自注意力和卷积 自注意力 ? 1 × 1 卷积 卷积 自注意力 \overset{1\times 1 卷积 }\leftrightarrows 卷积 自注意力?1×1卷积?卷积

项目	分解阶段1	分解阶段2
kernel大小为k×k的传统卷积	k 2 k^2 k2个单独的 1×1 卷积	然后进行移位和求和操作。
自注意力	查询、键和值的投影解释为多个 1×1 卷积	然后通过计算注意力权重和聚合值。
具有某种相似	此阶段占据更多计算

卷积(下图为stride=1的情况)
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卷积核	输入	输出
K ∈ R C o u t × C i n × k × k K ∈ R^{C_{out} ×C_{in}×k×k} K∈RCout?×Cin?×k×k	F ∈ R C i n × H × W F ∈ R^{C_{in}×H×W} F∈RCin?×H×W	G ∈ R C o u t × H × W G ∈ R^{C_{out}×H×W} G∈RCout?×H×W
像素表示(位置(i, j)的像素)	f i j ∈ R C i n f_{ij} ∈ R^{C_{in}} fij?∈RCin?	g i j ∈ R C o u t g_{ij} ∈ R^{C_{out}} gij?∈RCout?

注：这好像不是标准的卷积，它的特征不会随着卷积减小。
标准的卷积公式：
g i j = ∑ p , q K p , q f i + p ? ? k / 2 ? , j + q ? ? k / 2 ? (1) g_{ij}=\sum_{p,q}K_{p,q}f_{i+p-\left\lfloor k/2 \right\rfloor,j+q-\left\lfloor k/2 \right\rfloor} \tag{1} gij?=p,q∑?Kp,q?fi+p??k/2?,j+q??k/2??(1)
核在位置 ( p , q ) 的权重 : K p , q ∈ R C o u t × C i n , p , q ∈ { 0 , 1 , . . . , k ? 1 } 核在位置 (p, q) 的权重:K_{p,q }∈ R^{C_{out}×C_{in}} , p, q ∈ \{0, 1, . . . , k?1\} 核在位置(p,q)的权重:Kp,q?∈RCout?×Cin?,p,q∈{0,1,...,k?1}
式 (1) 重写为式（2）和（3）: 不同内核位置的特征映射的和
g i j = ∑ p , q g i j ( p , q ) (2) g_{ij}=\sum_{p,q}g_{ij}^{(p,q)} \tag{2} gij?=p,q∑?gij(p,q)?(2)
g i j ( p , q ) = K p , q f i + p ? ? k / 2 ? , j + q ? ? k / 2 ? (3) g_{ij}^{(p,q)} = K_{p,q}f_{i+p-\left\lfloor k/2 \right\rfloor,j+q-\left\lfloor k/2 \right\rfloor} \tag{3} gij(p,q)?=Kp,q?fi+p??k/2?,j+q??k/2??(3)
定义平移算子 f ~ ? S h i f t ( f , Δ x , Δ y ) \widetilde f \triangleq Shift(f,\Delta x,\Delta y) f ??Shift(f,Δx,Δy)
f ~ i , j = f i + Δ x , j + Δ y , ? i , j (4) \widetilde f_{i,j} = f_{i+\Delta x,j+\Delta y},\forall i,j \tag{4} f ?i,j?=fi+Δx,j+Δy?,?i,j(4)
如此将卷积写为
g i j ( p , q ) = K p , q f i + p ? ? k / 2 ? , j + q ? ? k / 2 ? = S h i f t ( K p , q f i , j , p ? ? k / 2 ? , q ? ? k / 2 ? ) (5) g_{ij}^{(p,q)} = K_{p,q}f_{i+p-\left\lfloor k/2 \right\rfloor,j+q-\left\lfloor k/2 \right\rfloor}= Shift(K_{p,q}f_{i,j} ,p-\left\lfloor k/2 \right\rfloor,q-\left\lfloor k/2 \right\rfloor)\tag{5} gij(p,q)?=Kp,q?fi+p??k/2?,j+q??k/2??=Shift(Kp,q?fi,j?,p??k/2?,q??k/2?)(5)
最终将标准卷积转为以下表述

标准卷积：
第一阶段	输入 feature map 对于某一位置 (p, q) ，按照核权重，进行线性投影，（标准的 1 × 1 卷积)
第二阶段	特征映射会根据内核位置进行平移，相加聚合。

