CNN是一种多层神经网络，基于人工神经网络，在人工神经网络前，用滤波器进行特征抽取，使用卷积核作为特征抽取器，自动训练特征抽取器，就是说卷积核以及阈值参数这些都需要由网络去学习。
图像可以直接作为网络的输入，避免了传统识别算法中复杂的特征提取和数据重建过程。
一般卷积神经网络的结构：
前面feature extraction部分体现了CNN的特点，feature extraction部分最后的输出可以作为分类器的输入。这个分类器你可以用softmax或RBF等等。
局部感受野与权值共享
权值共享指每一个map都由一个卷积核去做卷积得到。
权值共享减少了权值数量，降低了网络复杂度。
Conv layer
假设层为s层特征图像Xil-1（pooling层），l层为卷积C层，卷积结果为Xjl
参数kij为两层间卷积核（滤波器，kernals），由于s层有m个特征，c层有n个特征，所以一共有m*n个卷积核。bj为卷基层每个结果特征对应的一个偏置项bias，f为非线性变换函数sigmoid函数；Mj为选择s层特征输入的个数，即选择多少个s层的图像特征作为输入；由于选择s层的特征个数方法不同，主要分为三种卷积选择方式
1、全部选择：
s层的全部特征都作为输入， Mj=m。如上图所示。
2、自动稀疏选择：
在卷积计算前加入稀疏稀疏aij，通过稀疏规则限制（论文后面），使算法自动选取部分s层特征作为输入，具体个数不确定。
3、部分选择：
按照一定的规则，固定的选取2个或者3个作为输入。
Subsampling layer
下采样降低feature map的空间分辨率，从而实现一定程度的shift 和 distortion invariance。利用图像局部相关性的原理，对图像进行子抽样，可以减少数据处理量同时保留有用信息。这个过程也被称为pooling，常用pooling 有两种：mean-pooling、max-pooling。
down(Xjl-1)表示下采样操作，就是对一个小区域进行pooling操作，使数据降维。
有的算法还会对mean-pooling、max-pooling后的输出进行非线性映射
f可以是sigmoid函数。
输出层
最后一个卷积-采样层的输出可作为输出层的输入，输出层是全连接层，可采用svm、bp算法、softmax等分类算法。
以LeNet5为例，其结构如下：
C1,C3,C5 : Convolutional layer.
5 × 5 Convolution matrix.
S2 , S4 : Subsampling layer.
Subsampling by factor 2.
F6 : Fully connected layer
OUTPUT : RBF
C层和S层成对出现，C承担特征抽取，S承担抗变形。C元中涉及两个重要参数，即感受野与阈值参数，前者确定输入连接的数目，后者则控制对特征子模式的反应程度。
C1层是一个卷积层（通过卷积运算，可以增强图像的某种特征，并且降低噪音），由6个特征图Feature Map构成。Feature Map中每个神经元与输入中5*5的邻域相连。特征图的大小为28*28。C1有156个可训练参数（每个滤波器5*5=25个unit参数和一个bias参数，一共6个滤波器，共(5*5+1)*6=156个参数），共156*(28*28)=122,304个连接。
S2层是一个下采样层，有6个14*14的特征图。特征图中的每个单元与C1中相对应特征图的2*2邻域相连接。S2层每个单元的4个输入相加，乘以一个可训练参数，再加上一个可训练偏置。结果通过sigmoid函数计算。可训练系数和偏置控制着sigmoid函数的非线性程度。如果系数比较小，那么运算近似于线性运算，亚采样相当于模糊图像。如果系数比较大，根据偏置的大小亚采样可以被看成是有噪声的“或”运算或者有噪声的“与”运算。每个单元的2*2感受野并不重叠，因此S2中每个特征图的大小是C1中特征图大小的1/4（行和列各1/2）。S2层有12个可训练参数和5880个连接。
C3层也是一个卷积层，它同样通过5x5的卷积核去卷积层S2，然后得到的特征map就只有10x10个神经元，但是它有16种不同的卷积核，所以就存在16个特征map了。这里需要注意的一点是：C3中的每个特征map是连接到S2中的所有6个或者几个特征map的，表示本层的特征map是上一层提取到的特征map的不同组合。
S4层是一个下采样层，由16个5*5大小的特征图构成。特征图中的每个单元与C3中相应特征图的2*2邻域相连接，跟C1和S2之间的连接一样。S4层有32个可训练参数（每个特征图1个因子和一个偏置）和2000个连接。
C5层是一个卷积层，有120个特征图。每个单元与S4层的全部16个单元的5*5邻域相连。由于S4层特征图的大小也为5*5（同滤波器一样），故C5特征图的大小为1*1：这构成了S4和C5之间的全连接。之所以仍将C5标示为卷积层而非全相联层，是因为如果LeNet-5的输入变大，而其他的保持不变，那么此时特征图的维数就会比1*1大。C5层有48120个可训练连接。
F6层有84个单元（之所以选这个数字的原因来自于输出层的设计），与C5层全相连。有10164个可训练参数。如同经典神经网络，F6层计算输入向量和权重向量之间的点积，再加上一个偏置。然后将其传递给sigmoid函数产生单元i的一个状态。
最后，输出层由欧式径向基函数（Euclidean Radial Basis Function）单元组成，每类一个单元，每个有84个输入。换句话说，每个输出RBF单元计算输入向量和参数向量之间的欧式距离。输入离参数向量越远，RBF输出的越大。一个RBF输出可以被理解为衡量输入模式和与RBF相关联类的一个模型的匹配程度的惩罚项。用概率术语来说，RBF输出可以被理解为F6层配置空间的高斯分布的负log-likelihood。给定一个输入模式，损失函数应能使得F6的配置与RBF参数向量（即模式的期望分类）足够接近。这些单元的参数是人工选取并保持固定的（至少初始时候如此）。这些参数向量的成分被设为-1或1。虽然这些参数可以以-1和1等概率的方式任选，或者构成一个纠错码，但是被设计成一个相应字符类的7*12大小（即84）的格式化图片。
