在http://blog.csdn.net/fengbingchun/article/details/50814710中给出了CNN的简单实现，这里对每一步的实现作个说明：
共7层：依次为输入层、C1层、S2层、C3层、S4层、C5层、输出层。C代表卷积层(特征提取)。S代表降採样层或池化层(Pooling)，输出层为全连接层。
1. 各层权值、偏置(阈值)初始化：
各层权值、偏置个数计算例如以下：
(1)、输入层：预处理后的32*32图像数据。无权值和偏置；
(2)、C1层：卷积窗大小5*5，输出特征图数量6，卷积窗种类1*6=6。输出特征图大小28*28，因此可训练參数(权值+偏置)：(5*5*1)*6+6=150+6。
(3)、S2层：卷积窗大小2*2。输出下採样图数量6，卷积窗种类6，输出下採样图大小14*14，因此可训练參数(权值+偏置)：1*6+6=6+6。
(4)、C3层：卷积窗大小5*5。输出特征图数量16。卷积窗种类6*16=96，输出特征图大小10*10。因此可训练參数(权值+偏置)：(5*5*6)*16+16=2400+16。
(5)、S4层：卷积窗大小2*2。输出下採样图数量16，卷积窗种类16，输出下採样图大小5*5，因此可训练參数(权值+偏置)：1*16+16=16+16。
(6)、C5层：卷积窗大小5*5。输出特征图数量120，卷积窗种类16*120=1920，输出特征图大小1*1，因此可训练參数(权值+偏置)：(5*5*16)*120+120=48000+120；
(7)、输出层：卷积窗大小1*1，输出特征图数量10。卷积窗种类120*10=1200，输出特征图大小1*1，因此可训练參数(权值+偏置)：(1*120)*10+10=1200+10.
代码段例如以下：
#define num_map_input_CNN 1 //输入层map个数 #define num_map_C1_CNN 6 //C1层map个数 #define num_map_S2_CNN 6 //S2层map个数 #define num_map_C3_CNN 16 //C3层map个数 #define num_map_S4_CNN 16 //S4层map个数 #define num_map_C5_CNN 120 //C5层map个数 #define num_map_output_CNN 10 //输出层map个数 #define len_weight_C1_CNN 150 //C1层权值数，(5*5*1)*6=150 #define len_bias_C1_CNN 6 //C1层阈值数，6 #define len_weight_S2_CNN 6 //S2层权值数,1*6=6 #define len_bias_S2_CNN 6 //S2层阈值数,6 #define len_weight_C3_CNN 2400 //C3层权值数，(5*5*6)*16=2400 #define len_bias_C3_CNN 16 //C3层阈值数,16 #define len_weight_S4_CNN 16 //S4层权值数。1*16=16 #define len_bias_S4_CNN 16 //S4层阈值数。16 #define len_weight_C5_CNN 48000 //C5层权值数，(5*5*16)*120=48000 #define len_bias_C5_CNN 120 //C5层阈值数，120 #define len_weight_output_CNN 1200 //输出层权值数。(1*120)*10=1200 #define len_bias_output_CNN 10 //输出层阈值数，10 #define num_neuron_input_CNN 1024 //输入层神经元数，(32*32)*1=1024 #define num_neuron_C1_CNN 4704 //C1层神经元数，(28*28)*6=4704 #define num_neuron_S2_CNN 1176 //S2层神经元数。(14*14)*6=1176 #define num_neuron_C3_CNN 1600 //C3层神经元数。(10*10)*16=1600 #define num_neuron_S4_CNN 400 //S4层神经元数。(5*5)*16=400 #define num_neuron_C5_CNN 120 //C5层神经元数。(1*1)*120=120 #define num_neuron_output_CNN 10 //输出层神经元数，(1*1)*10=10
权值、偏置初始化：
(1)、权值使用函数uniform_real_distribution均匀分布初始化。tiny-cnn中每次初始化权值数值都同样。这里作了调整，使每次初始化的权值均不同。每层权值初始化大小范围都不一样；
(2)、全部层的偏置均初始化为0.
代码段例如以下：
double CNN::uniform_rand(double min, double max)
{
//static std::mt19937 gen(1);
std::random_device rd;
std::mt19937 gen(rd());
std::uniform_real_distribution<double> dst(min, max);
return dst(gen);
}
bool CNN::uniform_rand(double* src, int len, double min, double max)
{
for (int i = 0; i < len; i++) {
src[i] = uniform_rand(min, max);
}
return true;
}
bool CNN::initWeightThreshold()
{
srand(time(0) + rand());
const double scale = 6.0;
double min_ = -std::sqrt(scale / (25.0 + 150.0));
double max_ = std::sqrt(scale / (25.0 + 150.0));
uniform_rand(weight_C1, len_weight_C1_CNN, min_, max_);
for (int i = 0; i < len_bias_C1_CNN; i++) {
bias_C1[i] = 0.0;
}
min_ = -std::sqrt(scale / (4.0 + 1.0));
max_ = std::sqrt(scale / (4.0 + 1.0));
uniform_rand(weight_S2, len_weight_S2_CNN, min_, max_);
for (int i = 0; i < len_bias_S2_CNN; i++) {
bias_S2[i] = 0.0;
}
min_ = -std::sqrt(scale / (150.0 + 400.0));
max_ = std::sqrt(scale / (150.0 + 400.0));
uniform_rand(weight_C3, len_weight_C3_CNN, min_, max_);
for (int i = 0; i < len_bias_C3_CNN; i++) {
bias_C3[i] = 0.0;
}
min_ = -std::sqrt(scale / (4.0 + 1.0));
max_ = std::sqrt(scale / (4.0 + 1.0));
uniform_rand(weight_S4, len_weight_S4_CNN, min_, max_);
for (int i = 0; i < len_bias_S4_CNN; i++) {
bias_S4[i] = 0.0;
}
min_ = -std::sqrt(scale / (400.0 + 3000.0));
max_ = std::sqrt(scale / (400.0 + 3000.0));
uniform_rand(weight_C5, len_weight_C5_CNN, min_, max_);
for (int i = 0; i < len_bias_C5_CNN; i++) {
bias_C5[i] = 0.0;
}
min_ = -std::sqrt(scale / (120.0 + 10.0));
max_ = std::sqrt(scale / (120.0 + 10.0));
uniform_rand(weight_output, len_weight_output_CNN, min_, max_);
for (int i = 0; i < len_bias_output_CNN; i++) {
