卷积神经网络CNN（LeNet）的Theano实现

一、LeNet 结构

二、卷积层

ConvOp是在Theano中实现卷积层的主要主力。 ConvOp由theano.tensor.signal.conv2d使用，它具有两个输入：
1. input：对应于小批量输入图像的4D tensor。 tensor的形状如下：[mini-batch大小，输入的feature maps数量，图像height，图像width]。
2. W: 对应于权重矩阵W的4D tensor。tensor的形状是：[层 m的feature map的数量，层m-1处的feature map的数量，filter height，filter width]

举个简单的栗子：
输入： 3个feature map（一个500*500大小的RGB图片）
卷积：2个9*9*3大小的卷积核
输出：2个feature map

代码如下：

import theano
from theano import tensor as T
from theano.tensor.nnet import conv2d

import numpy

import pylab
from PIL import Image

rng = numpy.random.RandomState(23455)

# instantiate 4D tensor for input
input = T.tensor4(name='input')

# initialize shared variable for weights.
w_shp = (2, 3, 9, 9)# [#f_map of the m layer,#f_map of the m-1 layer, n_filter,filter_height,filter_width]
w_bound = numpy.sqrt(3 * 9 * 9)
W = theano.shared(numpy.asarray(
    rng.uniform(
        low=-1.0 / w_bound,
        high=1.0 / w_bound,
        size=w_shp),
    dtype=input.dtype), name='W')

# initialize shared variable for bias (1D tensor) with random values
# IMPORTANT: biases are usually initialized to zero. However in this
# particular application, we simply apply the convolutional layer to
# an image without learning the parameters. We therefore initialize
# them to random values to "simulate" learning.
b_shp = (2,)
b = theano.shared(numpy.asarray(
    rng.uniform(low=-.5, high=.5, size=b_shp),
    dtype=input.dtype), name='b')

# build symbolic expression that computes the convolution of input with
# filters in w
conv_out = conv2d(input, W)   #4d tensor dimension [batch_size, n_filter,f_map_height,f_map_width]  [1,2,112,112]

# build symbolic expression to add bias and apply activation function, i.e. produce neural net layer output
output = T.nnet.sigmoid(conv_out + b.dimshuffle('x', 0, 'x', 'x'))

# create theano function to compute filtered images
f = theano.function([input], output,allow_input_downcast=True)


# open random image of dimensions 639x516
#path=r'E:\\py\\theano\\3wolfmoon.jpg'
img = Image.open('cat.jpg')
# dimensions are (height, width, channel)
img = numpy.asarray(img, dtype='float64') / 256.

# put image in 4D tensor of shape (1, 3, height, width)
img_ = img.transpose(2, 0, 1).reshape(1, 3, 500, 500)
filtered_img = f(img_)

# plot original image and first and second components of output
pylab.subplot(1, 3, 1); pylab.axis('off'); pylab.imshow(img)
pylab.gray();
# recall that the convOp output (filtered image) is actually a "minibatch",
# of size 1 here, so we take index 0 in the first dimension:
pylab.subplot(1, 3, 2); pylab.axis('off'); pylab.imshow(filtered_img[0, 0, :, :])
pylab.subplot(1, 3, 3); pylab.axis('off'); pylab.imshow(filtered_img[0, 1, :, :])
pylab.show()

运行结果如下：

可以看到，随机生成的卷积核也能达到类似边缘滤波的效果！

三、MaxPooling

Maxpooling 实际上是下采样的过程，在每个n*n的子块中取最大的值输出。

主要作用有两点：
1. 去除最大值以外的值，使得后续计算量减少
2. 提供平移不变性，原因如下：

每张图的底层是卷积层的输出，上层是池化后的输出。

上图是原本的，下图是向右平移一位后的。可以看到虽然底层的每个数值都变了，但是池化后的输出中只有一半数值变化了。

因为maxpooling只对最大值敏感，只要左边新引入的数值不比原来的最大值大，右边移出池化区域的数值不是原本的最大值，那么maxpooling的结果就不会变。

Maxpooling通过theano.tensor.signal.pool.pool_2d完成

输入：
1. input . N维tensor
2. 下采样因子，比如[2,2]
3. ignore_border: 一般设为True

再举个栗子：

from theano.tensor.signal import pool

input = T.dtensor4('input')
maxpool_shape = (2, 2)
pool_out = pool.pool_2d(input, maxpool_shape, ignore_border=True)
f = theano.function([input],pool_out)

invals = numpy.random.RandomState(1).rand(3, 2, 5, 5)
print 'With ignore_border set to True:'
print 'invals[0, 0, :, :] =\n', invals[0, 0, :, :]
print 'output[0, 0, :, :] =\n', f(invals)[0, 0, :, :]

pool_out = pool.pool_2d(input, maxpool_shape, ignore_border=False)
f = theano.function([input],pool_out)
print 'With ignore_border set to False:'
print 'invals[1, 0, :, :] =\n ', invals[1, 0, :, :]
print 'output[1, 0, :, :] =\n ', f(invals)[1, 0, :, :]

