本文只设计简单的介绍和pytorch实现，数学原理部分请移步知乎“不刷知乎”大佬的文章：半小时理解变分自编码器
本文部分观点引自上述知乎文章。
降维是减少描述数据的特征数量的过程。可以通过选择（仅保留一些现有特征）或通过提取（基于旧特征组合来生成数量更少的新特征）来进行降维。降维在许多需要低维数据（数据可视化，数据存储，繁重的计算…）的场景中很有用。
PCA的想法是构建m个新的独立特征，这些特征是n个旧特征的线性组合，并使得这些新特征所定义的子空间上的数据投影尽可能接近初始数据（就欧几里得距离而言）。换句话说，PCA寻找初始空间的最佳线性子空间（由新特征的正交基定义），以使投影到该子空间上的近似数据的误差尽可能小。
简单来说就是使用神经网络做编码器（Encoder）和解码器（Decoder）。
自编码器的缺点：
隐空间的规则性可以通过两个主要属性表示：
简单来说，为了保证隐空间的规则性，VAE的编码器不是将输入编码为隐空间中的单个点，而是将其编码为隐空间中的概率分布。然后解码时按照此概率分布从隐空间中采样进行解码。训练过程：
如何计算KL散度：
将输入编码为具有一定方差而不是单个点的分布的原因是这样可以非常自然地表达隐空间规则化：编码器返回的分布被强制接近标准正态分布。
本节实现AE和VAE对MNIST数据集的编码与解码（重现）。
'''
定义自编码器网络结构
'''
import torch
from torch import nn
class AE(nn.Module):
def __init__(self):
super(AE, self).__init__()
# [b, 784] => [b, 20]
self.encoder = nn.Sequential(
nn.Linear(784, 256),
nn.ReLU(),
nn.Linear(256, 64),
nn.ReLU(),
nn.Linear(64, 20),
nn.ReLU()
)
# [b, 20] => [b, 784]
self.decoder = nn.Sequential(
nn.Linear(20, 64),
nn.ReLU(),
nn.Linear(64, 256),
nn.ReLU(),
nn.Linear(256, 784),
nn.Sigmoid()
)
def forward(self, x):
"""
:param x: [b, 1, 28, 28]
:return:
"""
batchsz = x.size(0)
# flatten（打平）
x = x.view(batchsz, 784)
# encoder
x = self.encoder(x)
# decoder
x = self.decoder(x)
# reshape
x = x.view(batchsz, 1, 28, 28)
return x
'''
此处需要安装并开启visdom
安装：pip install visdom
开启：python -m visdom.server
'''
import torch
from torch.utils.data import DataLoader
from torch import nn, optim
from torchvision import transforms, datasets
from ae import AE
import visdom
def main():
'''import mnist dataset'''
mnist_train = datasets.MNIST('mnist', True, transform=transforms.Compose([
transforms.ToTensor()
]), download=True)
mnist_train = DataLoader(mnist_train, batch_size=32, shuffle=True)
mnist_test = datasets.MNIST('mnist', False, transform=transforms.Compose([
transforms.ToTensor()
]), download=True)
mnist_test = DataLoader(mnist_test, batch_size=32, shuffle=True)
'''show shape of data'''
x, label_unuse = iter(mnist_train).next()
print('x:', x.shape) # torch.Size([32, 1, 28, 28])
'''定义神经网络相关内容'''
device = torch.device('cuda')
model = AE().to(device)
# model = VAE().to(device)
criteon = nn.MSELoss()
optimizer = optim.Adam(model.parameters(), lr=1e-3)
print(model)
'''可视化'''
viz = visdom.Visdom()
for epoch in range(1000):
'''train'''
for batchidx, (x, label_unuse) in enumerate(mnist_train):
# [b, 1, 28, 28]
x = x.to(device)
x_hat = model(x)
loss = criteon(x_hat, x)
# backprop
optimizer.zero_grad()
loss.backward()
optimizer.step()
print(epoch, 'loss:', loss.item())
'''test'''
x, label_unuse = iter(mnist_test).next()
x = x.to(device)
with torch.no_grad():
x_hat = model(x)
'''show test result'''
viz.images(x, nrow=8, win='x', opts=dict(title='x'))