bias_output[i] = 0.0;
}
return true;
}2. 载入MNIST数据：
关于MNIST的介绍能够參考：http://blog.csdn.net/fengbingchun/article/details/49611549
使用MNIST库作为训练集和測试集。训练样本集为60000个，測试样本集为10000个。
(1)、MNIST库中图像原始大小为28*28，这里缩放为32*32，数据取值范围为[-1,1]，扩充值均取-1，作为输入层输入数据。
代码段例如以下：
static void readMnistImages(std::string filename, double* data_dst, int num_image)
{
const int width_src_image = 28;
const int height_src_image = 28;
const int x_padding = 2;
const int y_padding = 2;
const double scale_min = -1;
const double scale_max = 1;
std::ifstream file(filename, std::ios::binary);
assert(file.is_open());
int magic_number = 0;
int number_of_images = 0;
int n_rows = 0;
int n_cols = 0;
file.read((char*)&magic_number, sizeof(magic_number));
magic_number = reverseInt(magic_number);
file.read((char*)&number_of_images, sizeof(number_of_images));
number_of_images = reverseInt(number_of_images);
assert(number_of_images == num_image);
file.read((char*)&n_rows, sizeof(n_rows));
n_rows = reverseInt(n_rows);
file.read((char*)&n_cols, sizeof(n_cols));
n_cols = reverseInt(n_cols);
assert(n_rows == height_src_image && n_cols == width_src_image);
int size_single_image = width_image_input_CNN * height_image_input_CNN;
for (int i = 0; i < number_of_images; ++i) {
int addr = size_single_image * i;
for (int r = 0; r < n_rows; ++r) {
for (int c = 0; c < n_cols; ++c) {
unsigned char temp = 0;
file.read((char*)&temp, sizeof(temp));
data_dst[addr + width_image_input_CNN * (r + y_padding) + c + x_padding] = (temp / 255.0) * (scale_max - scale_min) + scale_min;
}
}
}
}(2)、对于Label,输出层有10个节点，相应位置的节点值设为0.8。其他节点设为-0.8，作为输出层数据。
代码段例如以下：
static void readMnistLabels(std::string filename, double* data_dst, int num_image)
{
const double scale_max = 0.8;
std::ifstream file(filename, std::ios::binary);
assert(file.is_open());
int magic_number = 0;
int number_of_images = 0;
file.read((char*)&magic_number, sizeof(magic_number));
magic_number = reverseInt(magic_number);
file.read((char*)&number_of_images, sizeof(number_of_images));
number_of_images = reverseInt(number_of_images);
assert(number_of_images == num_image);
for (int i = 0; i < number_of_images; ++i) {
unsigned char temp = 0;
file.read((char*)&temp, sizeof(temp));
data_dst[i * num_map_output_CNN + temp] = scale_max;
}
}static void readMnistLabels(std::string filename, double* data_dst, int num_image)
{
const double scale_max = 0.8;
std::ifstream file(filename, std::ios::binary);
assert(file.is_open());
int magic_number = 0;
int number_of_images = 0;
file.read((char*)&magic_number, sizeof(magic_number));
magic_number = reverseInt(magic_number);
file.read((char*)&number_of_images, sizeof(number_of_images));
number_of_images = reverseInt(number_of_images);
assert(number_of_images == num_image);
for (int i = 0; i < number_of_images; ++i) {
unsigned char temp = 0;
file.read((char*)&temp, sizeof(temp));
data_dst[i * num_map_output_CNN + temp] = scale_max;
}
}3. 前向传播：主要计算每层的神经元值。当中C1层、C3层、C5层操作过程同样。S2层、S4层操作过程同样。
(1)、输入层：神经元数为(32*32)*1=1024。
(2)、C1层：神经元数为(28*28)*6=4704，分别用每个5*5的卷积图像去乘以32*32的图像，获得一个28*28的图像。即相应位置相加再求和，stride长度为1；一共6个5*5的卷积图像，然后对每个神经元加上一个阈值。最后再通过tanh激活函数对每一神经元进行运算得到终于每个神经元的结果。
激活函数的作用：它是用来增加非线性因素的，解决线性模型所不能解决的问题。提供网络的非线性建模能力。
假设没有激活函数。那么该网络仅能够表达线性映射。此时即便有再多的隐藏层，其整个网络跟单层神经网络也是等价的。因此也能够觉得，唯独增加了激活函数之后，深度神经网络才具备了分层的非线性映射学习能力。
代码段例如以下：
double CNN::activation_function_tanh(double x)
{
double ep = std::exp(x);
double em = std::exp(-x);
return (ep - em) / (ep + em);
}
bool CNN::Forward_C1()
{
init_variable(neuron_C1, 0.0, num_neuron_C1_CNN);
for (int o = 0; o < num_map_C1_CNN; o++) {
for (int inc = 0; inc < num_map_input_CNN; inc++) {
int addr1 = get_index(0, 0, num_map_input_CNN * o + inc, width_kernel_conv_CNN, height_kernel_conv_CNN, num_map_C1_CNN * num_map_input_CNN);
int addr2 = get_index(0, 0, inc, width_image_input_CNN, height_image_input_CNN, num_map_input_CNN);