卷积核和偏置的初始化
卷积核和偏置在开始时要随机初始化，这些参数是要在训练过程中学习的。
CNN涉及的算法
这张图来自博客http://blog.csdn.net/wds555/article/details/44100581，总结得很好
CNN网络算法流程
/***********训练阶段************/
/*********************************/
for t = 0; t < repeat; t++)
{
epochsNum = trainset.size() / batchSize;
epochsNum = trainset.size() / batchSize;
for i = 0; i < epochsNum; i++
{
随机抽取batchSize个[0,size)的数batch[batchSize]作为本次批量训练的训练集 //size为总体trainSet的大小
for j=0;j<batchSize;j++
{
train(trainset.getRecord(batch[j]))
}
跑完一个batch后更新权重updateParas();
}
动态调整准学习速率
}
训练单个样本
train(Record record) {
forward(record);
boolean result = backPropagation(record);
return result;
}
forward(record);
boolean result = backPropagation(record);
return result;
}
反向传输
*/
private boolean backPropagation(Record record) {
boolean result = setOutLayerErrors(record);
setHiddenLayerErrors();
return result;
}
*/
private boolean backPropagation(Record record) {
boolean result = setOutLayerErrors(record);
setHiddenLayerErrors();
return result;
}
"""This tutorial introduces the LeNet5 neural network architecture
using Theano. LeNet5 is a convolutional neural network, good for
classifying images. This tutorial shows how to build the architecture,
and comes with all the hyper-parameters you need to reproduce the
paper's MNIST results.
This implementation simplifies the model in the following ways:
- LeNetConvPool doesn't implement location-specific gain and bias parameters
- LeNetConvPool doesn't implement pooling by average, it implements pooling
by max.
- Digit classification is implemented with a logistic regression rather than
an RBF network
- LeNet5 was not fully-connected convolutions at second layer
References:
- Y. LeCun, L. Bottou, Y. Bengio and P. Haffner:
Gradient-Based Learning Applied to Document
Recognition, Proceedings of the IEEE, 86(11):2278-2324, November 1998.
http://yann.lecun.com/exdb/publis/pdf/lecun-98.pdf
"""
import os
import sys
import time
import numpy
import theano
import theano.tensor as T
from theano.tensor.signal import downsample
from theano.tensor.nnet import conv
from logistic_sgd import LogisticRegression, load_data
from mlp import HiddenLayer
class LeNetConvPoolLayer(object):
"""Pool Layer of a convolutional network """
def __init__(self, rng, input, filter_shape, image_shape, poolsize=(2, 2)):
"""
Allocate a LeNetConvPoolLayer with shared variable internal parameters.
