Octave卷积学习笔记

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Octave卷积

Octave卷积的主题思想来自于图片的分频思想,首先认为图像可进行分频:

  • 低频部分:图像低频部分保存图像的大体信息,信息数据量较少
  • 高频部分:图像高频部分保留图像的细节信息,信息数据量较大

由此,认为卷积神经网络中的feature map也可以进行分频,可按channel分为高频部分和低频部分,如图所示:

feature_map.png

对于一个feature map,将其按通道分为两个部分,分别为低频通道和高频通道。随后将低频通道的长宽各缩减一半,则将一个feature map分为了高频和低频两个部分,即为Octave卷积处理的基本feature map,使用X表示,该类型X可表示为X = [X^H,X^L],其中X^H为高频部分,X^L为低频部分。

为了处理这种结构的feature map,其使用了如下所示的Octave卷积操作:

octave_conv.png

首先考虑低频部分输入X^L,该部分进行两个部分的操作:

  • X^L \to X^H:从低频到高频,首先使用指定卷积核W^{L \to H}进行卷积,随后进行Upample操作生成与高频部分长宽相同的Tensor,最终产生Y^{L\to H} = Upsample(Conv(X^L,W^{L \to H}),2)
  • X^L \to X^L:从低频到低频,这个部分为直接进行卷积操作Y^{L \to L} = Conv(X^L,W^{L \to L})

随后考虑高频部分,与低频部分类似有两个部分的操作:

  • X^H \to X^H:从高频到高频,直接进行卷积操作Y^{H \to H} = Conv(X^H,W^{H \to H})
  • X^H \to X^L:从高频到低频,首先进行stride和kernel均为2的平均值池化,再进行卷积操作,生成与Y^L通道数相同的feature map,最终产生Y^{H \to L} = conv(avgpool(X^H,2),W^{H \to L}))

最终,有Y^L = Y^{H \to L} + Y^{L \to L}Y^H = Y^{H \to H} +Y^{L \to H},因此可以总结如下公式:
Y^L = Y^{H \to L} + Y^{L \to L} = Y^{H \to L} = conv(avgpool(X^H,2),W^{H \to L})) + Conv(X^L,W^{L \to L}) \\ Y^H = Y^{H \to H} +Y^{L \to H} = Conv(X^H,W^{H \to H}) + Upsample(Conv(X^L,W^{L \to H}),2)
因此有四个部分的权值:

来源/去向 \to H \to L
H W^{H \to H} W^{H \to L}
L W^{L \to H} W^{L \to L}

另外进行使用时,在网络的输入和输出需要将两个频率上的Tensor聚合,做法如下:

  • 输入部分,取X = [X,0],即有X^H = XX^L = 0,仅进行H \to LH \to H操作,输出输出的低频仅有X生成,即Y^H = Y^{H \to H}Y^L = Y^{H \to L}
  • 输出部分,取X = [X^H,X^L]\alpha = 0。即仅进行L \to HH \to H的操作,最终输出为Y = Y^{L \to H} + Y^{H \to H}

性能分析

以下计算均取原Tensor尺寸为CI \times W \times H,卷积尺寸为CO \times CI \times K \times K,输出Tensor尺寸为CO \times W \times H(stride=1,padding设置使feature map尺寸不变)。

计算量分析

Octave卷积的最大优势在于减小计算量,取参数\alpha为低频通道占总通道的比例。首先考虑直接卷积的计算量,对于输出feature map中的每个数据,需要进行CI \times K \times K次乘加计算,因此总的计算量为:
C_{conv} = (CO \times W \times H) \times (CI \times K \times K)
现考虑Octave卷积,有四个卷积操作:

  • L \to L卷积:C_{L \to L} = \alpha^2 \times (CO \times \frac{W}{2} \times \frac{H}{2}) \times (CI \times K \times K) = \frac{\alpha^2}{4} \times C_{conv}
  • L \to H卷积:C_{L \to H} = ((1 - \alpha) \times CO \times \frac{W}{2} \times \frac{H}{2}) \times ( \alpha \times CI \times K \times K) = \frac{\alpha(1-\alpha)}{4} \times C_{conv}
  • H \to L卷积:C_{H \to L} = (\alpha \times CO \times \frac{W}{2} \times \frac{H}{2}) \times ((1 - \alpha) \times CI \times K \times K) = \frac{\alpha(1-\alpha)}{4} \times C_{conv}
  • H \to H卷积:C_{H \to H} = ((1 - \alpha) \times CO \times W \times H) \times ((1 - \alpha) \times CI \times K \times K) = (1 - \alpha)^2 \times C_{conv}

总上,可以得出计算量有:
\frac{C_{octave}}{C_{conv}} = \frac{\alpha^2 + 2\alpha(1-\alpha) + 4 (1 - \alpha)^2}{4} = 1 - \frac{3}{4}\alpha(2- \alpha)
\alpha \in [0,1]中单调递减,当取\alpha = 1时,有\frac{C_{octave}}{C_{conv}} = \frac{1}{4}

参数量分析

原卷积的参数量为:
W_{conv} = CO \times CI \times K \times K
Octave卷积将该部分分为四个,对于每个卷积有:

  • L \to L卷积:W_{L \to L} =(\alpha\times CO) \times (\alpha \times CI) \times K \times K = \alpha^2 \times W_{conv}
  • L \to H卷积:W_{L \to H} =((1-\alpha) \times CO) \times (\alpha \times CI) \times K \times K = \alpha(1 - \alpha) \times W_{conv}
  • H \to L卷积:W_{H \to L} =(\alpha \times CO) \times ((1-\alpha) \times CI) \times K \times K = \alpha(1 - \alpha) \times W_{conv}
  • H \to H卷积:W_{H \to L} =((1-\alpha) \times CO) \times ((1-\alpha) \times CI) \times K \times K = (1 - \alpha)^2 \times W_{conv}

因此共有参数量:
C_{octave} = (\alpha^2 + 2\alpha(1 - \alpha) + (1 - \alpha)^2) \times C_{conv} = C_{conv}
由此,参数量没有发生变化,该方法无法减少参数量。

Octave卷积实现

Octave卷积模块

以下实现了一个兼容普通卷积的Octave卷积模块,针对不同的高频低频feature map的通道数,分为以下几种情况:

  • Lout_channel != 0 and Lin_channel != 0:通用Octave卷积,需要四个卷积参数
  • Lout_channel == 0 and Lin_channel != 0:输出Octave卷积,输入有低频部分,输出无低频部分,仅需要两个卷积参数
  • Lout_channel != 0 and Lin_channel == 0:输入Octave卷积,输入无低频部分,输出有低频部分,仅需要两个卷积参数
  • Lout_channel == 0 and Lin_channel == 0:退化为普通卷积,输入输出均无低频部分,仅有一个卷积参数
class OctaveConv(pt.nn.Module):

    def __init__(self,Lin_channel,Hin_channel,Lout_channel,Hout_channel,
            kernel,stride,padding):
        super(OctaveConv, self).__init__()
        if Lout_channel != 0 and Lin_channel != 0:
            self.convL2L = pt.nn.Conv2d(Lin_channel,Lout_channel, kernel,stride,padding)
            self.convH2L = pt.nn.Conv2d(Hin_channel,Lout_channel, kernel,stride,padding)
            self.convL2H = pt.nn.Conv2d(Lin_channel,Hout_channel, kernel,stride,padding)
            self.convH2H = pt.nn.Conv2d(Hin_channel,Hout_channel, kernel,stride,padding)
        elif Lout_channel == 0 and Lin_channel != 0:
            self.convL2L = None
            self.convH2L = None
            self.convL2H = pt.nn.Conv2d(Lin_channel,Hout_channel, kernel,stride,padding)
            self.convH2H = pt.nn.Conv2d(Hin_channel,Hout_channel, kernel,stride,padding)
        elif Lout_channel != 0 and Lin_channel == 0:
            self.convL2L = None
            self.convH2L = pt.nn.Conv2d(Hin_channel,Lout_channel, kernel,stride,padding)
            self.convL2H = None
            self.convH2H = pt.nn.Conv2d(Hin_channel,Hout_channel, kernel,stride,padding)
        else:
            self.convL2L = None
            self.convH2L = None
            self.convL2H = None
            self.convH2H = pt.nn.Conv2d(Hin_channel,Hout_channel, kernel,stride,padding)
        self.upsample = pt.nn.Upsample(scale_factor=2)
        self.pool = pt.nn.AvgPool2d(2)