int addr3 = get_index(0, 0, o, width_image_C1_CNN, height_image_C1_CNN, num_map_C1_CNN);
const double* pw = &weight_C1[0] + addr1;
const double* pi = data_single_image + addr2;
double* pa = &neuron_C1[0] + addr3;
for (int y = 0; y < height_image_C1_CNN; y++) {
for (int x = 0; x < width_image_C1_CNN; x++) {
const double* ppw = pw;
const double* ppi = pi + y * width_image_input_CNN + x;
double sum = 0.0;
for (int wy = 0; wy < height_kernel_conv_CNN; wy++) {
for (int wx = 0; wx < width_kernel_conv_CNN; wx++) {
sum += *ppw++ * ppi[wy * width_image_input_CNN + wx];
}
}
pa[y * width_image_C1_CNN + x] += sum;
}
}
}
int addr3 = get_index(0, 0, o, width_image_C1_CNN, height_image_C1_CNN, num_map_C1_CNN);
double* pa = &neuron_C1[0] + addr3;
double b = bias_C1[o];
for (int y = 0; y < height_image_C1_CNN; y++) {
for (int x = 0; x < width_image_C1_CNN; x++) {
pa[y * width_image_C1_CNN + x] += b;
}
}
}
for (int i = 0; i < num_neuron_C1_CNN; i++) {
neuron_C1[i] = activation_function_tanh(neuron_C1[i]);
}
return true;
}(3)、S2层：神经元数为(14*14)*6=1176，对C1中6个28*28的特征图生成6个14*14的下採样图，相邻四个神经元分别乘以同一个权值再进行相加求和，再求均值即除以4，然后再加上一个阈值，最后再通过tanh激活函数对每一神经元进行运算得到终于每个神经元的结果。
代码段例如以下：
bool CNN::Forward_S2()
{
init_variable(neuron_S2, 0.0, num_neuron_S2_CNN);
double scale_factor = 1.0 / (width_kernel_pooling_CNN * height_kernel_pooling_CNN);
assert(out2wi_S2.size() == num_neuron_S2_CNN);
assert(out2bias_S2.size() == num_neuron_S2_CNN);
for (int i = 0; i < num_neuron_S2_CNN; i++) {
const wi_connections& connections = out2wi_S2[i];
neuron_S2[i] = 0;
for (int index = 0; index < connections.size(); index++) {
neuron_S2[i] += weight_S2[connections[index].first] * neuron_C1[connections[index].second];
}
neuron_S2[i] *= scale_factor;
neuron_S2[i] += bias_S2[out2bias_S2[i]];
}
for (int i = 0; i < num_neuron_S2_CNN; i++) {
neuron_S2[i] = activation_function_tanh(neuron_S2[i]);
}
return true;
}(4)、C3层：神经元数为(10*10)*16=1600。C3层实现方式与C1层全然同样。由S2中的6个14*14下採样图生成16个10*10特征图，对于生成的每个10*10的特征图，是由6个5*5的卷积图像去乘以6个14*14的下採样图，然后相应位置相加求和，然后对每个神经元加上一个阈值，最后再通过tanh激活函数对每一神经元进行运算得到终于每个神经元的结果。
也可依照Y.Lecun给出的表进行计算。即对于生成的每个10*10的特征图，是由n个5*5的卷积图像去乘以n个14*14的下採样图，当中n是小于6的，即不全然连接。这样做的原因：第一，不全然的连接机制将连接的数量保持在合理的范围内。第二，也是最重要的，其破坏了网络的对称性。因为不同的特征图有不同的输入，所以迫使他们抽取不同的特征。
代码段例如以下：
// connection table [Y.Lecun, 1998 Table.1]
#define O true
#define X false
static const bool tbl[6][16] = {
O, X, X, X, O, O, O, X, X, O, O, O, O, X, O, O,
O, O, X, X, X, O, O, O, X, X, O, O, O, O, X, O,
O, O, O, X, X, X, O, O, O, X, X, O, X, O, O, O,
X, O, O, O, X, X, O, O, O, O, X, X, O, X, O, O,
X, X, O, O, O, X, X, O, O, O, O, X, O, O, X, O,
X, X, X, O, O, O, X, X, O, O, O, O, X, O, O, O
};
#undef O
#undef X
bool CNN::Forward_C3()
{
init_variable(neuron_C3, 0.0, num_neuron_C3_CNN);
for (int o = 0; o < num_map_C3_CNN; o++) {
for (int inc = 0; inc < num_map_S2_CNN; inc++) {
if (!tbl[inc][o]) continue;
int addr1 = get_index(0, 0, num_map_S2_CNN * o + inc, width_kernel_conv_CNN, height_kernel_conv_CNN, num_map_C3_CNN * num_map_S2_CNN);
int addr2 = get_index(0, 0, inc, width_image_S2_CNN, height_image_S2_CNN, num_map_S2_CNN);
int addr3 = get_index(0, 0, o, width_image_C3_CNN, height_image_C3_CNN, num_map_C3_CNN);
const double* pw = &weight_C3[0] + addr1;
const double* pi = &neuron_S2[0] + addr2;
double* pa = &neuron_C3[0] + addr3;
for (int y = 0; y < height_image_C3_CNN; y++) {
for (int x = 0; x < width_image_C3_CNN; x++) {
const double* ppw = pw;
const double* ppi = pi + y * width_image_S2_CNN + x;
double sum = 0.0;
for (int wy = 0; wy < height_kernel_conv_CNN; wy++) {
for (int wx = 0; wx < width_kernel_conv_CNN; wx++) {
sum += *ppw++ * ppi[wy * width_image_S2_CNN + wx];
}
}
pa[y * width_image_C3_CNN + x] += sum;
}
}
}
int addr3 = get_index(0, 0, o, width_image_C3_CNN, height_image_C3_CNN, num_map_C3_CNN);