运行结果：

With ignore_border set to True:
    invals[0, 0, :, :] =
    [[  4.17022005e-01   7.20324493e-01   1.14374817e-04   3.02332573e-01 1.46755891e-01]
     [  9.23385948e-02   1.86260211e-01   3.45560727e-01   3.96767474e-01 5.38816734e-01]
     [  4.19194514e-01   6.85219500e-01   2.04452250e-01   8.78117436e-01 2.73875932e-02]
     [  6.70467510e-01   4.17304802e-01   5.58689828e-01   1.40386939e-01 1.98101489e-01]
     [  8.00744569e-01   9.68261576e-01   3.13424178e-01   6.92322616e-01 8.76389152e-01]]
    output[0, 0, :, :] =
    [[ 0.72032449  0.39676747]
     [ 0.6852195   0.87811744]]

With ignore_border set to False:
    invals[1, 0, :, :] =
    [[ 0.01936696  0.67883553  0.21162812  0.26554666  0.49157316]
     [ 0.05336255  0.57411761  0.14672857  0.58930554  0.69975836]
     [ 0.10233443  0.41405599  0.69440016  0.41417927  0.04995346]
     [ 0.53589641  0.66379465  0.51488911  0.94459476  0.58655504]
     [ 0.90340192  0.1374747   0.13927635  0.80739129  0.39767684]]
    output[1, 0, :, :] =
    [[ 0.67883553  0.58930554  0.69975836]
     [ 0.66379465  0.94459476  0.58655504]
     [ 0.90340192  0.80739129  0.39767684]]

四、CNN结构

具体采取的cnn结构如下图：

layer0： conv+pool
layer1： conv+pool
layer3： tanh
layer4： softmax

全部代码下载：csdn下载链接

这里贴一下build model的代码：

def evaluate_lenet(learning_rate=0.1, n_epochs=200,
                   dataset='mnist.pkl.gz',
                   nkerns=[20, 50], batch_size=500):

    rng = numpy.random.RandomState(23455)

    # load data
    datasets = load_data(dataset)

    train_setx, train_sety = datasets[0]
    valid_setx, valid_sety = datasets[1]
    test_setx, test_sety = datasets[2]

    # compute number of minibatches for train/valid/test
    n_train_batches = train_setx.get_value(borrow=True).shape[0] // batch_size
    n_valid_batches = valid_setx.get_value(borrow=True).shape[0] // batch_size
    n_test_batches = test_setx.get_value(borrow=True).shape[0] // batch_size

    # declare variables of the data
    index = T.lscalar()  # index to a minibatch

    x = T.matrix('x')  # data representing  images
    y = T.ivector('y')  # data representing tags(classes)

    ###############
    # build model #
    ###############
    print('... building the model')

    # reshape the image data into 4D tensor of the shape(batch_size,number of
    # feature map, image height,image width) in order to be compatible with
    # our LeNetConvPoolLayer.
    # (28,28) is the size of MNIST images
    layer0_input = x.reshape((batch_size, 1, 28, 28))

    layer0 = LeNetConvPoolLayer(
        rng,
        input=layer0_input,
        filter_shape=(nkerns[0], 1, 5, 5),
        image_shape=(batch_size, 1, 28, 28),
    )

    layer1 = LeNetConvPoolLayer(
        rng,
        input=layer0.output,
        filter_shape=(nkerns[1], nkerns[0], 5, 5),
        image_shape=(batch_size, nkerns[0], 12, 12)
    )

    layer2_input = layer1.output.flatten(2)

    layer2 = HiddenLayer(
        rng,
        input=layer2_input,
        n_in=nkerns[1] * 4 * 4,
        n_out=500,
        activation=T.tanh
    )

    layer3 = LogisticRegression(input=layer2.output, n_in=500, n_out=10)

    cost = layer3.negative_log_likelihood(y)

    params = layer3.params + layer2.params + layer1.params + layer0.params

    grads = T.grad(cost, params)

    updates = [
        (param_i, param_i - learning_rate * grad_i)
        # param and its corresponding grad in pairs
        for param_i, grad_i in zip(params, grads)
    ]

    train_model = theano.function(
        [index],
        cost,
        updates=updates,
        givens={
            x: train_setx[index * batch_size:(index + 1) * batch_size],
            y: train_sety[index * batch_size:(index + 1) * batch_size]
        }
    )

五、 结果