两种网络层实现的数学细节。
一、前向计算和反向传播数学过程讲解
这里讲解的是平均池化层，最大池化层见本文第三小节
二、测试代码
数据和上面完全一致，自行打印验证即可。
1、前向传播
import tensorflow as tf
import numpy as np
# 输入张量为3×3的二维矩阵
M = np.array([
[[1], [-1], [0]],
[[-1], [2], [1]],
[[0], [2], [-2]]
])
# 定义卷积核权重和偏置项。由权重可知我们只定义了一个2×2×1的卷积核
filter_weight = tf.get_variable('weights', [2, 2, 1, 1], initializer=tf.constant_initializer([
[1, -1],
[0, 2]]))
biases = tf.get_variable('biases', [1], initializer=tf.constant_initializer(1))
# 调整输入格式符合TensorFlow要求
M = np.asarray(M, dtype='float32')
M = M.reshape(1, 3, 3, 1)
# 计算输入张量通过卷积核和池化滤波器计算后的结果
x = tf.placeholder('float32', [1, None, None, 1])
# 我们使用了带Padding，步幅为2的卷积操作，因为filter_weight的深度确定了卷积核的数量
conv = tf.nn.conv2d(x, filter_weight, strides=[1, 2, 2, 1], padding='SAME')
bias = tf.nn.bias_add(conv, biases)
# 使用带Padding，步幅为2的平均池化操作
pool = tf.nn.avg_pool(x, ksize=[1, 2, 2, 1], strides=[1, 2, 2, 1], padding='VALID')
# 执行计算图
with tf.Session() as sess:
tf.global_variables_initializer().run()
convoluted_M = sess.run(bias, feed_dict={x: M})
pooled_M = sess.run(pool, feed_dict={x: M})
print("convoluted_M: \n", convoluted_M)
print("pooled_M: \n", pooled_M)2、反向传播
import tensorflow as tf
import numpy as np
# 输入张量为3×3的二维矩阵
M = np.array([
[[1], [-1], [0]],
[[-1], [2], [1]],
[[0], [2], [-2]]
])
# 定义卷积核权重和偏置项。由权重可知我们只定义了一个2×2×1的卷积核
filter_weight = tf.get_variable('weights', [2, 2, 1, 1], initializer=tf.constant_initializer([
[1, -1],
[0, 2]]))
biases = tf.get_variable('biases', [1], initializer=tf.constant_initializer(1))
# 调整输入格式符合TensorFlow要求
M = np.asarray(M, dtype='float32')
M = M.reshape(1, 3, 3, 1)
# 计算输入张量通过卷积核和池化滤波器计算后的结果
x = tf.placeholder('float32', [1, None, None, 1])
# 我们使用了带Padding，步幅为2的卷积操作，因为filter_weight的深度确定了卷积核的数量
conv = tf.nn.conv2d(x, filter_weight, strides=[1, 2, 2, 1], padding='SAME')
bias = tf.nn.bias_add(conv, biases)
d_filter = tf.gradients(bias,filter_weight)
d_biases = tf.gradients(bias,biases)
d_conv = tf.gradients(bias,conv)
d_conv_x = tf.gradients(conv,x)
d_conv_w = tf.gradients(conv,filter_weight)
# d_bias_x = tf.gradients(bias,x)
# 使用带Padding，步幅为2的平均池化操作
pool = tf.nn.avg_pool(x, ksize=[1, 2, 2, 1], strides=[1, 2, 2, 1], padding='SAME')
d_pool = tf.gradients(pool,x)
# 执行计算图
with tf.Session() as sess:
tf.global_variables_initializer().run()
# convoluted_M = sess.run(bias, feed_dict={x: M})
# pooled_M = sess.run(pool, feed_dict={x: M})
#
# print("convoluted_M: \n", convoluted_M)
# print("pooled_M: \n", pooled_M)
print("d_filter:\n", sess.run(d_filter, feed_dict={x: M}))
print("d_biases:\n", sess.run(d_biases, feed_dict={x: M}))
print("d_conv:\n", sess.run(d_conv, feed_dict={x: M}))
print("d_conv_x:\n", sess.run(d_conv_x, feed_dict={x: M}))
print("d_conv_w:\n", sess.run(d_conv_w, feed_dict={x: M}))四、CS31n上实现的卷积层池化层API
1、卷积层
卷积层向前传播示意图：
def conv_forward_naive(x, w, b, conv_param):
"""
A naive implementation of the forward pass for a convolutional layer.
The input consists of N data points, each with C channels, height H and width
W. We convolve each input with F different filters, where each filter spans
all C channels and has height HH and width HH.
Input:
- x: Input data of shape (N, C, H, W)
- w: Filter weights of shape (F, C, HH, WW)
- b: Biases, of shape (F,)
- conv_param: A dictionary with the following keys:
- 'stride': The number of pixels between adjacent receptive fields in the
horizontal and vertical directions.
- 'pad': The number of pixels that will be used to zero-pad the input.