double* pa = &neuron_C3[0] + addr3;
double b = bias_C3[o];
for (int y = 0; y < height_image_C3_CNN; y++) {
for (int x = 0; x < width_image_C3_CNN; x++) {
pa[y * width_image_C3_CNN + x] += b;
}
}
}
for (int i = 0; i < num_neuron_C3_CNN; i++) {
neuron_C3[i] = activation_function_tanh(neuron_C3[i]);
}
return true;
}(5)、S4层：神经元数为(5*5)*16=400，S4层实现方式与S2层全然同样。由C3中16个10*10的特征图生成16个5*5下採样图，相邻四个神经元分别乘以同一个权值再进行相加求和，再求均值即除以4，然后再加上一个阈值。最后再通过tanh激活函数对每一神经元进行运算得到终于每个神经元的结果。
代码段例如以下：
bool CNN::Forward_S4()
{
double scale_factor = 1.0 / (width_kernel_pooling_CNN * height_kernel_pooling_CNN);
init_variable(neuron_S4, 0.0, num_neuron_S4_CNN);
assert(out2wi_S4.size() == num_neuron_S4_CNN);
assert(out2bias_S4.size() == num_neuron_S4_CNN);
for (int i = 0; i < num_neuron_S4_CNN; i++) {
const wi_connections& connections = out2wi_S4[i];
neuron_S4[i] = 0.0;
for (int index = 0; index < connections.size(); index++) {
neuron_S4[i] += weight_S4[connections[index].first] * neuron_C3[connections[index].second];
}
neuron_S4[i] *= scale_factor;
neuron_S4[i] += bias_S4[out2bias_S4[i]];
}
for (int i = 0; i < num_neuron_S4_CNN; i++) {
neuron_S4[i] = activation_function_tanh(neuron_S4[i]);
}
return true;
}(6)、C5层：神经元数为(1*1)*120=120，也可看为全连接层，C5层实现方式与C1、C3层全然同样。由S4中16个5*5下採样图生成120个1*1特征图，对于生成的每个1*1的特征图，是由16个5*5的卷积图像去乘以16个5*5的下採用图，然后相加求和，然后对每个神经元加上一个阈值，最后再通过tanh激活函数对每一神经元进行运算得到终于每个神经元的结果。
代码段例如以下：
bool CNN::Forward_C5()
{
init_variable(neuron_C5, 0.0, num_neuron_C5_CNN);
for (int o = 0; o < num_map_C5_CNN; o++) {
for (int inc = 0; inc < num_map_S4_CNN; inc++) {
int addr1 = get_index(0, 0, num_map_S4_CNN * o + inc, width_kernel_conv_CNN, height_kernel_conv_CNN, num_map_C5_CNN * num_map_S4_CNN);
int addr2 = get_index(0, 0, inc, width_image_S4_CNN, height_image_S4_CNN, num_map_S4_CNN);
int addr3 = get_index(0, 0, o, width_image_C5_CNN, height_image_C5_CNN, num_map_C5_CNN);
const double *pw = &weight_C5[0] + addr1;
const double *pi = &neuron_S4[0] + addr2;
double *pa = &neuron_C5[0] + addr3;
for (int y = 0; y < height_image_C5_CNN; y++) {
for (int x = 0; x < width_image_C5_CNN; x++) {
const double *ppw = pw;
const double *ppi = pi + y * width_image_S4_CNN + x;
double sum = 0.0;
for (int wy = 0; wy < height_kernel_conv_CNN; wy++) {
for (int wx = 0; wx < width_kernel_conv_CNN; wx++) {
sum += *ppw++ * ppi[wy * width_image_S4_CNN + wx];
}
}
pa[y * width_image_C5_CNN + x] += sum;
}
}
}
int addr3 = get_index(0, 0, o, width_image_C5_CNN, height_image_C5_CNN, num_map_C5_CNN);
double *pa = &neuron_C5[0] + addr3;
double b = bias_C5[o];
for (int y = 0; y < height_image_C5_CNN; y++) {
for (int x = 0; x < width_image_C5_CNN; x++) {
pa[y * width_image_C5_CNN + x] += b;
}
}
}
for (int i = 0; i < num_neuron_C5_CNN; i++) {
neuron_C5[i] = activation_function_tanh(neuron_C5[i]);
}
return true;
}(7)、输出层：神经元数为(1*1)*10=10。为全连接层。输出层中的每个神经元均是由C5层中的120个神经元乘以相相应的权值。然后相加求和；然后对每个神经元加上一个阈值。最后再通过tanh激活函数对每一神经元进行运算得到终于每个神经元的结果。
代码段例如以下：
bool CNN::Forward_output()
{
init_variable(neuron_output, 0.0, num_neuron_output_CNN);
for (int i = 0; i < num_neuron_output_CNN; i++) {
neuron_output[i] = 0.0;
for (int c = 0; c < num_neuron_C5_CNN; c++) {
neuron_output[i] += weight_output[c * num_neuron_output_CNN + i] * neuron_C5[c];
}
neuron_output[i] += bias_output[i];
}
for (int i = 0; i < num_neuron_output_CNN; i++) {
neuron_output[i] = activation_function_tanh(neuron_output[i]);
}
return true;
}4. 反向传播：主要计算每层权值和偏置的误差以及每层神经元的误差；当中输入层、S2层、S4层操作过程同样。C1层、C3层操作过程同样。
(1)、输出层：计算输出层神经元误差；通过mse损失函数的导数函数和tanh激活函数的导数函数来计算输出层神经元误差，即a、已计算出的输出层神经元值减去相应label值，b、1.0减去输出层神经元值的平方，c、a与c的乘积和。
损失函数作用：在统计学中损失函数是一种衡量损失和错误(这样的损失与”错误地”预计有关)程度的函数。损失函数在实践中最重要的运用。在于协助我们通过过程的改善而持续降低目标值的变异。并不是只追求符合逻辑。
在深度学习中，对于损失函数的收敛特性。我们期望是当误差越大的时候。收敛(学习)速度应该越快。成为损失函数须要满足两点要求：非负性；预測值和期望值接近时，函数值趋于0.
