into high and low frequency components in vision transformers. With this technique, it claims to address the problem of attention complexity without the loss of information as in Fourier or DWT based Transformers. This claim is based on the invertible property of Dual-time Complex Wavelet Transforms. This paper claims to achieve state-of-art image classification performance on the ImageNet dataset with a significant reduction in a number of parameters and FLOPS, and comparable results in instance segmentation. Qualitative and quantitative results are provided.\n\nSTRENGTHS:\n1. The most attractive of this paper is the usage of Dual-time Complex Wavelet Transforms. The reviewer appreciates that the authors noticed the information loss in previous Fourier or DWT based Transformers, and proposed to adopt Dual-time Complex Wavelet Transforms to solve such a problem. It is invertible (though also redundant) and hence would not lead to a loss.\n\n2. The paper provides a lot of technical details, and also the codes to reproduce the results.\n\n3. Extensive experiments are conducted in the paper.\n\nWEAKNESSES:\n1. Clarity of Technical Contribution\n\nThe paper's presentation raises some concerns about clarity and comprehension. The key technical contribution of the work is obscured by convoluted explanations and distracting statements.\n\nFor example, L81-L93, at the beginning of the method section, this paper detailedly talks about something like \"In Vanilla SVT, Given an image, we split it into patches of size 16*16. We use a linear  projection layer to get the embedding feature for each patch ......\" These sentences seem to depict unique contributions of the proposed method but, upon closer inspection, merely explain standard operations in the Vision Transformer (ViT). Similarly, lines 102-109 are bewildering, appearing repetitive and unclear about the authors' intended message.\n\nMoreover, the paper repeatedly emphasizes \"tensor multiplication\" and \"Einstein multiplication\" across various sections. These appear to be mere implementation details, not substantial technical contributions.\n\nThe manuscript's current style is closer a technical report rather than an academic paper. The authors are strongly encouraged to (a) explicitly state the key contribution of this work in the rebuttal and method section, and (b) condense lines 81-126 into a succinct, high-level introduction of the method, ideally within 10 lines.\n\n2. Claims need support\n\nSome of the claims in the paper, while potentially plausible, lack sufficient support/proof. For example, L52, \"the ability to separate low-frequency and high-frequency components of an image is also important\". In L100, \"SVT has improved robustness compared to most other transformers, which will also be established in the performance studies.\" These could be true, but the authors did not prove the robustness is improved.\n\n3. Minors\n\nThere are some minor typos/problems to be fixed. For example,  L130, \"Where\" to \"where\". In page 7, some table captions are bold and some are not.\n\nQUESTIONS:\nOverall the reviewer thinks using DTCWT in vision transformers is worth exploring, and this is why the reviewer still vote for borderline accept at this stage. If the authors cannot provide appropriate explanation/discussion about their contribution in the rebuttal period, the reviewer would downvote the score.\n\nBesides, the reviewer feel some details provided by the main paper are not interesting/exciting, e.g., Table 1. For future work, the reviewer would suggest the authors to explore more insightful things. For instance, DTCWT enjoys invertibility but has redundancy at the same time. How would the network deal with such a redundancy? Moreover, providing more quantitive or qualitative comparisons to Fourier or DWT based Transformers to analyse how the invertibility of DTCWT helps vision transformers understand visual contents, instead of just reporting the final numbers for the tasks like classification.\n\n____________________________________________________\n\n\nThe author response solves the main concerns. Under the assumption that the authors would revise the manuscript as they promised, the reviewer agreed to raise the score to weak accept.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes SVT, a novel vision transformer model that addresses the challenges of attention complexity and capturing fine-grained information in images. SVT utilizes a spectral scattering network and the Dual-time Complex Wavelet Transforms (DTCWT) to decompose image features into low-frequency and high-frequency components. The paper also introduces an efficient feature mixing technique using Einstein multiplication in the high-frequency components and tensor multiplication in the low-frequency components. Experimental results demonstrate that SVT achieves state-of-the-art performance on the ImageNet dataset with reduced parameters and computational complexity.\n\nSTRENGTHS:\n1.The proposed SVT model presents an innovative approach to addressing attention complexity and capturing fine-grained information in images. The use of spectral scattering and DTCWT decomposition enables efficient representation and separation of frequency components.\n2.The feature mixing technique using Einstein multiplication is a novel contribution that efficiently combines token and channel features, leading to improved performance.\n3.The experimental results on the ImageNet dataset and other vision tasks demonstrate the superiority of SVT compared to existing vision transformers, such as LiTv2 and iFormer. The significant reduction in parameters and computational complexity further enhances the practicality and scalability of SVT.\n\nWEAKNESSES:\n1. The English writing in this paper needs to be carefully reviewed as there are several grammar errors. For example, line 79, lesser should be less, line 83, \"Given\" should be in lower case \"given\". And in Figure 4's illustration, some parts are missing. There are many more errors in your paper writing.\n2. I find the approach in this paper to be interesting but overly tricky. As we have seen with LLMs, many highly technical improvements become less significant when supported by the large parameter size and data volume of such models. Moreover, in the experimental section, the performance improvement brought by SVT does not appear to be significant. From my perspective, simple and elegant techniques often yield more meaningful improvements compared to complex transformations.\n\nQUESTIONS:\nIn the experiments, why not compare your method with more popular  CNN-based methods like Yolo  and Faster-RCNN?\n\nLIMITATIONS:\nAuthor has presented some limitations of the proposed methods and future plan.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces the Scattering Vision Transformer (SVT), which utilizes a spectrally scattering network to capture fine-grained information in images and addresses the invertibility issue. SVT incorporates a novel spectral mixing technique using Einstein multiplication for efficient channel and token mixing. The approach achieves state-of-the-art performance on the ImageNet dataset, significantly reducing the number of parameters and FLOPS. It also demonstrates competitive results in other vision tasks, including transfer learning on standard datasets.\n\nSTRENGTHS:\n## Novelty\n\nThe use of scattering networks and Fourier-like frequency processing is novel and innovative. The paper addresses a significant problem related to the texture processing ability of vision transformers, showcasing desirable novelty and originality.\n\n## Quality & Clarity\n\nThe paper is well-organized, providing clear preliminaries, assumptions, definitions, and solutions. The experiments are detailed and concrete.\n\n## Significance\n\nSVT effectively separates low-frequency and high-frequency image components while reducing computational complexity through the Einstein multiplication-based mixing technique. It achieves state-of-the-art performance on image classification and instance segmentation tasks and shows comparable results in object detection tasks.\n\nWEAKNESSES:\nThe computational costs and complexity limit the number of directional orientations used in SVT. Currently, SVT employs six orientations, but increasing the number of orientations would capture more semantic information at the expense of higher computational complexity. Optimization possibilities should be explored. Additionally, SVT's performance in domains such as speech and NLP remains unexplored.\n\nQUESTIONS:\nNone\n\nLIMITATIONS:\nNone", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-09"}