    def forward(self,Lx,Hx):
        if self.convL2L is not None:
            L2Ly = self.convL2L(Lx)
        else:
            L2Ly = 0
        if self.convL2H is not None:
            L2Hy = self.upsample(self.convL2H(Lx))
        else:
            L2Hy = 0
        if self.convH2L is not None:
            H2Ly = self.convH2L(self.pool(Hx))
        else:
            H2Ly = 0
        if self.convH2H is not None:
            H2Hy = self.convH2H(Hx)
        else:
            H2Hy = 0
        return L2Ly+H2Ly,L2Hy+H2Hy

在前项传播的过程中,根据是否有对应的卷积操作参数判断是否进行卷积,若不进行卷积,将输出置为0。前向传播时,输入为低频和高频两个feature map,输出为低频和高频两个feature map,输入情况和参数配置应与通道数的配置匹配。

其他部分

使用MNIST数据集,构建了一个三层卷积+两层全连接层的神经网络,使用Adam优化器训练,代价函数使用交叉熵函数,训练3轮,最后在测试集上进行测试。

import torch as pt
import torchvision as ptv
# download dataset
train_dataset = ptv.datasets.MNIST("./",train=True,download=True,transform=ptv.transforms.ToTensor())
test_dataset = ptv.datasets.MNIST("./",train=False,download=True,transform=ptv.transforms.ToTensor())
train_loader = pt.utils.data.DataLoader(train_dataset,batch_size=64,shuffle=True)
test_loader = pt.utils.data.DataLoader(test_dataset,batch_size=64,shuffle=True)

# build network
class mnist_model(pt.nn.Module):

    def __init__(self):
        super(mnist_model, self).__init__()
        self.conv1 = OctaveConv(0,1,8,8,kernel=3,stride=1,padding=1)        
        self.conv2 = OctaveConv(8,8,16,16,kernel=3,stride=1,padding=1)      
        self.conv3 = OctaveConv(16,16,0,64,kernel=3,stride=1,padding=1)
        self.pool =  pt.nn.MaxPool2d(2)
        self.relu = pt.nn.ReLU()
        self.fc1 = pt.nn.Linear(64*7*7,256)
        self.fc2 = pt.nn.Linear(256,10)

    def forward(self,x):
        out = [self.pool(self.relu(i)) for i in self.conv1(0,x)]
        out = self.conv2(*out)
        _,out = self.conv3(*out)
        out = self.fc1(self.pool(self.relu(out)).view(-1,64*7*7))
        return self.fc2(out)


net = mnist_model().cuda()
# print(net)
# prepare training
def acc(outputs,label):
    _,data = pt.max(outputs,dim=1)
    return pt.mean((data.float()==label.float()).float()).item()

lossfunc = pt.nn.CrossEntropyLoss().cuda()
optimizer = pt.optim.Adam(net.parameters())

# train
for _ in range(3):
    for i,(data,label) in enumerate(train_loader) :
        optimizer.zero_grad()
        # print(i,data,label)
        data,label = data.cuda(),label.cuda()
        outputs = net(data)
        loss = lossfunc(outputs,label)
        loss.backward()

        optimizer.step()
        if i % 100 == 0:
            print(i,loss.cpu().data.item(),acc(outputs,label))

# test
acc_list = []
for i,(data,label) in enumerate(test_loader):
    data,label = data.cuda(),label.cuda()
    outputs = net(data)
    acc_list.append(acc(outputs,label))
print("Test:",sum(acc_list)/len(acc_list))

# save
pt.save(net,"./model.pth")

最终获得模型的准确率为0.988

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