代码段例如以下：
double CNN::loss_function_mse_derivative(double y, double t)
{
return (y - t);
}
void CNN::loss_function_gradient(const double* y, const double* t, double* dst, int len)
{
for (int i = 0; i < len; i++) {
dst[i] = loss_function_mse_derivative(y[i], t[i]);
}
}
double CNN::activation_function_tanh_derivative(double x)
{
return (1.0 - x * x);
}
double CNN::dot_product(const double* s1, const double* s2, int len)
{
double result = 0.0;
for (int i = 0; i < len; i++) {
result += s1[i] * s2[i];
}
return result;
}
bool CNN::Backward_output()
{
init_variable(delta_neuron_output, 0.0, num_neuron_output_CNN);
double dE_dy[num_neuron_output_CNN];
init_variable(dE_dy, 0.0, num_neuron_output_CNN);
loss_function_gradient(neuron_output, data_single_label, dE_dy, num_neuron_output_CNN); // 损失函数: mean squared error(均方差)
// delta = dE/da = (dE/dy) * (dy/da)
for (int i = 0; i < num_neuron_output_CNN; i++) {
double dy_da[num_neuron_output_CNN];
init_variable(dy_da, 0.0, num_neuron_output_CNN);
dy_da[i] = activation_function_tanh_derivative(neuron_output[i]);
delta_neuron_output[i] = dot_product(dE_dy, dy_da, num_neuron_output_CNN);
}
return true;
}(2)、C5层：计算C5层神经元误差、输出层权值误差、输出层偏置误差；通过输出层神经元误差乘以输出层权值。求和。结果再乘以C5层神经元的tanh激活函数的导数(即1-C5层神经元值的平方)。获得C5层每个神经元误差。通过输出层神经元误差乘以C5层神经元获得输出层权值误差；输出层偏置误差即为输出层神经元误差。
代码段例如以下：
bool CNN::muladd(const double* src, double c, int len, double* dst)
{
for (int i = 0; i < len; i++) {
dst[i] += (src[i] * c);
}
return true;
}
bool CNN::Backward_C5()
{
init_variable(delta_neuron_C5, 0.0, num_neuron_C5_CNN);
init_variable(delta_weight_output, 0.0, len_weight_output_CNN);
init_variable(delta_bias_output, 0.0, len_bias_output_CNN);
for (int c = 0; c < num_neuron_C5_CNN; c++) {
// propagate delta to previous layer
// prev_delta[c] += current_delta[r] * W_[c * out_size_ + r]
delta_neuron_C5[c] = dot_product(&delta_neuron_output[0], &weight_output[c * num_neuron_output_CNN], num_neuron_output_CNN);
delta_neuron_C5[c] *= activation_function_tanh_derivative(neuron_C5[c]);
}
// accumulate weight-step using delta
// dW[c * out_size + i] += current_delta[i] * prev_out[c]
for (int c = 0; c < num_neuron_C5_CNN; c++) {
muladd(&delta_neuron_output[0], neuron_C5[c], num_neuron_output_CNN, &delta_weight_output[0] + c * num_neuron_output_CNN);
}
for (int i = 0; i < len_bias_output_CNN; i++) {
delta_bias_output[i] += delta_neuron_output[i];
}
return true;
}(3)、S4层：计算S4层神经元误差、C5层权值误差、C5层偏置误差；通过C5层权值乘以C5层神经元误差。求和。结果再乘以S4层神经元的tanh激活函数的导数(即1-S4神经元的平方)，获得S4层每个神经元误差。通过S4层神经元乘以C5层神经元误差，求和，获得C5层权值误差。C5层偏置误差即为C5层神经元误差。
代码段例如以下：
bool CNN::Backward_S4()
{
init_variable(delta_neuron_S4, 0.0, num_neuron_S4_CNN);
init_variable(delta_weight_C5, 0.0, len_weight_C5_CNN);
init_variable(delta_bias_C5, 0.0, len_bias_C5_CNN);
// propagate delta to previous layer
for (int inc = 0; inc < num_map_S4_CNN; inc++) {
for (int outc = 0; outc < num_map_C5_CNN; outc++) {
int addr1 = get_index(0, 0, num_map_S4_CNN * outc + inc, width_kernel_conv_CNN, height_kernel_conv_CNN, num_map_S4_CNN * num_map_C5_CNN);
int addr2 = get_index(0, 0, outc, width_image_C5_CNN, height_image_C5_CNN, num_map_C5_CNN);
int addr3 = get_index(0, 0, inc, width_image_S4_CNN, height_image_S4_CNN, num_map_S4_CNN);
const double* pw = &weight_C5[0] + addr1;
const double* pdelta_src = &delta_neuron_C5[0] + addr2;
double* pdelta_dst = &delta_neuron_S4[0] + addr3;
for (int y = 0; y < height_image_C5_CNN; y++) {
for (int x = 0; x < width_image_C5_CNN; x++) {
const double* ppw = pw;
const double ppdelta_src = pdelta_src[y * width_image_C5_CNN + x];
double* ppdelta_dst = pdelta_dst + y * width_image_S4_CNN + x;
for (int wy = 0; wy < height_kernel_conv_CNN; wy++) {
for (int wx = 0; wx < width_kernel_conv_CNN; wx++) {
ppdelta_dst[wy * width_image_S4_CNN + wx] += *ppw++ * ppdelta_src;
}
}
}
}
}
}
for (int i = 0; i < num_neuron_S4_CNN; i++) {
delta_neuron_S4[i] *= activation_function_tanh_derivative(neuron_S4[i]);
}
// accumulate dw
for (int inc = 0; inc < num_map_S4_CNN; inc++) {
for (int outc = 0; outc < num_map_C5_CNN; outc++) {
for (int wy = 0; wy < height_kernel_conv_CNN; wy++) {
for (int wx = 0; wx < width_kernel_conv_CNN; wx++) {
int addr1 = get_index(wx, wy, inc, width_image_S4_CNN, height_image_S4_CNN, num_map_S4_CNN);
int addr2 = get_index(0, 0, outc, width_image_C5_CNN, height_image_C5_CNN, num_map_C5_CNN);
int addr3 = get_index(wx, wy, num_map_S4_CNN * outc + inc, width_kernel_conv_CNN, height_kernel_conv_CNN, num_map_S4_CNN * num_map_C5_CNN);
double dst = 0.0;
const double* prevo = &neuron_S4[0] + addr1;
const double* delta = &delta_neuron_C5[0] + addr2;
for (int y = 0; y < height_image_C5_CNN; y++) {
dst += dot_product(prevo + y * width_image_S4_CNN, delta + y * width_image_C5_CNN, width_image_C5_CNN);
}
delta_weight_C5[addr3] += dst;
}
}
}
}
// accumulate db
