图融合GCN（Graph Convolutional Networks）

数据其实是图（graph），图在生活中无处不在，如社交网络，知识图谱，蛋白质结构等。本文介绍GNN（Graph Neural Networks）中的分支：GCN（Graph Convolutional Networks）。

 GCN的PyTorch实现

虽然GCN从数学上较难理解，但是，实现是非常简单的，值得注意的一点是，一般情况下邻接矩阵是稀疏矩阵，所以，在实现矩阵乘法时，采用稀疏运算会更高效。首先，图卷积层的实现：

import torch
import torch.nn as nn


class GraphConvolution(nn.Module):
"""GCN layer"""

def __init__(self, in_features, out_features, bias=True):
super(GraphConvolution, self).__init__()
self.in_features = in_features
self.out_features = out_features
self.weight = nn.Parameter(torch.Tensor(in_features, out_features))
if bias:
self.bias = nn.Parameter(torch.Tensor(out_features))
else:
self.register_parameter('bias', None)

self.reset_parameters()

def reset_parameters(self):
nn.init.kaiming_uniform_(self.weight)
if self.bias isnotNone:
nn.init.zeros_(self.bias)

def forward(self, input, adj):
support = torch.mm(input, self.weight)
output = torch.spmm(adj, support)
if self.bias isnotNone:
return output + self.bias
else:
return output

def extra_repr(self):
return'in_features={}, out_features={}, bias={}'.format(
self.in_features, self.out_features, self.bias isnotNone
)
对于GCN，只需要将图卷积层堆积起来就可以，这里，实现一个两层的GCN：
class GCN(nn.Module):
"""a simple two layer GCN"""
def __init__(self, nfeat, nhid, nclass):
super(GCN, self).__init__()
self.gc1 = GraphConvolution(nfeat, nhid)
self.gc2 = GraphConvolution(nhid, nclass)

def forward(self, input, adj):
h1 = F.relu(self.gc1(input, adj))
logits = self.gc2(h1, adj)
return logits

这里的激活函数采用ReLU，后面，将用这个网络实现一个图中节点的半监督分类任务。

数据的提取，只需要load就可以：

# https://github.com/tkipf/pygcn/blob/master/pygcn/utils.py
adj, features, labels, idx_train, idx_val, idx_test = load_data(path="./data/cora/")

值得注意的有两点，一是论文引用应该是单向图，但是在网络时，要先将其转成无向图，或者说建立双向引用，这个对模型训练结果影响较大：

# build symmetric adjacency matrix
adj = adj + adj.T.multiply(adj.T > adj) - adj.multiply(adj.T > adj)

另外，官方实现中对邻接矩阵采用的是普通均值归一化，当然，也可以采用对称归一化方式：

def normalize_adj(adj):
    """compute L=D^-0.5 * (A+I) * D^-0.5"""
    adj += sp.eye(adj.shape[0])
    degree = np.array(adj.sum(1))
    d_hat = sp.diags(np.power(degree, -0.5).flatten())
    norm_adj = d_hat.dot(adj).dot(d_hat)
    return norm_adj

这里，只采用图中140个有标签样本对GCN进行训练，每个epoch计算出这些节点特征，然后计算loss：

    loss_history = []
    val_acc_history = []
    for epoch in range(epochs):
        model.train()
        logits = model(features, adj)
        loss = criterion(logits[idx_train], labels[idx_train])
       
        train_acc = accuracy(logits[idx_train], labels[idx_train])
       
        optimizer.zero_grad()
        loss.backward()
        optimizer.step()
       
        val_acc = test(idx_val)
        loss_history.append(loss.item())
        val_acc_history.append(val_acc.item())
        print("Epoch {:03d}: Loss {:.4f}, TrainAcc {:.4}, ValAcc {:.4f}".format(
            epoch, loss.item(), train_acc.item(), val_acc.item()))

只需要训练200个epoch，就可以在测试集上达到80%左右的分类准确，GCN的强大可想而知：

 融合BN和Conv层

在PyTorch中实现这个融合操作：nn.Conv2d参数:

• filter weights,W: conv.weight;
• bias,b: conv.bias;

nn.BatchNorm2d参数:

具体的实现代码如下（Google Colab, https://colab.research.google.com/drive/1mRyq_LlJW4u_rArzzhEe_T6tmEWoNN1K）:

import torch
    import torchvision
   
    def fuse(conv, bn):
   
        fused = torch.nn.Conv2d(
            conv.in_channels,
            conv.out_channels,
            kernel_size=conv.kernel_size,
            stride=conv.stride,
            padding=conv.padding,
            bias=True
        )
   
        # setting weights
        w_conv = conv.weight.clone().view(conv.out_channels, -1)
        w_bn = torch.diag(bn.weight.div(torch.sqrt(bn.eps+bn.running_var)))
        fused.weight.copy_( torch.mm(w_bn, w_conv).view(fused.weight.size()) )
       
        # setting bias
        if conv.bias isnotNone:
            b_conv = conv.bias
        else:
            b_conv = torch.zeros( conv.weight.size(0) )
        b_bn = bn.bias - bn.weight.mul(bn.running_mean).div(
                              torch.sqrt(bn.running_var + bn.eps)
                            )
        fused.bias.copy_( b_conv + b_bn )
   
        return fused
   
    # Testing
    # we need to turn off gradient calculation because we didn't write it
    torch.set_grad_enabled(False)
    x = torch.randn(16, 3, 256, 256)
    resnet18 = torchvision.models.resnet18(pretrained=True)
    # removing all learning variables, etc
    resnet18.eval()
    model = torch.nn.Sequential(
        resnet18.conv1,
        resnet18.bn1
    )
    f1 = model.forward(x)
    fused = fuse(model[0], model[1])
    f2 = fused.forward(x)
    d = (f1 - f2).mean().item()
    print("error:",d)

参考链接：

1. Semi-Supervised Classification with Graph Convolutional Networks https://arxiv.org/abs/1609.02907
2. How to do Deep Learning on Graphs with Graph Convolutional Networks https://towardsdatascience.com/how-to-do-deep-learning-on-graphs-with-graph-convolutional-networks-7d2250723780
3. Graph Convolutional Networks http://tkipf.github.io/graph-convolutional-networks
4. Graph Convolutional Networks in PyTorch https://github.com/tkipf/pygcn
5. 回顾频谱图卷积的经典工作：从ChebNet到GCN https://www.jianshu.com/p/2fd5a2454781
6. 图数据集之cora数据集介绍- 用pyton处理 - 可用于GCN任务 https://blog.csdn.net/yeziand01/article/details/93374216
7. Speeding up model with fusing batch normalization and convolution (http://learnml.today/speeding-up-model-with-fusing-batch-normalization-and-convolution-3)