:type rng: numpy.random.RandomState
:param rng: a random number generator used to initialize weights
:type input: theano.tensor.dtensor4
:param input: symbolic image tensor, of shape image_shape
:type filter_shape: tuple or list of length 4
:param filter_shape: (number of filters, num input feature maps,
filter height, filter width)
:type image_shape: tuple or list of length 4
:param image_shape: (batch size, num input feature maps,
image height, image width)
:type poolsize: tuple or list of length 2
:param poolsize: the downsampling (pooling) factor (#rows, #cols)
"""
assert image_shape[1] == filter_shape[1]
self.input = input
# there are "num input feature maps * filter height * filter width"
# inputs to each hidden unit
fan_in = numpy.prod(filter_shape[1:])
# each unit in the lower layer receives a gradient from:
# "num output feature maps * filter height * filter width" /
# pooling size
fan_out = (filter_shape[0] * numpy.prod(filter_shape[2:]) /
numpy.prod(poolsize))
# initialize weights with random weights
W_bound = numpy.sqrt(6. / (fan_in + fan_out))
self.W = theano.shared(
numpy.asarray(
rng.uniform(low=-W_bound, high=W_bound, size=filter_shape),
dtype=theano.config.floatX
),
borrow=True
)
# the bias is a 1D tensor -- one bias per output feature map
b_values = numpy.zeros((filter_shape[0],), dtype=theano.config.floatX)
self.b = theano.shared(value=b_values, borrow=True)
# convolve input feature maps with filters
conv_out = conv.conv2d(
input=input,
filters=self.W,
filter_shape=filter_shape,
image_shape=image_shape
)
# downsample each feature map individually, using maxpooling
pooled_out = downsample.max_pool_2d(
input=conv_out,
ds=poolsize,
ignore_border=True
)
# add the bias term. Since the bias is a vector (1D array), we first
# reshape it to a tensor of shape (1, n_filters, 1, 1). Each bias will
# thus be broadcasted across mini-batches and feature map
# width & height
self.output = T.tanh(pooled_out + self.b.dimshuffle('x', 0, 'x', 'x'))
# store parameters of this layer
self.params = [self.W, self.b]
def evaluate_lenet5(learning_rate=0.1, n_epochs=200,
dataset='mnist.pkl.gz',
nkerns=[20, 50], batch_size=500):
""" Demonstrates lenet on MNIST dataset
:type learning_rate: float
:param learning_rate: learning rate used (factor for the stochastic
gradient)
:type n_epochs: int
:param n_epochs: maximal number of epochs to run the optimizer
:type dataset: string
:param dataset: path to the dataset used for training /testing (MNIST here)
:type nkerns: list of ints
:param nkerns: number of kernels on each layer
"""
rng = numpy.random.RandomState(23455)
datasets = load_data(dataset)
train_set_x, train_set_y = datasets[0]
valid_set_x, valid_set_y = datasets[1]
test_set_x, test_set_y = datasets[2]
# compute number of minibatches for training, validation and testing
n_train_batches = train_set_x.get_value(borrow=True).shape[0]
n_valid_batches = valid_set_x.get_value(borrow=True).shape[0]
n_test_batches = test_set_x.get_value(borrow=True).shape[0]
n_train_batches /= batch_size
n_valid_batches /= batch_size
n_test_batches /= batch_size
# allocate symbolic variables for the data
index = T.lscalar() # index to a [mini]batch
# start-snippet-1
x = T.matrix('x') # the data is presented as rasterized images
y = T.ivector('y') # the labels are presented as 1D vector of
# [int] labels
######################
# BUILD ACTUAL MODEL #
######################
print '... building the model'
# Reshape matrix of rasterized images of shape (batch_size, 28 * 28)
# to a 4D tensor, compatible with our LeNetConvPoolLayer
# (28, 28) is the size of MNIST images.
layer0_input = x.reshape((batch_size, 1, 28, 28))
# Construct the first convolutional pooling layer:
# filtering reduces the image size to (28-5+1 , 28-5+1) = (24, 24)
# maxpooling reduces this further to (24/2, 24/2) = (12, 12)
# 4D output tensor is thus of shape (batch_size, nkerns[0], 12, 12)
layer0 = LeNetConvPoolLayer(
rng,
input=layer0_input,
image_shape=(batch_size, 1, 28, 28),
filter_shape=(nkerns[0], 1, 5, 5),
poolsize=(2, 2)
)
# Construct the second convolutional pooling layer
# filtering reduces the image size to (12-5+1, 12-5+1) = (8, 8)
# maxpooling reduces this further to (8/2, 8/2) = (4, 4)
# 4D output tensor is thus of shape (batch_size, nkerns[1], 4, 4)
layer1 = LeNetConvPoolLayer(
rng,
input=layer0.output,
image_shape=(batch_size, nkerns[0], 12, 12),
filter_shape=(nkerns[1], nkerns[0], 5, 5),
poolsize=(2, 2)
)