S t a g e I :g ~ i j ( p , q ) = K p , q f i jS t a g e I I : g i j ( p , q ) = S h i f t ( g i j ( p , q ) , p ? ? k / 2 ? , q ? ? k / 2 ? ) g i j = ∑ p , q g i j ( p , q )Stage I : \ \ \ \widetilde g_{ij}^{(p,q)} = K_{p,q}f_{ij} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ Stage II : g_{ij}^{(p,q)} = Shift( g_{ij}^{(p,q)} ,p-\left\lfloor k/2 \right\rfloor,q-\left\lfloor k/2 \right\rfloor ) \\ g_{ij}=\sum_{p,q} g_{ij}^{(p,q)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ StageI:g ?ij(p,q)?=Kp,q?fij?StageII:gij(p,q)?=Shift(gij(p,q)?,p??k/2?,q??k/2?)gij?=p,q∑?gij(p,q)?
自注意力（N 个 head）
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	输入	输出
整体表示	F ∈ R C i n × H × W F ∈ R^{C_{in}×H×W} F∈RCin?×H×W	G ∈ R C o u t × H × W G ∈ R^{C_{out}×H×W} G∈RCout?×H×W
像素表示(位置(i, j)的像素)	f i j ∈ R C i n f_{ij} ∈ R^{C_{in}} fij?∈RCin?	g i j ∈ R C o u t g_{ij} ∈ R^{C_{out}} gij?∈RCout?

g i j = N ∥ l = 1 ( ∑ a , b ∈ N k ( i , j ) A ( W q ( l ) f i j , W k ( l ) f a b ) W v ( l ) f a b ) (9) g_{ij}= \begin{array}{c}N\\\parallel\\{l=1}\end{array}(\sum_{a,b\in \mathcal N_k(i,j)}A(W_q^{(l)}f_{ij},W_k^{(l)}f_{ab}) W_v^{(l)}f_{ab}) \tag{9} gij?=N∥l=1?(a,b∈Nk?(i,j)∑?A(Wq(l)?fij?,Wk(l)?fab?)Wv(l)?fab?)(9)

∥ \parallel ∥	concatenation of the outputs of N attentionheads
N k ( i , j ) \mathcal N _k (i, j) Nk?(i,j)	以 (i, j) 为中心，空间宽度为 k 的像素的局部区域
A ( W q ( l ) f i j , W k ( l ) f a b ) A(W^{(l)}_q f_{ij} , W^{(l)}_kf_{ab}) A(Wq(l)?fij?,Wk(l)?fab?)	关于 N k ( i , j ) N _k (i, j) Nk?(i,j)的权重

常用A的计算方法(d是 W q ( l ) f i j W^{(l)}_q f_{ij} Wq(l)?fij?的维数)：

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S t a g e I :q i , j ( l ) = W q ( l ) f i j , k i , j ( l ) = W k ( l ) f i j , v i , j ( l ) = W v ( l ) f i j ,S t a g e I I : g i j = N ∥ l = 1 ( ∑ a , b ∈ N k ( i , j ) A ( q i , j ( l ) , k i , j ( l ) ) v i , j ( l ) ) Stage I : \ \ \ \ q_{i,j}^{(l)}=W_q^{(l)}f_{ij}, k_{i,j}^{(l)}=W_k^{(l)}f_{ij}, v_{i,j}^{(l)}=W_v^{(l)}f_{ij}, \ \ \ \\ Stage II :g_{ij}= \begin{array}{c}N\\\parallel\\{l=1}\end{array}(\sum_{a,b\in \mathcal N_k(i,j)}A( q_{i,j}^{(l)} , k_{i,j}^{(l)} ) v_{i,j}^{(l)}) StageI:qi,j(l)?=Wq(l)?fij?,ki,j(l)?=Wk(l)?fij?,vi,j(l)?=Wv(l)?fij?,StageII:gij?=N∥l=1?(a,b∈Nk?(i,j)∑?A(qi,j(l)?,ki,j(l)?)vi,j(l)?)
Method 4.1. Relating Self-Attention with Convolution
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综合：
第一阶段	输入特征通过三个 1×1 卷积进行投影，分别重塑成N 个片段。这样，得到了一个包含 3×N feature map 的丰富的中间特征集。
第二阶段	而在第二阶段，投影的特征映射会根据内核位置进行移动，并最终聚合在一起。

F o u t = α F a t t + β F c o n v F_{out} = \alpha F_{att}+\beta F_{conv} Fout?=αFatt?+βFconv?
trick 【深度学习|ACmix 自注意力和卷积集成 On the Integration of Self-Attention and Convolution+】将张量向各个方向移动实际上打破了数据局部性，并且难以实现向量化实现。这可能会极大地损害本文模块在推理时的实际效率。本文采用固定核的深度卷积来替代无效张量位移,如图所示：

