在中给出了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 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：
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 falsestatic 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 Xbool 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 & 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 & 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;} GitHub：