深度有趣 | 06 变分自编码器

简介

变分自编码器（Variational Autoencoder，VAE）是生成式模型（Generative Model）的一种，另外一种常见的生成式模型是生成式对抗网络（Generative Adversarial Network，GAN）

这里咱们介绍下VAE的原理，并用Keras实现

原理

咱们常常会有这样的需求：根据不少个样本，学会生成新的样本

以MNIST为例，在看过几千张手写数字图片以后，咱们能进行模仿，并生成一些相似的图片，这些图片在原始数据中并不存在，有一些变化可是看起来类似

换言之，须要学会数据x的分布，这样，根据数据的分布就能轻松地产生新样本

$$ P(X) $$

但数据分布的估计不是件容易的事情，尤为是当数据量不足的时候

可使用一个隐变量z，由z通过一个复杂的映射获得x，而且假设z服从高斯分布

$$ x=f(z;\theta) $$

所以只须要学习隐变量所服从高斯分布的参数，以及映射函数，便可获得原始数据的分布

为了学习隐变量所服从高斯分布的参数，须要获得z足够多的样本

然而z的样本并不能直接得到，所以还须要一个映射函数（条件几率分布），从已有的x样本中获得对应的z样本

$$ z=Q(z|x) $$

这看起来和自编码器很类似，从数据自己，经编码获得隐层表示，经解码还原

但VAE和AE的区别以下：

• AE中隐层表示的分布未知，而VAE中隐变量服从高斯分布
• AE中学习的是encoder和decoder，VAE中还学习了隐变量的分布，包括高斯分布的均值和方差
• AE只能从一个x，获得对应的重构x
• VAE能够产生新的z，从而获得新的x，即生成新的样本

损失函数

除了重构偏差，因为在VAE中咱们假设隐变量z服从高斯分布，所以encoder对应的条件几率分布，应当和高斯分布尽量类似

能够用相对熵，又称做KL散度（Kullback–Leibler Divergence），来衡量两个分布的差别，或者说距离，但相对熵是非对称的

$$ D(f\parallel g)=\int f(x)\log\frac{f(x)}{g(x)}dx $$

实现

这里以MNIST为例，学习隐变量z所服从高斯分布的均值和方差两个参数，从而能够重新的z生成原始数据中没有的x

encoder和decoder各用两层全链接层，简单一些，主要为了说明VAE的实现

加载库

# -*- coding: utf-8 -*-

import numpy as np
import matplotlib.pyplot as plt

from keras.layers import Input, Dense, Lambda
from keras.models import Model
from keras import backend as K
from keras import objectives
from keras.datasets import mnist

定义一些常数

batch_size = 100
original_dim = 784
intermediate_dim = 256
latent_dim = 2
epochs = 50

encoder部分，两层全链接层，隐层表示包括均值和方差

x = Input(shape=(original_dim,))
h = Dense(intermediate_dim, activation='relu')(x)
z_mean = Dense(latent_dim)(h)
z_log_var = Dense(latent_dim)(h)

Lambda层不参与训练，只参与计算，用于后面产生新的z

def sampling(args):
    z_mean, z_log_var = args
    epsilon = K.random_normal(shape=(batch_size, latent_dim), mean=0.)
    return z_mean + K.exp(z_log_var / 2) * epsilon

z = Lambda(sampling, output_shape=(latent_dim,))([z_mean, z_log_var])

decoder部分，两层全链接层，x_decoded_mean为重构的输出

decoder_h = Dense(intermediate_dim, activation='relu')
decoder_mean = Dense(original_dim, activation='sigmoid')
h_decoded = decoder_h(z)
x_decoded_mean = decoder_mean(h_decoded)

自定义总的损失函数并编译模型

def vae_loss(x, x_decoded_mean):
    xent_loss = original_dim * objectives.binary_crossentropy(x, x_decoded_mean)
    kl_loss = -0.5 * K.sum(1 + z_log_var - K.square(z_mean) - K.exp(z_log_var), axis=-1)
    return xent_loss + kl_loss

vae = Model(x, x_decoded_mean)
vae.compile(optimizer='rmsprop', loss=vae_loss)

加载数据并训练，CPU训练的速度还算能忍

(x_train, y_train), (x_test, y_test) = mnist.load_data()

x_train = x_train.astype('float32') / 255.
x_test = x_test.astype('float32') / 255.
x_train = x_train.reshape((len(x_train), np.prod(x_train.shape[1:])))
x_test = x_test.reshape((len(x_test), np.prod(x_test.shape[1:])))

vae.fit(x_train, x_train,
        shuffle=True,
        epochs=epochs,
        batch_size=batch_size,
        validation_data=(x_test, x_test))

定义一个encoder，看看MNIST中的数据在隐层中变成了什么样子

encoder = Model(x, z_mean)

x_test_encoded = encoder.predict(x_test, batch_size=batch_size)
plt.figure(figsize=(6, 6))
plt.scatter(x_test_encoded[:, 0], x_test_encoded[:, 1], c=y_test)
plt.colorbar()
plt.show()

结果以下，说明在二维的隐层中，不一样的数字被很好地分开了

再定义一个生成器，从隐层到输出，用于产生新的样本

decoder_input = Input(shape=(latent_dim,))
_h_decoded = decoder_h(decoder_input)
_x_decoded_mean = decoder_mean(_h_decoded)
generator = Model(decoder_input, _x_decoded_mean)

