【机器学习】LFM（Latent Factor Model）

                            LFM（Latent Factor Model）

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参考了[Key_Ky博客](%28http://www.cnblogs.com/Key-Ky/p/3579363.html%29)的潜在矩阵分解的代码，实践了一下。[图及公式取自Harry Huang博客](http://blog.csdn.net/harryhuang1990/article/details/9924377)

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                                矩阵分解图

                            目标函数（最小平方误差）：

                            随机梯度求解目标函数中的参数

梯度：

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import numpy as npimport matplotlib.pyplot as pltimport randomimport mathclass LFM:    '''    LFM 使用随机梯度下降法，求解LFM参数    '''    data_address = ''    datasets = []    np_training_datasets =np.zeros(1)    decompose_u = np.zeros(1)    decompose_v = np.zeros(1)    factor = 0    size_training_datasets_x = 0    size_training_datasets_y = 0    alpha = 0.1    iter_num = 20    Lambda = 0.1    epsilon = 0.01    delta_error = []    def __init__(self,data_address,factor,iter_num = 20,alpha = 0.1,Lambda = 0.1,epsilon = 0.01):        '''            @summary: 初始化参数        '''        self.data_address = data_address        self.factor = factor        self.alpha = alpha        self.iter_num = iter_num        self.Lambda = Lambda        self.epsilon = epsilon    def loadData(self):        '''        @summary: 加载原始数据        '''        input_file = open(self.data_address,'r')        for line in input_file:            tmp = line[:-1].split()            self.datasets.append([int(i) for i in tmp])        input_file.close()        self.np_training_datasets = np.array(self.datasets)    def initModel(self):        '''        @summary: 初始化U,V的潜在因子矩阵        '''        [x,y] = self.np_training_datasets.shape        self.size_training_datasets_x = x        self.size_training_datasets_y = y        self.decompose_u = np.ones([x,self.factor])        self.decompose_v = np.ones([self.factor,y])    def fNormcalc(self,matrix):        '''        @summary: 计算矩阵的F范数，即所有元素的平方和再开方        '''        [x,y] = matrix.shape        f_norm = 0        for i in range(x):            for j in range(y):                f_norm += pow(matrix[i][j],2)        f_norm = math.sqrt(f_norm)        return f_norm    #构建目标函数    def c_error_cacl(self):        '''        @summary: 构建目标函数，即误差平方和，以及加上正则化项，防止过拟合        '''        error_sum = 0        for i in range(self.size_training_datasets_x):            for j in range(self.size_training_datasets_y):                if self.np_training_datasets[i][j] != 0 :                    #即如果用户i对商品j有评分                    eui=0                    for m in range(self.factor):#                         eui += eui + self.decompose_u[i][m] * self.decompose_v[m][j]   #预测的评分！！！！！！！！！！                        eui += self.decompose_u[i][m] * self.decompose_v[m][j]                    error_sum += pow(self.np_training_datasets[i][j] - eui,2) + self.Lambda * pow(self.fNormcalc(self.decompose_u),2) + self.Lambda * pow(self.fNormcalc(self.decompose_v),2)        return error_sum    #随机梯度下降法，迭代    def iterator(self):        for step in range(self.iter_num):            old_error = 0.5 * self.c_error_cacl()   #目标函数1/2可以再梯度下不用乘以2，方便计算            print 'old_error ',old_error            for i in range(self.size_training_datasets_x):                for j in range(self.size_training_datasets_y):                    if self.np_training_datasets[i][j] != 0 :                        for f in range(self.factor):                            eui = 0                            for m in range(self.factor):                                eui = eui + self.decompose_u[i][m] * self.decompose_v[m][j]                            self.decompose_u[i][f] += self.alpha * ((self.np_training_datasets[i][j] - eui) * self.decompose_v[f][j] - self.Lambda * self.decompose_u[i][f])                            self.decompose_v[f][j] += self.alpha * ((self.np_training_datasets[i][j] - eui) * self.decompose_u[i][f] - self.Lambda * self.decompose_v[f][j])            new_error = 0.5 * self.c_error_cacl()            print 'new_error ',new_error            if abs(new_error - old_error) < self.epsilon:                break            self.delta_error.append(abs(new_error - old_error)) #保存每一次迭代的误差if __name__=='__main__':#     randomdata('F://rating.txt')    lfm=LFM(r'F:\rating1.txt',3,100)    lfm.loadData()    lfm.initModel()    lfm.iterator()    print lfm.decompose_u    print lfm.decompose_v    ex = range(len(lfm.delta_error))    plt.figure(1)    plt.plot(ex,lfm.delta_error)    plt.show()

简单的测试数据集：

0 1 2 0 0 4 0
0 0 0 5 0 6 0
0 0 0 0 0 0 0
0 0 0 0 9 0 0
0 0 0 0 0 0 0
10 0 9 8 0 0 0

分解的P矩阵：
[[ 0.36717743 1.07801356 0.83258288]
[ 1.22030144 1.19854111 1.16571532]
[ 1. 1. 1. ]
[ 2.05226677 1.46459895 1.31047188]
[ 1. 1. 1. ]
[ 2.75487483 1.50212454 1.32204588]]

分解的Q矩阵：
[[ 2.04842494 0.47546457 2.41670178 1.63844667 2.33548542 1.71443626 1. ]
[ 1.61164606 0.42750604 0.79754534 1.23739612 1.5793569 1.76219035 1. ]
[ 1.38761605 0.50476132 0.77759064 1.15922173 1.36918913 1.47754311 1. ]]

误差函数曲线：

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参考文献：
Key_Ky博客：http://www.cnblogs.com/Key-Ky/p/3579363.html
Harry Huang：http://blog.csdn.net/harryhuang1990/article/details/9924377