for (int outc = 0; outc < num_map_C5_CNN; outc++) {
int addr2 = get_index(0, 0, outc, width_image_C5_CNN, height_image_C5_CNN, num_map_C5_CNN);
const double* delta = &delta_neuron_C5[0] + addr2;
for (int y = 0; y < height_image_C5_CNN; y++) {
for (int x = 0; x < width_image_C5_CNN; x++) {
delta_bias_C5[outc] += delta[y * width_image_C5_CNN + x];
}
}
}
return true;
}(4)、C3层：计算C3层神经元误差、S4层权值误差、S4层偏置误差。通过S4层权值乘以S4层神经元误差。求和，结果再乘以C3层神经元的tanh激活函数的导数(即1-S4神经元的平方)，然后再乘以1/4。获得C3层每个神经元误差；通过C3层神经元乘以S4神经元误差，求和。再乘以1/4。获得S4层权值误差；通过S4层神经元误差求和，来获得S4层偏置误差。
代码段例如以下：
bool CNN::Backward_C3()
{
init_variable(delta_neuron_C3, 0.0, num_neuron_C3_CNN);
init_variable(delta_weight_S4, 0.0, len_weight_S4_CNN);
init_variable(delta_bias_S4, 0.0, len_bias_S4_CNN);
double scale_factor = 1.0 / (width_kernel_pooling_CNN * height_kernel_pooling_CNN);
assert(in2wo_C3.size() == num_neuron_C3_CNN);
assert(weight2io_C3.size() == len_weight_S4_CNN);
assert(bias2out_C3.size() == len_bias_S4_CNN);
for (int i = 0; i < num_neuron_C3_CNN; i++) {
const wo_connections& connections = in2wo_C3[i];
double delta = 0.0;
for (int j = 0; j < connections.size(); j++) {
delta += weight_S4[connections[j].first] * delta_neuron_S4[connections[j].second];
}
delta_neuron_C3[i] = delta * scale_factor * activation_function_tanh_derivative(neuron_C3[i]);
}
for (int i = 0; i < len_weight_S4_CNN; i++) {
const io_connections& connections = weight2io_C3[i];
double diff = 0;
for (int j = 0; j < connections.size(); j++) {
diff += neuron_C3[connections[j].first] * delta_neuron_S4[connections[j].second];
}
delta_weight_S4[i] += diff * scale_factor;
}
for (int i = 0; i < len_bias_S4_CNN; i++) {
const std::vector<int>& outs = bias2out_C3[i];
double diff = 0;
for (int o = 0; o < outs.size(); o++) {
diff += delta_neuron_S4[outs[o]];
}
delta_bias_S4[i] += diff;
}
return true;
}(5)、S2层：计算S2层神经元误差、C3层权值误差、C3层偏置误差。通过C3层权值乘以C3层神经元误差。求和，结果再乘以S2层神经元的tanh激活函数的导数(即1-S2神经元的平方)，获得S2层每个神经元误差；通过S2层神经元乘以C3层神经元误差。求和，获得C3层权值误差；C3层偏置误差即为C3层神经元误差和。
代码段例如以下：
bool CNN::Backward_S2()
{
init_variable(delta_neuron_S2, 0.0, num_neuron_S2_CNN);
init_variable(delta_weight_C3, 0.0, len_weight_C3_CNN);
init_variable(delta_bias_C3, 0.0, len_bias_C3_CNN);
// propagate delta to previous layer
for (int inc = 0; inc < num_map_S2_CNN; inc++) {
for (int outc = 0; outc < num_map_C3_CNN; outc++) {
if (!tbl[inc][outc]) continue;
int addr1 = get_index(0, 0, num_map_S2_CNN * outc + inc, width_kernel_conv_CNN, height_kernel_conv_CNN, num_map_S2_CNN * num_map_C3_CNN);
int addr2 = get_index(0, 0, outc, width_image_C3_CNN, height_image_C3_CNN, num_map_C3_CNN);
int addr3 = get_index(0, 0, inc, width_image_S2_CNN, height_image_S2_CNN, num_map_S2_CNN);
const double *pw = &weight_C3[0] + addr1;
const double *pdelta_src = &delta_neuron_C3[0] + addr2;;
double* pdelta_dst = &delta_neuron_S2[0] + addr3;
for (int y = 0; y < height_image_C3_CNN; y++) {
for (int x = 0; x < width_image_C3_CNN; x++) {
const double* ppw = pw;
const double ppdelta_src = pdelta_src[y * width_image_C3_CNN + x];
double* ppdelta_dst = pdelta_dst + y * width_image_S2_CNN + x;
for (int wy = 0; wy < height_kernel_conv_CNN; wy++) {
for (int wx = 0; wx < width_kernel_conv_CNN; wx++) {
ppdelta_dst[wy * width_image_S2_CNN + wx] += *ppw++ * ppdelta_src;
}
}
}
}
}
}
for (int i = 0; i < num_neuron_S2_CNN; i++) {
delta_neuron_S2[i] *= activation_function_tanh_derivative(neuron_S2[i]);
}
// accumulate dw
for (int inc = 0; inc < num_map_S2_CNN; inc++) {
for (int outc = 0; outc < num_map_C3_CNN; outc++) {
if (!tbl[inc][outc]) continue;
for (int wy = 0; wy < height_kernel_conv_CNN; wy++) {
for (int wx = 0; wx < width_kernel_conv_CNN; wx++) {
int addr1 = get_index(wx, wy, inc, width_image_S2_CNN, height_image_S2_CNN, num_map_S2_CNN);
int addr2 = get_index(0, 0, outc, width_image_C3_CNN, height_image_C3_CNN, num_map_C3_CNN);
int addr3 = get_index(wx, wy, num_map_S2_CNN * outc + inc, width_kernel_conv_CNN, height_kernel_conv_CNN, num_map_S2_CNN * num_map_C3_CNN);
double dst = 0.0;
const double* prevo = &neuron_S2[0] + addr1;
const double* delta = &delta_neuron_C3[0] + addr2;
for (int y = 0; y < height_image_C3_CNN; y++) {
dst += dot_product(prevo + y * width_image_S2_CNN, delta + y * width_image_C3_CNN, width_image_C3_CNN);
}
delta_weight_C3[addr3] += dst;
}
}
}
}
// accumulate db
for (int outc = 0; outc < len_bias_C3_CNN; outc++) {
int addr1 = get_index(0, 0, outc, width_image_C3_CNN, height_image_C3_CNN, num_map_C3_CNN);
const double* delta = &delta_neuron_C3[0] + addr1;
for (int y = 0; y < height_image_C3_CNN; y++) {
for (int x = 0; x < width_image_C3_CNN; x++) {
delta_bias_C3[outc] += delta[y * width_image_C3_CNN + x];
}
}
}
return true;
}(6)、C1层：计算C1层神经元误差、S2层权值误差、S2层偏置误差；通过S2层权值乘以S2层神经元误差，求和。结果再乘以C1层神经元的tanh激活函数的导数(即1-C1神经元的平方)，然后再乘以1/4，获得C1层每个神经元误差；通过C1层神经元乘以S2神经元误差，求和。再乘以1/4，获得S2层权值误差；通过S2层神经元误差求和，来获得S4层偏置误差。
