Normalization

local_response_normalization

local_response_normalization出现在论文”ImageNet Classification with deep Convolutional Neural Networks”中,论文中说,这种normalization对于泛化是有好处的.

bix,y=aix,y(k+α∑min(0,i+n/2)j=max(0,i−n/2)(ajx,y)2)β

b_{x,y}^i = \frac{a_{x,y}^i}{ (k+\alpha\sum_{j=max(0,i-n/2)}^{min(0,i+n/2)}(a_{x,y}^j)^2)^\beta} 经过了一个conv2d或pooling后,我们获得了[batch_size, height, width, channels]这样一个tensor.现在,将channels称之为层,不考虑batch_size

• ii代表第ii层
• aix,ya_{x,y}^i就代表 第ii层的 (x,y)位置所对应的值
• nn个相邻feature maps.
• k...α...n...βk...\alpha ... n ...\beta是hyper parameters
• 可以看出,这个函数的功能就是, aix,ya_{x,y}^i需要用他的相邻的map的同位置的值进行normalization 在alexnet中, k=2,n=5,α=10−4,β=0.75k=2, n=5, \alpha=10^{-4}, \beta=0.75

tf.nn.local_response_normalization(input, depth_radius=None, bias=None, alpha=None, beta=None, name=None)
'''
Local Response Normalization.
The 4-D input tensor is treated as a 3-D array of 1-D vectors (along the last dimension), and each vector is normalized independently. Within a given vector, each component is divided by the weighted, squared sum of inputs within depth_radius. In detail,
'''
"""
input: A Tensor. Must be one of the following types: float32, half. 4-D.
depth_radius: An optional int. Defaults to 5. 0-D. Half-width of the 1-D normalization window.
bias: An optional float. Defaults to 1. An offset (usually positive to avoid dividing by 0).
alpha: An optional float. Defaults to 1. A scale factor, usually positive.
beta: An optional float. Defaults to 0.5. An exponent.
name: A name for the operation (optional).
"""

• depth_radius: 就是公式里的n/2n/2
• bias : 公式里的kk
• input: 将conv2d或pooling 的输出输入就行了[batch_size, height, width, channels]
• return :[batch_size, height, width, channels], 正则化后

batch_normalization

论文地址 batch_normalization, 故名思意,就是以batch为单位进行normalization - 输入:mini_batch: In={x1,x2,..,xm}In=\{x^1,x^2,..,x^m\} - γ,β\gamma,\beta,需要学习的参数,都是向量 - ϵ\epsilon: 一个常量 - 输出: Out={y1,y2,...,ym}Out=\{y^1, y^2, ..., y^m\} 算法如下: (1)mini_batch mean:

μIn←1m∑i=1mxi

\mu_{In} \leftarrow \frac{1}{m}\sum_{i=1}^m x_i (2)mini_batch variance

σ2In=1m∑i=1m(xi−μIn)2

\sigma_{In}^2=\frac{1}{m}\sum_{i=1}^m(x^i-\mu_In)^2 (3)Normalize

x^i=xi−μInσ2In+ϵ−−−−−−√

\hat x^i=\frac{x^i-\mu_{In}}{\sqrt{\sigma_{In}^2 + \epsilon}} (4)scale and shift

yi=γx^i+β

y^i=\gamma\hat x^i + \beta 可以看出,batch_normalization之后,数据的维数没有任何变化,只是数值发生了变化 OutOut作为下一层的输入 函数: tf.nn.batch_normalization()

def batch_normalization(x,
                        mean,
                        variance,
                        offset,
                        scale,
                        variance_epsilon,
                        name=None):

Args:

• x: Input Tensor of arbitrary dimensionality.
• mean: A mean Tensor.
• variance: A variance Tensor.
• offset: An offset Tensor, often denoted β\beta in equations, or None. If present, will be added to the normalized tensor.
• scale: A scale Tensor, often denoted γ\gamma in equations, or None. If present, the scale is applied to the normalized tensor.
• variance_epsilon: A small float number to avoid dividing by 0.
• name: A name for this operation (optional).
• Returns: the normalized, scaled, offset tensor. 对于卷积,x:[bathc,height,width,depth] 对于卷积,我们要feature map中共享 γi\gamma_i 和 βi\beta_i ,所以 γ,β\gamma, \beta的维度是[depth]

现在,我们需要一个函数 返回mean和variance, 看下面.

tf.nn.moments()

def moments(x, axes, shift=None, name=None, keep_dims=False):
# for simple batch normalization pass `axes=[0]` (batch only).

对于卷积的batch_normalization, x 为[batch_size, height, width, depth],axes=[0,1,2],就会输出(mean,variance), mean 与 variance 均为标量。