viz.images(x_hat, nrow=8, win='x_hat', opts=dict(title='x_hat'))
if __name__ == '__main__':
main()
'''
定义变分自编码器网络结构
'''
import torch
from torch import nn
class VAE(nn.Module):
def __init__(self):
super(VAE, self).__init__()
# [b, 784] => [b, 20]
# u: [b, 10]
# sigma: [b, 10]
self.encoder = nn.Sequential(
nn.Linear(784, 256),
nn.ReLU(),
nn.Linear(256, 64),
nn.ReLU(),
nn.Linear(64, 20),
nn.ReLU()
)
# [b, 10] => [b, 784]
self.decoder = nn.Sequential(
nn.Linear(10, 64),
nn.ReLU(),
nn.Linear(64, 256),
nn.ReLU(),
nn.Linear(256, 784),
nn.Sigmoid()
)
self.criteon = nn.MSELoss()
def forward(self, x):
"""
:param x: [b, 1, 28, 28]
:return:
"""
batchsz = x.size(0)
# flatten
x = x.view(batchsz, 784)
# encoder
# [b, 20], including mean and sigma
h_ = self.encoder(x)
# [b, 20] => [b, 10] and [b, 10]
mu, sigma = h_.chunk(2, dim=1)
# reparametrize trick, epison~N(0, 1)
h = mu + sigma * torch.randn_like(sigma) # 随机抽样
# decoder
x_hat = self.decoder(h)
# reshape
x_hat = x_hat.view(batchsz, 1, 28, 28)
kld = 0.5 * torch.sum(
torch.pow(mu, 2) +
torch.pow(sigma, 2) -
torch.log(1e-8 + torch.pow(sigma, 2)) - 1
) / (batchsz*28*28) # 计算与标准正态分布相比的散度
return x_hat, kld
'''
此处需要安装并开启visdom
安装：pip install visdom
开启：python -m visdom.server
'''
import torch
from torch.utils.data import DataLoader
from torch import nn, optim
from torchvision import transforms, datasets
from vae import VAE
import visdom
def main():
'''import data set'''
mnist_train = datasets.MNIST('mnist', True, transform=transforms.Compose([
transforms.ToTensor()
]), download=True)
mnist_train = DataLoader(mnist_train, batch_size=32, shuffle=True)
mnist_test = datasets.MNIST('mnist', False, transform=transforms.Compose([
transforms.ToTensor()
]), download=True)
mnist_test = DataLoader(mnist_test, batch_size=32, shuffle=True)
'''show data shape'''
x, _ = iter(mnist_train).next()
print('x:', x.shape) # torch.Size([32, 1, 28, 28])
'''def model'''
device = torch.device('cuda')
model = VAE().to(device)
criteon = nn.MSELoss()
optimizer = optim.Adam(model.parameters(), lr=1e-3)
print(model)
viz = visdom.Visdom()
for epoch in range(1000):
'''train'''
for batchidx, (x, _) in enumerate(mnist_train):
# [b, 1, 28, 28]
x = x.to(device)
x_hat, kld = model(x)
loss = criteon(x_hat, x)
if kld is not None:
elbo = - loss - 1.0 * kld
loss = - elbo
# backprop
optimizer.zero_grad()
loss.backward()
optimizer.step()
print(epoch, 'loss:', loss.item(), 'kld:', kld.item())
'''test'''
x, _ = iter(mnist_test).next()
x = x.to(device)
with torch.no_grad():
x_hat, kld = model(x)
'''show test result'''
viz.images(x, nrow=8, win='x', opts=dict(title='x'))
viz.images(x_hat, nrow=8, win='x_hat', opts=dict(title='x_hat'))
if __name__ == '__main__':
main()