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如果将卷积核 (核大小 k = 3) 表示为（15），相应的输出可以表示为（16）和（17）。

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如图中的(a)到(b)部分，即用一个卷积来表示位移。

（c）部分	解释
l	为了进一步结合不同方向特征的总和，本文将所有输入特征和卷积核分别串联起来，将移位操作表示为单个组卷积
ll	卷积核为可学习权值，平移核为初始化
lll	多组卷积核来匹配卷积和自注意力路径的输出通道维数

网络的结构(或组成部分)

ResNet( (conv1): Conv2d(3, 64, kernel_size=(7, 7), stride=(2, 2), padding=(3, 3), bias=False) (bn1): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) (maxpool): MaxPool2d(kernel_size=3, stride=2, padding=1, dilation=1, ceil_mode=False) (layer1): Sequential( (0): Bottleneck( (conv1): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 16, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(144, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=16, bias=False) ) (bn2): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(64, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) (downsample): Sequential( (0): Conv2d(64, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (1): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) ) ) (1): Bottleneck( (conv1): Conv2d(256, 64, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 16, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(144, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=16, bias=False) ) (bn2): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(64, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) (2): Bottleneck( (conv1): Conv2d(256, 64, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 16, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(144, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=16, bias=False) ) (bn2): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(64, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) ) (layer2): Sequential( (0): Bottleneck( (conv1): Conv2d(256, 128, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(128, 128, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(128, 128, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(128, 128, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 32, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=2) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(288, 128, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), groups=32, bias=False) ) (bn2): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(128, 512, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) (downsample): Sequential( (0): Conv2d(256, 512, kernel_size=(1, 1), stride=(2, 2), bias=False) (1): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) ) ) (1): Bottleneck( (conv1): Conv2d(512, 128, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(128, 128, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(128, 128, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(128, 128, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 32, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(288, 128, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=32, bias=False) ) (bn2): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(128, 512, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) (2): Bottleneck( (conv1): Conv2d(512, 128, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(128, 128, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(128, 128, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(128, 128, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 32, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(288, 128, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=32, bias=False) ) (bn2): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(128, 512, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) (3): Bottleneck( (conv1): Conv2d(512, 128, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(128, 128, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(128, 128, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(128, 128, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 32, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(288, 128, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=32, bias=False) ) (bn2): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(128, 512, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) ) (layer3): Sequential( (0): Bottleneck( (conv1): Conv2d(512, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 64, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=2) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(576, 256, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), groups=64, bias=False) ) (bn2): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(256, 1024, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(1024, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) (downsample): Sequential( (0): Conv2d(512, 1024, kernel_size=(1, 1), stride=(2, 2), bias=False) (1): BatchNorm2d(1024, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) ) ) (1): Bottleneck( (conv1): Conv2d(1024, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 64, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(576, 256, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=64, bias=False) ) (bn2): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(256, 1024, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(1024, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) (2): Bottleneck( (conv1): Conv2d(1024, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 64, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(576, 256, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=64, bias=False) ) (bn2): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(256, 1024, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(1024, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) (3): Bottleneck( (conv1): Conv2d(1024, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 64, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(576, 256, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=64, bias=False) ) (bn2): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(256, 1024, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(1024, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) (4): Bottleneck( (conv1): Conv2d(1024, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 64, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(576, 256, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=64, bias=False) ) (bn2): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(256, 1024, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(1024, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) (5): Bottleneck( (conv1): Conv2d(1024, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(256, 256, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 64, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(576, 256, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=64, bias=False) ) (bn2): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(256, 1024, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(1024, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) ) (layer4): Sequential( (0): Bottleneck( (conv1): Conv2d(1024, 512, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(512, 512, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(512, 512, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(512, 512, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 128, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=2) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(1152, 512, kernel_size=(3, 3), stride=(2, 2), padding=(1, 1), groups=128, bias=False) ) (bn2): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(512, 2048, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(2048, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) (downsample): Sequential( (0): Conv2d(1024, 2048, kernel_size=(1, 1), stride=(2, 2), bias=False) (1): BatchNorm2d(2048, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) ) ) (1): Bottleneck( (conv1): Conv2d(2048, 512, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(512, 512, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(512, 512, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(512, 512, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 128, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(1152, 512, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=128, bias=False) ) (bn2): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(512, 2048, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(2048, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) (2): Bottleneck( (conv1): Conv2d(2048, 512, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(512, 512, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(512, 512, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(512, 512, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 128, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(1152, 512, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=128, bias=False) ) (bn2): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(512, 2048, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(2048, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) ) (avgpool): AdaptiveAvgPool2d(output_size=(1, 1)) (fc): Linear(in_features=2048, out_features=1000, bias=True) )

