深入解析变分自编码器（VAE）：理论、数学原理、实现与应用

import torch
import torch.nn as nn
import torch.optim as optim
from torch.utils.data import DataLoader
from torchvision import datasets, transforms
from torchvision.utils import save_image
import os

class VAE(nn.Module):
    def __init__(self, input_dim=784, hidden_dim=400, latent_dim=20):
        super(VAE, self).__init__()
        # 编码器部分
        self.encoder = nn.Sequential(
            nn.Linear(input_dim, hidden_dim),
            nn.ReLU()
        )
        self.fc_mu = nn.Linear(hidden_dim, latent_dim)
        self.fc_logvar = nn.Linear(hidden_dim, latent_dim)
        
        # 解码器部分
        self.decoder = nn.Sequential(
            nn.Linear(latent_dim, hidden_dim),
            nn.ReLU(),
            nn.Linear(hidden_dim, input_dim),
            nn.Sigmoid()
        )
    
    def encode(self, x):
        h = self.encoder(x)
        mu = self.fc_mu(h)
        logvar = self.fc_logvar(h)
        return mu, logvar
    
    def reparameterize(self, mu, logvar):
        std = torch.exp(0.5 * logvar)
        eps = torch.randn_like(std)  # 采样自标准正态分布
        return mu + eps * std
    
    def decode(self, z):
        return self.decoder(z)
    
    def forward(self, x):
        mu, logvar = self.encode(x)
        z = self.reparameterize(mu, logvar)
        recon_x = self.decode(z)
        return recon_x, mu, logvar

def vae_loss(recon_x, x, mu, logvar):
    # 重构误差使用二元交叉熵
    BCE = nn.functional.binary_cross_entropy(recon_x, x, reduction='sum')
    # KL散度计算
    KLD = -0.5 * torch.sum(1 + logvar - mu.pow(2) - logvar.exp())
    return BCE + KLD

transform = transforms.Compose([
    transforms.ToTensor(),
    transforms.Normalize((0.5,), (0.5,))
])

train_dataset = datasets.MNIST(root='./data', train=True, transform=transform, download=True)
train_loader = DataLoader(train_dataset, batch_size=128, shuffle=True)

device = torch.device('cuda' if torch.cuda.is_available() else'cpu')
vae = VAE().to(device)
optimizer = optim.Adam(vae.parameters(), lr=1e-3)

epochs = 10
ifnot os.path.exists('./results'):
    os.makedirs('./results')

for epoch in range(1, epochs + 1):
    vae.train()
    train_loss = 0
    for batch_idx, (data, _) in enumerate(train_loader):
        data = data.view(-1, 784).to(device)
        optimizer.zero_grad()
        recon_batch, mu, logvar = vae(data)
        loss = vae_loss(recon_batch, data, mu, logvar)
        loss.backward()
        train_loss += loss.item()
        optimizer.step()
    
    average_loss = train_loss / len(train_loader.dataset)
    print(f'Epoch {epoch}, Average Loss: {average_loss:.4f}')
    
    # 生成样本并保存
    with torch.no_grad():
        z = torch.randn(64, 20).to(device)
        sample = vae.decode(z).cpu()
        save_image(sample.view(64, 1, 28, 28), f'./results/sample_epoch_{epoch}.png')

vae.eval()
with torch.no_grad():
    z = torch.randn(16, 20).to(device)
    generated = vae.decode(z).cpu()
    save_image(generated.view(16, 1, 28, 28), 'generated_digits.png')