Returns a tuple of:
- out: Output data, of shape (N, F, H', W') where H' and W' are given by
H' = 1 + (H + 2 * pad - HH) / stride
W' = 1 + (W + 2 * pad - WW) / stride
- cache: (x, w, b, conv_param)
"""
out = None
#############################################################################
# TODO: Implement the convolutional forward pass. #
# Hint: you can use the function np.pad for padding. #
############################################################################
pad = conv_param['pad']
stride = conv_param['stride']
N, C, H, W = x.shape
F, _, HH, WW = w.shape
H0 = 1 + (H + 2 * pad - HH) / stride
W0 = 1 + (W + 2 * pad - WW) / stride
x_pad = np.pad(x, ((0,0),(0,0),(pad,pad),(pad,pad)),'constant') # 填充后的输入
out = np.zeros((N,F,H0,W0)) # 初始化的输出
# 以输出的每一个像素点为单位写出其前传表达式
for n in range(N):
for f in range(F):
for h0 in range(H0):
for w0 in range(W0):
out[n,f,h0,w0] = np.sum(x_pad[n,:,h0*stride:HH+h0*stride,w0*stride:WW+w0*stride] * w[f]) + b[f]
#############################################################################
# END OF YOUR CODE #
#############################################################################
cache = (x, w, b, conv_param)
return out, cache卷积层反向传播示意图：
def conv_backward_naive(dout, cache):
"""
A naive implementation of the backward pass for a convolutional layer.
Inputs:
- dout: Upstream derivatives.
- cache: A tuple of (x, w, b, conv_param) as in conv_forward_naive
Returns a tuple of:
- dx: Gradient with respect to x
- dw: Gradient with respect to w
- db: Gradient with respect to b
"""
dx, dw, db = None, None, None
#############################################################################
# TODO: Implement the convolutional backward pass. #
#############################################################################
x, w, b, conv_param = cache
pad = conv_param['pad']
stride = conv_param['stride']
N, C, H, W = x.shape
F, _, HH, WW = w.shape
_, _, H0, W0 = out.shape
x_pad = np.pad(x, [(0,0), (0,0), (pad,pad), (pad,pad)], 'constant')
dx, dw = np.zeros_like(x), np.zeros_like(w)
dx_pad = np.pad(dx, [(0,0), (0,0), (pad,pad), (pad,pad)], 'constant')
# 计算b的梯度(F,)
db = np.sum(dout, axis=(0,2,3)) # dout:(N,F,H0,W0)
# 以每一个dout点为基准计算其两个输入矩阵x(:,:,窗,窗)和w(f)的梯度，注意由于这两个矩阵都是多次参与运算，所以都是累加的关系
for n in range(N):
for f in range(F):
for h0 in range(H0):
for w0 in range(W0):
x_win = x_pad[n,:,h0*stride:h0*stride+HH,w0*stride:w0*stride+WW]
dw[f] += x_win * dout[n,f,h0,w0]
dx_pad[n,:,h0*stride:h0*stride+HH,w0*stride:w0*stride+WW] += w[f] * dout[n,f,h0,w0]
dx = dx_pad[:,:,pad:pad+H,pad:pad+W]
#############################################################################
# END OF YOUR CODE #
#############################################################################
return dx, dw, db2、最大池化层
池化层向前传播：
和卷积层类似，但是更简单一点，只要在对应feature map的原输入上取个窗口然后池化之即可，
def max_pool_forward_naive(x, pool_param):
HH, WW = pool_param['pool_height'], pool_param['pool_width']
s = pool_param['stride']
N, C, H, W = x.shape
H_new = 1 + (H - HH) / s
W_new = 1 + (W - WW) / s
out = np.zeros((N, C, H_new, W_new))
for i in xrange(N):
for j in xrange(C):
for k in xrange(H_new):
for l in xrange(W_new):
window = x[i, j, k*s:HH+k*s, l*s:WW+l*s]
out[i, j, k, l] = np.max(window)
cache = (x, pool_param)
return out, cache池化层反向传播：
反向传播的时候也是还原窗口，除最大值处继承上层梯度外（也就是说本层梯度为零），其他位置置零。
池化层没有过滤器，只有dx梯度，且x的窗口不像卷积层会重叠，所以不用累加，
def max_pool_backward_naive(dout, cache):
x, pool_param = cache
HH, WW = pool_param['pool_height'], pool_param['pool_width']
s = pool_param['stride']
N, C, H, W = x.shape
H_new = 1 + (H - HH) / s
W_new = 1 + (W - WW) / s
dx = np.zeros_like(x)
for i in xrange(N):
for j in xrange(C):
for k in xrange(H_new):
for l in xrange(W_new):
window = x[i, j, k*s:HH+k*s, l*s:WW+l*s]
m = np.max(window)
dx[i, j, k*s:HH+k*s, l*s:WW+l*s] = (window == m) * dout[i, j, k, l]
return dx五、实际框架实现方法
实际框架当然不会才用这种大循环的手段实现卷积操作，矩阵化运算才是正路。
1、Theano
常见的一种拆法是将二维 input 展开成一维向量（[in_h * in_w] -> [out_h * out_w]），将卷积核展开为（[in_h * in_w, out_h * out_w]），
上面仅讨论了2维输入，其实由于 input 的 channels 和 kernal 的 channels 数一致，所以情况延申起来原理并无改变。最后的运算如下：
y = C·xT
2、Caffe
caffe的卷积矩阵化如下，其直接把 input 的各个通道的值放在了一个矩阵种，将各个 kernals 的各个通道值放入同一个矩阵，一次解决所有运算，感觉比上面的做法高明了一点（不过都是很厉害的算法）。
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