代码段例如以下：
bool CNN::Backward_C1()
{
init_variable(delta_neuron_C1, 0.0, num_neuron_C1_CNN);
init_variable(delta_weight_S2, 0.0, len_weight_S2_CNN);
init_variable(delta_bias_S2, 0.0, len_bias_S2_CNN);
double scale_factor = 1.0 / (width_kernel_pooling_CNN * height_kernel_pooling_CNN);
assert(in2wo_C1.size() == num_neuron_C1_CNN);
assert(weight2io_C1.size() == len_weight_S2_CNN);
assert(bias2out_C1.size() == len_bias_S2_CNN);
for (int i = 0; i < num_neuron_C1_CNN; i++) {
const wo_connections& connections = in2wo_C1[i];
double delta = 0.0;
for (int j = 0; j < connections.size(); j++) {
delta += weight_S2[connections[j].first] * delta_neuron_S2[connections[j].second];
}
delta_neuron_C1[i] = delta * scale_factor * activation_function_tanh_derivative(neuron_C1[i]);
}
for (int i = 0; i < len_weight_S2_CNN; i++) {
const io_connections& connections = weight2io_C1[i];
double diff = 0.0;
for (int j = 0; j < connections.size(); j++) {
diff += neuron_C1[connections[j].first] * delta_neuron_S2[connections[j].second];
}
delta_weight_S2[i] += diff * scale_factor;
}
for (int i = 0; i < len_bias_S2_CNN; i++) {
const std::vector<int>& outs = bias2out_C1[i];
double diff = 0;
for (int o = 0; o < outs.size(); o++) {
diff += delta_neuron_S2[outs[o]];
}
delta_bias_S2[i] += diff;
}
return true;
}(7)、输入层：计算输入层神经元误差、C1层权值误差、C1层偏置误差；通过C1层权值乘以C1层神经元误差。求和。结果再乘以输入层神经元的tanh激活函数的导数(即1-输入层神经元的平方)，获得输入层每个神经元误差；通过输入层层神经元乘以C1层神经元误差，求和。获得C1层权值误差；C1层偏置误差即为C1层神经元误差和。
bool CNN::Backward_input()
{
init_variable(delta_neuron_input, 0.0, num_neuron_input_CNN);
init_variable(delta_weight_C1, 0.0, len_weight_C1_CNN);
init_variable(delta_bias_C1, 0.0, len_bias_C1_CNN);
// propagate delta to previous layer
for (int inc = 0; inc < num_map_input_CNN; inc++) {
for (int outc = 0; outc < num_map_C1_CNN; outc++) {
int addr1 = get_index(0, 0, num_map_input_CNN * outc + inc, width_kernel_conv_CNN, height_kernel_conv_CNN, num_map_C1_CNN);
int addr2 = get_index(0, 0, outc, width_image_C1_CNN, height_image_C1_CNN, num_map_C1_CNN);
int addr3 = get_index(0, 0, inc, width_image_input_CNN, height_image_input_CNN, num_map_input_CNN);
const double* pw = &weight_C1[0] + addr1;
const double* pdelta_src = &delta_neuron_C1[0] + addr2;
double* pdelta_dst = &delta_neuron_input[0] + addr3;
for (int y = 0; y < height_image_C1_CNN; y++) {
for (int x = 0; x < width_image_C1_CNN; x++) {
const double* ppw = pw;
const double ppdelta_src = pdelta_src[y * width_image_C1_CNN + x];
double* ppdelta_dst = pdelta_dst + y * width_image_input_CNN + x;
for (int wy = 0; wy < height_kernel_conv_CNN; wy++) {
for (int wx = 0; wx < width_kernel_conv_CNN; wx++) {
ppdelta_dst[wy * width_image_input_CNN + wx] += *ppw++ * ppdelta_src;
}
}
}
}
}
}
for (int i = 0; i < num_neuron_input_CNN; i++) {
delta_neuron_input[i] *= activation_function_identity_derivative(data_single_image[i]/*neuron_input[i]*/);
}
// accumulate dw
for (int inc = 0; inc < num_map_input_CNN; inc++) {
for (int outc = 0; outc < num_map_C1_CNN; outc++) {
for (int wy = 0; wy < height_kernel_conv_CNN; wy++) {
for (int wx = 0; wx < width_kernel_conv_CNN; wx++) {
int addr1 = get_index(wx, wy, inc, width_image_input_CNN, height_image_input_CNN, num_map_input_CNN);
int addr2 = get_index(0, 0, outc, width_image_C1_CNN, height_image_C1_CNN, num_map_C1_CNN);
int addr3 = get_index(wx, wy, num_map_input_CNN * outc + inc, width_kernel_conv_CNN, height_kernel_conv_CNN, num_map_C1_CNN);
double dst = 0.0;
const double* prevo = data_single_image + addr1;//&neuron_input[0]
const double* delta = &delta_neuron_C1[0] + addr2;
for (int y = 0; y < height_image_C1_CNN; y++) {
dst += dot_product(prevo + y * width_image_input_CNN, delta + y * width_image_C1_CNN, width_image_C1_CNN);
}
delta_weight_C1[addr3] += dst;
}
}
}
}
// accumulate db
for (int outc = 0; outc < len_bias_C1_CNN; outc++) {
int addr1 = get_index(0, 0, outc, width_image_C1_CNN, height_image_C1_CNN, num_map_C1_CNN);
const double* delta = &delta_neuron_C1[0] + addr1;
for (int y = 0; y < height_image_C1_CNN; y++) {
for (int x = 0; x < width_image_C1_CNN; x++) {
delta_bias_C1[outc] += delta[y * width_image_C1_CNN + x];
}
}
}
return true;
}5. 更新各层权值、偏置：通过之前计算的各层权值、各层权值误差。各层偏置、各层偏置误差以及学习率来更新各层权值和偏置。
代码段例如以下：
void CNN::update_weights_bias(const double* delta, double* e_weight, double* weight, int len)
{
for (int i = 0; i < len; i++) {
e_weight[i] += delta[i] * delta[i];
weight[i] -= learning_rate_CNN * delta[i] / (std::sqrt(e_weight[i]) + eps_CNN);
}
}
bool CNN::UpdateWeights()
{
update_weights_bias(delta_weight_C1, E_weight_C1, weight_C1, len_weight_C1_CNN);
update_weights_bias(delta_bias_C1, E_bias_C1, bias_C1, len_bias_C1_CNN);
update_weights_bias(delta_weight_S2, E_weight_S2, weight_S2, len_weight_S2_CNN);