一个layer的结构

Sequential( (0): Bottleneck( (conv1): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 16, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(144, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=16, bias=False) ) (bn2): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(64, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) (downsample): Sequential( (0): Conv2d(64, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (1): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) ) ) (1): Bottleneck( (conv1): Conv2d(256, 64, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 16, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(144, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=16, bias=False) ) (bn2): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(64, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) (2): Bottleneck( (conv1): Conv2d(256, 64, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 16, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(144, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=16, bias=False) ) (bn2): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(64, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) ) )

一个Bottleneck(包含ACmix)的结构

(0): Bottleneck( (conv1): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn1): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv2): ACmix( (conv1): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv2): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv3): Conv2d(64, 64, kernel_size=(1, 1), stride=(1, 1)) (conv_p): Conv2d(2, 16, kernel_size=(1, 1), stride=(1, 1)) (pad_att): ReflectionPad2d((3, 3, 3, 3)) (unfold): Unfold(kernel_size=7, dilation=1, padding=0, stride=1) (softmax): Softmax(dim=1) (fc): Conv2d(12, 9, kernel_size=(1, 1), stride=(1, 1), bias=False) (dep_conv): Conv2d(144, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=16, bias=False) ) (bn2): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (conv3): Conv2d(64, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (bn3): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) (relu): ReLU(inplace=True) (downsample): Sequential( (0): Conv2d(64, 256, kernel_size=(1, 1), stride=(1, 1), bias=False) (1): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True) ) )

模块ACmix

class ACmix(nn.Module): def __init__(self, in_planes, out_planes, kernel_att=7, head=4, kernel_conv=3, stride=1, dilation=1): super(ACmix, self).__init__() self.in_planes = in_planes self.out_planes = out_planes self.head = head self.kernel_att = kernel_att self.kernel_conv = kernel_conv self.stride = stride self.dilation = dilation self.rate1 = torch.nn.Parameter(torch.Tensor(1)) self.rate2 = torch.nn.Parameter(torch.Tensor(1)) self.head_dim = self.out_planes // self.headself.conv1 = nn.Conv2d(in_planes, out_planes, kernel_size=1) self.conv2 = nn.Conv2d(in_planes, out_planes, kernel_size=1) self.conv3 = nn.Conv2d(in_planes, out_planes, kernel_size=1) self.conv_p = nn.Conv2d(2, self.head_dim, kernel_size=1)self.padding_att = (self.dilation * (self.kernel_att - 1) + 1) // 2 self.pad_att = torch.nn.ReflectionPad2d(self.padding_att) self.unfold = nn.Unfold(kernel_size=self.kernel_att, padding=0, stride=self.stride) self.softmax = torch.nn.Softmax(dim=1)self.fc = nn.Conv2d(3*self.head, self.kernel_conv * self.kernel_conv, kernel_size=1, bias=False) self.dep_conv = nn.Conv2d(self.kernel_conv * self.kernel_conv * self.head_dim, out_planes, kernel_size=self.kernel_conv, bias=True, groups=self.head_dim, padding=1, stride=stride)self.reset_parameters()def reset_parameters(self): init_rate_half(self.rate1) init_rate_half(self.rate2) kernel = torch.zeros(self.kernel_conv * self.kernel_conv, self.kernel_conv, self.kernel_conv) for i in range(self.kernel_conv * self.kernel_conv): kernel[i, i//self.kernel_conv, i%self.kernel_conv] = 1. kernel = kernel.squeeze(0).repeat(self.out_planes, 1, 1, 1) self.dep_conv.weight = nn.Parameter(data=https://www.it610.com/article/kernel, requires_grad=True) self.dep_conv.bias = init_rate_0(self.dep_conv.bias)def forward(self, x): q, k, v = self.conv1(x), self.conv2(x), self.conv3(x) scaling = float(self.head_dim) ** -0.5 b, c, h, w = q.shape h_out, w_out = h//self.stride, w//self.stride# ### att # ## positional encoding pe = self.conv_p(position(h, w, x.is_cuda))q_att = q.view(b*self.head, self.head_dim, h, w) * scaling k_att = k.view(b*self.head, self.head_dim, h, w) v_att = v.view(b*self.head, self.head_dim, h, w)if self.stride> 1: q_att = stride(q_att, self.stride) q_pe = stride(pe, self.stride) else: q_pe = peunfold_k = self.unfold(self.pad_att(k_att)).view(b*self.head, self.head_dim, self.kernel_att*self.kernel_att, h_out, w_out) # b*head, head_dim, k_att^2, h_out, w_out unfold_rpe = self.unfold(self.pad_att(pe)).view(1, self.head_dim, self.kernel_att*self.kernel_att, h_out, w_out) # 1, head_dim, k_att^2, h_out, w_outatt = (q_att.unsqueeze(2)*(unfold_k + q_pe.unsqueeze(2) - unfold_rpe)).sum(1) # (b*head, head_dim, 1, h_out, w_out) * (b*head, head_dim, k_att^2, h_out, w_out) -> (b*head, k_att^2, h_out, w_out) att = self.softmax(att)out_att = self.unfold(self.pad_att(v_att)).view(b*self.head, self.head_dim, self.kernel_att*self.kernel_att, h_out, w_out) out_att = (att.unsqueeze(1) * out_att).sum(2).view(b, self.out_planes, h_out, w_out)## conv f_all = self.fc(torch.cat([q.view(b, self.head, self.head_dim, h*w), k.view(b, self.head, self.head_dim, h*w), v.view(b, self.head, self.head_dim, h*w)], 1)) f_conv = f_all.permute(0, 2, 1, 3).reshape(x.shape[0], -1, x.shape[-2], x.shape[-1])out_conv = self.dep_conv(f_conv)return self.rate1 * out_att + self.rate2 * out_conv