用网格化的方法产生一些二维数据，做为新的z输入到生成器，并将生成的x展现出来

n = 20
digit_size = 28
figure = np.zeros((digit_size * n, digit_size * n))
grid_x = np.linspace(-4, 4, n)
grid_y = np.linspace(-4, 4, n)

for i, xi in enumerate(grid_x):
    for j, yi in enumerate(grid_y):
        z_sample = np.array([[yi, xi]])
        x_decoded = generator.predict(z_sample)
        digit = x_decoded[0].reshape(digit_size, digit_size)
        figure[(n - i - 1) * digit_size: (n - i) * digit_size,
               j * digit_size: (j + 1) * digit_size] = digit

plt.figure(figsize=(10, 10))
plt.imshow(figure)
plt.show()

结果以下，和以前看到的隐层图是一致的，甚至能看到一些数字之间的过渡态

因为包含一些随机因素，因此每次生成的结果会存在一些差别

若是将全链接层换成CNN，应该能够获得更好的表示结果

拓展

掌握以上内容后，用相同的方法，能够在FashionMNIST这个数据集上再跑一遍，数据集规模和MNIST彻底相同

只需改动四行便可

from keras.datasets import fashion_mnist

(x_train, y_train), (x_test, y_test) = fashion_mnist.load_data()

grid_x = np.linspace(-3, 3, n)
grid_y = np.linspace(-3, 3, n)

完整代码以下

# -*- coding: utf-8 -*-

import numpy as np
import matplotlib.pyplot as plt

from keras.layers import Input, Dense, Lambda
from keras.models import Model
from keras import backend as K
from keras import objectives
from keras.datasets import fashion_mnist

batch_size = 100
original_dim = 784
intermediate_dim = 256
latent_dim = 2
epochs = 50

x = Input(shape=(original_dim,))
h = Dense(intermediate_dim, activation='relu')(x)
z_mean = Dense(latent_dim)(h)
z_log_var = Dense(latent_dim)(h)

def sampling(args):
    z_mean, z_log_var = args
    epsilon = K.random_normal(shape=(batch_size, latent_dim), mean=0.)
    return z_mean + K.exp(z_log_var / 2) * epsilon

z = Lambda(sampling, output_shape=(latent_dim,))([z_mean, z_log_var])

decoder_h = Dense(intermediate_dim, activation='relu')
decoder_mean = Dense(original_dim, activation='sigmoid')
h_decoded = decoder_h(z)
x_decoded_mean = decoder_mean(h_decoded)

def vae_loss(x, x_decoded_mean):
    xent_loss = original_dim * objectives.binary_crossentropy(x, x_decoded_mean)
    kl_loss = -0.5 * K.sum(1 + z_log_var - K.square(z_mean) - K.exp(z_log_var), axis=-1)
    return xent_loss + kl_loss

vae = Model(x, x_decoded_mean)
vae.compile(optimizer='rmsprop', loss=vae_loss)

(x_train, y_train), (x_test, y_test) = fashion_mnist.load_data()

x_train = x_train.astype('float32') / 255.
x_test = x_test.astype('float32') / 255.
x_train = x_train.reshape((len(x_train), np.prod(x_train.shape[1:])))
x_test = x_test.reshape((len(x_test), np.prod(x_test.shape[1:])))

vae.fit(x_train, x_train,
        shuffle=True,
        epochs=epochs,
        batch_size=batch_size,
        validation_data=(x_test, x_test))

encoder = Model(x, z_mean)

x_test_encoded = encoder.predict(x_test, batch_size=batch_size)
plt.figure(figsize=(6, 6))
plt.scatter(x_test_encoded[:, 0], x_test_encoded[:, 1], c=y_test)
plt.colorbar()
plt.show()

decoder_input = Input(shape=(latent_dim,))
_h_decoded = decoder_h(decoder_input)
_x_decoded_mean = decoder_mean(_h_decoded)
generator = Model(decoder_input, _x_decoded_mean)

n = 20
digit_size = 28
figure = np.zeros((digit_size * n, digit_size * n))
grid_x = np.linspace(-3, 3, n)
grid_y = np.linspace(-3, 3, n)

for i, xi in enumerate(grid_x):
    for j, yi in enumerate(grid_y):
        z_sample = np.array([[yi, xi]])
        x_decoded = generator.predict(z_sample)
        digit = x_decoded[0].reshape(digit_size, digit_size)
        figure[(n - i - 1) * digit_size: (n - i) * digit_size,
               j * digit_size: (j + 1) * digit_size] = digit

plt.figure(figsize=(10, 10))
plt.imshow(figure)
plt.show()

咱们来看一下隐层的表示，一样起到了很好的分类效果

而后再来生成一些图形，能够看到不一样种类衣服之间的过渡

参考

• Auto-Encoding Variational Bayes：https://arxiv.org/pdf/1312.6114.pdf
• Tutorial on Variational Autoencoders：https://arxiv.org/pdf/1606.05908.pdf
• Building Autoencoders in Keras：https://blog.keras.io/building-autoencoders-in-keras.html
• Fashion-MNIST database of fashion articles：https://keras.io/datasets/#fashion-mnist-database-of-fashion-articles
• 【啄米平常】变分编码器VAE：https://zhuanlan.zhihu.com/p/25269592

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