update_weights_bias(delta_bias_S2, E_bias_S2, bias_S2, len_bias_S2_CNN);
update_weights_bias(delta_weight_C3, E_weight_C3, weight_C3, len_weight_C3_CNN);
update_weights_bias(delta_bias_C3, E_bias_C3, bias_C3, len_bias_C3_CNN);
update_weights_bias(delta_weight_S4, E_weight_S4, weight_S4, len_weight_S4_CNN);
update_weights_bias(delta_bias_S4, E_bias_S4, bias_S4, len_bias_S4_CNN);
update_weights_bias(delta_weight_C5, E_weight_C5, weight_C5, len_weight_C5_CNN);
update_weights_bias(delta_bias_C5, E_bias_C5, bias_C5, len_bias_C5_CNN);
update_weights_bias(delta_weight_output, E_weight_output, weight_output, len_weight_output_CNN);
update_weights_bias(delta_bias_output, E_bias_output, bias_output, len_bias_output_CNN);
return true;
}6. 測试准确率是否达到要求或已达到循环次数：依次循环3至5中操作，依据训练集数量。每循环60000次时，通过计算的权值和偏置。来对10000个測试集进行測试，假设准确率达到0.985或者达到迭代次数上限100次时。保存权值和偏置。
代码段例如以下：
bool CNN::train()
{
out2wi_S2.clear();
out2bias_S2.clear();
out2wi_S4.clear();
out2bias_S4.clear();
in2wo_C3.clear();
weight2io_C3.clear();
bias2out_C3.clear();
in2wo_C1.clear();
weight2io_C1.clear();
bias2out_C1.clear();
calc_out2wi(width_image_C1_CNN, height_image_C1_CNN, width_image_S2_CNN, height_image_S2_CNN, num_map_S2_CNN, out2wi_S2);
calc_out2bias(width_image_S2_CNN, height_image_S2_CNN, num_map_S2_CNN, out2bias_S2);
calc_out2wi(width_image_C3_CNN, height_image_C3_CNN, width_image_S4_CNN, height_image_S4_CNN, num_map_S4_CNN, out2wi_S4);
calc_out2bias(width_image_S4_CNN, height_image_S4_CNN, num_map_S4_CNN, out2bias_S4);
calc_in2wo(width_image_C3_CNN, height_image_C3_CNN, width_image_S4_CNN, height_image_S4_CNN, num_map_C3_CNN, num_map_S4_CNN, in2wo_C3);
calc_weight2io(width_image_C3_CNN, height_image_C3_CNN, width_image_S4_CNN, height_image_S4_CNN, num_map_C3_CNN, num_map_S4_CNN, weight2io_C3);
calc_bias2out(width_image_C3_CNN, height_image_C3_CNN, width_image_S4_CNN, height_image_S4_CNN, num_map_C3_CNN, num_map_S4_CNN, bias2out_C3);
calc_in2wo(width_image_C1_CNN, height_image_C1_CNN, width_image_S2_CNN, height_image_S2_CNN, num_map_C1_CNN, num_map_C3_CNN, in2wo_C1);
calc_weight2io(width_image_C1_CNN, height_image_C1_CNN, width_image_S2_CNN, height_image_S2_CNN, num_map_C1_CNN, num_map_C3_CNN, weight2io_C1);
calc_bias2out(width_image_C1_CNN, height_image_C1_CNN, width_image_S2_CNN, height_image_S2_CNN, num_map_C1_CNN, num_map_C3_CNN, bias2out_C1);
int iter = 0;
for (iter = 0; iter < num_epochs_CNN; iter++) {
std::cout << "epoch: " << iter + 1;
for (int i = 0; i < num_patterns_train_CNN; i++) {
data_single_image = data_input_train + i * num_neuron_input_CNN;
data_single_label = data_output_train + i * num_neuron_output_CNN;
Forward_C1();
Forward_S2();
Forward_C3();
Forward_S4();
Forward_C5();
Forward_output();
Backward_output();
Backward_C5();
Backward_S4();
Backward_C3();
Backward_S2();
Backward_C1();
Backward_input();
UpdateWeights();
}
double accuracyRate = test();
std::cout << ", accuray rate: " << accuracyRate << std::endl;
if (accuracyRate > accuracy_rate_CNN) {
saveModelFile("E:/GitCode/NN_Test/data/cnn.model");
std::cout << "generate cnn model" << std::endl;
break;
}
}
if (iter == num_epochs_CNN) {
saveModelFile("E:/GitCode/NN_Test/data/cnn.model");
std::cout << "generate cnn model" << std::endl;
}
return true;
}
double CNN::test()
{
int count_accuracy = 0;
for (int num = 0; num < num_patterns_test_CNN; num++) {
data_single_image = data_input_test + num * num_neuron_input_CNN;
data_single_label = data_output_test + num * num_neuron_output_CNN;
Forward_C1();
Forward_S2();
Forward_C3();
Forward_S4();
Forward_C5();
Forward_output();
int pos_t = -1;
int pos_y = -2;
double max_value_t = -9999.0;
double max_value_y = -9999.0;
for (int i = 0; i < num_neuron_output_CNN; i++) {
if (neuron_output[i] > max_value_y) {
max_value_y = neuron_output[i];
pos_y = i;
}
if (data_single_label[i] > max_value_t) {
max_value_t = data_single_label[i];
pos_t = i;
}
}
if (pos_y == pos_t) {
++count_accuracy;
}
Sleep(1);
}
return (count_accuracy * 1.0 / num_patterns_test_CNN);
}7. 对输入的图像数据进行识别：载入已保存的权值和偏置，对输入的数据进行识别。过程相当于前向传播。
代码段例如以下：
int CNN::predict(const unsigned char* data, int width, int height)
{
assert(data && width == width_image_input_CNN && height == height_image_input_CNN);
const double scale_min = -1;
const double scale_max = 1;
double tmp[width_image_input_CNN * height_image_input_CNN];
for (int y = 0; y < height; y++) {
for (int x = 0; x < width; x++) {
tmp[y * width + x] = (data[y * width + x] / 255.0) * (scale_max - scale_min) + scale_min;
}
}
data_single_image = &tmp[0];
Forward_C1();
Forward_S2();
Forward_C3();
Forward_S4();
Forward_C5();
Forward_output();
int pos = -1;
double max_value = -9999.0;
for (int i = 0; i < num_neuron_output_CNN; i++) {
if (neuron_output[i] > max_value) {
max_value = neuron_output[i];
pos = i;
}
}
return pos;
}