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Theano tensor 模块:nnet 子模块

nnettensor 模块中与神经网络 Neural Networks 相关的子模块。

In [1]:

import theano
from theano import tensor as T
Using gpu device 1: Tesla C2075 (CNMeM is disabled)

Sigmoid 函数

共有三种 sigmoid

  • T.nnet.sigmoid(x)
  • T.nnet.ultra_sigmoid(x)
  • T.nnet.hard_sigmoid(x)

精度和时间:

sigmoid > ultra_fast_sigmoid > hard_sigmoid

函数图像:

In [2]:

x, y, b = T.dvectors('x', 'y', 'b')
W = T.dmatrix('W')
y = T.nnet.sigmoid(T.dot(W, x) + b)

print theano.pprint(y)
sigmoid(((W \dot x) + b))

其他

T.nnet.softplus(x) 返回

$$\operatorname{softplus}(x) = \log_e{\left(1 + \exp(x)\right)}$$

会解决在 1 附近自定义函数值不准的问题。

In [3]:

x,y,b = T.dvectors('x','y','b')
W = T.dmatrix('W')
y = T.nnet.softplus(T.dot(W,x) + b)

print theano.pprint(y)
softplus(((W \dot x) + b))

T.nnet.softplus(x) 返回

$$ \operatorname{softmax}{ij}(x) = \frac{\exp{x{ij}}}{\sum_k\exp(x_{ik})} $$

softmax 作用到矩阵时,它会按照行进行计算。

不过,下面 的代码计算性能上更加稳定:

e_x = exp(x - x.max(axis=1, keepdims=True))
out = e_x / e_x.sum(axis=1, keepdims=True)

In [4]:

x,y,b = T.dvectors('x','y','b')
W = T.dmatrix('W')
y = T.nnet.softmax(T.dot(W,x) + b)

print theano.pprint(y)
Softmax(((W \dot x) + b))

T.nnet.relu(x, alpha=0) 返回这样一个函数:

$$ f(x_i) = \left{ \begin{aligned} x_i, & \ x_i > 0 \ \alpha x_i, & \ otherwise \end{aligned}\right. $$

损失函数

T.nnet.binary_crossentropy(output, target) 二类交叉熵:

$$ \text{crossentropy}(t,o) = -(t\cdot log(o) + (1 - t) \cdot log(1 - o)) $$In [5]:

x, y, b, c = T.dvectors('x', 'y', 'b', 'c')
W = T.dmatrix('W')
V = T.dmatrix('V')
h = T.nnet.sigmoid(T.dot(W, x) + b)
x_recons = T.nnet.sigmoid(T.dot(V, h) + c)
recon_cost = T.nnet.binary_crossentropy(x_recons, x).mean()

T.nnet.categorical_crossentropy(coding_dist, true_dist) 多类交叉熵

$$ H(p,q) = - \sum_x p(x) \log(q(x)) $$