# 其中位置编码用到的函数 def position(H, W, is_cuda=True): if is_cuda: loc_w = torch.linspace(-1.0, 1.0, W).cuda().unsqueeze(0).repeat(H, 1) loc_h = torch.linspace(-1.0, 1.0, H).cuda().unsqueeze(1).repeat(1, W) else: loc_w = torch.linspace(-1.0, 1.0, W).unsqueeze(0).repeat(H, 1) loc_h = torch.linspace(-1.0, 1.0, H).unsqueeze(1).repeat(1, W) loc = torch.cat([loc_w.unsqueeze(0), loc_h.unsqueeze(0)], 0).unsqueeze(0) return loc

Layer 1 x [2, 64, 56, 56] q [2, 64, 56, 56] k [2, 64, 56, 56] v [2, 64, 56, 56] scaling = float(self.head_dim) ** -0.5 0.25 pe = self.conv_p(position(h, w, x.is_cuda)) 通过对线性数据的卷积得到位置编码pe [1, 16, 56, 56] attention中的放缩因子 q_att [8, 16, 56, 56] view k_att view view(b*self.head, self.head_dim, h, w) [8, 16, 56, 56] view v_att view [8, 16, 56, 56] [8, 16, 62, 62] pad [8, 784, 3136] Unfold(kernel_size=7, dilation=1, padding=0, stride=1) [8, 16, 49, 56, 56] view b*head, head_dim, k_att^2, h_out, w_out unfold_rpe [1, 16, 49, 56, 56] att = (q_att.unsqueeze(2)*(unfold_k + q_pe.unsqueeze(2) - unfold_rpe)).sum(1) q_pe copy 与左侧k_att类似的unfold处理 [8, 49, 56, 56] softmax unfold_k out_att att 与左侧k_att类似的unfold处理 out_att (att.unsqueeze(1) * out_att).sum(2) out_att view 1×1conv 1×1conv 1×1conv f_all [2, 9, 16, 3136] 由q,k,v view,cat,通过fc获得 f_conv permute+reshape Conv2d(144, 64, kernel_size=(3, 3), stride=(1, 1), padding=(1, 1), groups=16, bias=False) dep_conv out_conv 加权求和 out [2, 64, 56, 56] [2, 64, 56, 56] [【cvpr2022】自注意力和卷积集成!ACmix性能速度全面提升！](https://zhuanlan.zhihu.com/p/490226994)：  pdf
code
卷积层的变体和替代
在Shift: A Zero FLOP, Zero Parameter Alternative to Spatial Convolutions中提出了Shfit操作, 这一操作基本上可以看成具有特殊设计的固定的权重的中Conv. 当我们将DepthwiseConv中的每一个卷积核改造为只有一处取1, 其它处取0的不可更新的核时, 我们就可以进行Shift操作了. 在直观上, Shift操作先对对每一个特征图进行了平移, 再通过1x1卷积提取信息. 在实现上, 可以直接对内存进行操作, 因而是Zero Flop.
更多 https://zhuanlan.zhihu.com/p/530428399
https://zhuanlan.zhihu.com/p/458016349
如何理解Inductive bias

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