# the HiddenLayer being fully-connected, it operates on 2D matrices of
# shape (batch_size, num_pixels) (i.e matrix of rasterized images).
# This will generate a matrix of shape (batch_size, nkerns[1] * 4 * 4),
# or (500, 50 * 4 * 4) = (500, 800) with the default values.
layer2_input = layer1.output.flatten(2)
# construct a fully-connected sigmoidal layer
layer2 = HiddenLayer(
rng,
input=layer2_input,
n_in=nkerns[1] * 4 * 4,
n_out=500,
activation=T.tanh
)
# classify the values of the fully-connected sigmoidal layer
layer3 = LogisticRegression(input=layer2.output, n_in=500, n_out=10)
# the cost we minimize during training is the NLL of the model
cost = layer3.negative_log_likelihood(y)
# create a function to compute the mistakes that are made by the model
test_model = theano.function(
[index],
layer3.errors(y),
givens={
x: test_set_x[index * batch_size: (index + 1) * batch_size],
y: test_set_y[index * batch_size: (index + 1) * batch_size]
}
)
validate_model = theano.function(
[index],
layer3.errors(y),
givens={
x: valid_set_x[index * batch_size: (index + 1) * batch_size],
y: valid_set_y[index * batch_size: (index + 1) * batch_size]
}
)
# create a list of all model parameters to be fit by gradient descent
params = layer3.params + layer2.params + layer1.params + layer0.params
# create a list of gradients for all model parameters
grads = T.grad(cost, params)
# train_model is a function that updates the model parameters by
# SGD Since this model has many parameters, it would be tedious to
# manually create an update rule for each model parameter. We thus
# create the updates list by automatically looping over all
# (params[i], grads[i]) pairs.
updates = [
(param_i, param_i - learning_rate * grad_i)
for param_i, grad_i in zip(params, grads)
]
train_model = theano.function(
[index],
cost,
updates=updates,
givens={
x: train_set_x[index * batch_size: (index + 1) * batch_size],
y: train_set_y[index * batch_size: (index + 1) * batch_size]
}
)
# end-snippet-1
###############
# TRAIN MODEL #
###############
print '... training'
# early-stopping parameters
patience = 10000 # look as this many examples regardless
patience_increase = 2 # wait this much longer when a new best is
# found
improvement_threshold = 0.995 # a relative improvement of this much is
# considered significant
validation_frequency = min(n_train_batches, patience / 2)
# go through this many
# minibatche before checking the network
# on the validation set; in this case we
# check every epoch
best_validation_loss = numpy.inf
best_iter = 0
test_score = 0.
start_time = time.clock()
epoch = 0
done_looping = False
while (epoch < n_epochs) and (not done_looping):
epoch = epoch + 1
for minibatch_index in xrange(n_train_batches):
iter = (epoch - 1) * n_train_batches + minibatch_index
if iter % 100 == 0:
print 'training @ iter = ', iter
cost_ij = train_model(minibatch_index)
if (iter + 1) % validation_frequency == 0:
# compute zero-one loss on validation set
validation_losses = [validate_model(i) for i
in xrange(n_valid_batches)]
this_validation_loss = numpy.mean(validation_losses)
print('epoch %i, minibatch %i/%i, validation error %f %%' %
(epoch, minibatch_index + 1, n_train_batches,
this_validation_loss * 100.))
# if we got the best validation score until now
if this_validation_loss < best_validation_loss:
#improve patience if loss improvement is good enough
if this_validation_loss < best_validation_loss *
improvement_threshold:
patience = max(patience, iter * patience_increase)
# save best validation score and iteration number
best_validation_loss = this_validation_loss
best_iter = iter
# test it on the test set
test_losses = [
test_model(i)
for i in xrange(n_test_batches)
]
test_score = numpy.mean(test_losses)
print((' epoch %i, minibatch %i/%i, test error of '
'best model %f %%') %
(epoch, minibatch_index + 1, n_train_batches,
test_score * 100.))
if patience <= iter:
done_looping = True
break
end_time = time.clock()
print('Optimization complete.')
print('Best validation score of %f %% obtained at iteration %i, '
'with test performance %f %%' %
(best_validation_loss * 100., best_iter + 1, test_score * 100.))
print >> sys.stderr, ('The code for file ' +
os.path.split(__file__)[1] +
' ran for %.2fm' % ((end_time - start_time) / 60.))
if __name__ == '__main__':
evaluate_lenet5()
def experiment(state, channel):
evaluate_lenet5(state.learning_rate, dataset=state.dataset)
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