SVD++分解
在现实情况下，用户的显式反馈严重不足，但一般拥有大量隐式反馈信息。所以在偏置svd基础上增加了用户的隐式反馈信息，该方法融合了用户的显式和隐式信息。
1.预测评分公式为
其中，有全局平均分，user的偏置信息，item的偏置信息，Ni为该用户评价过的所有item集合，
从隐式反馈出发，作为用户偏好的补充，ys为隐式偏好。
2.损失函数为平方误差+L2正则项，其中
是真实评分。
3.利用SGD训练更新，其中
。
4.程序，(数据movielens 100k，https://github.com/jiangnanboy/recommendation_methods/blob/master/com/sy/reco/recommendation/matrix_factorization/svdpp.py)
import numpy as np import math ''' 融合了BiasLFM以及用户的隐式行为 ''' class SVDPP(): ''' 初始化ratingMatrix,F, alpha, λ ratingMatrix:评分矩阵 F:隐因子数目 alpha:学习速率 λ:正则化参数，以防过拟合 ''' def __init__(self, ratingMatrix, F, alpha, λ): self.ratingMatrix = ratingMatrix self.F = F self.alpha = alpha self.λ = λ # 对U,I,Y矩阵初始化，随机填充，根据经验随机数与1/sqrt(F)成正比,bu向量与bi向量初始化为全0,u是所有有评分的全局平均 def __initPQ(self, userSum, itemSum): self.U = np.zeros((userSum, self.F)) self.I = np.zeros((itemSum, self.F)) self.Y=np.zeros((itemSum,self.F)) self.bu = np.zeros(userSum) self.bi = np.zeros(itemSum) self.meanV = np.mean(self.ratingMatrix[self.ratingMatrix > 0]) # 全局均值 for i in range(userSum): self.U[i] = [np.random.random() / math.sqrt(self.F) for x in range(self.F)] for i in range(itemSum): self.I[i] = [np.random.random() / math.sqrt(self.F) for x in range(self.F)] self.Y[i]=[np.random.random()/math.sqrt(self.F) for x in range(self.F)] # 迭代训练分解，max_iter:迭代次数 def iteration_train(self, max_iter): userSum = len(self.ratingMatrix) # 用户个数 itemSum = len(self.ratingMatrix[0]) # 项目个数 self.__initPQ(userSum, itemSum) # 初始化U,I,Y,bu,bi Z=np.zeros((userSum,self.F)) for step in range(max_iter): for user in range(userSum): Z[user]=self.U[user] ru=1/math.sqrt(len(self.U[user][self.U[user,:]>0]))#1/sqrt(Usum>0)用户评过分的项目个数 for item in range(itemSum): if self.ratingMatrix[user, item] > 0: # 未评分的项目不参与计算 for f in range(self.F): Z[user,f]+=self.Y[item,f]*ru #I更新中的部分(1/sqrt(Usum))*self.Y sum=np.zeros(self.F) for item in range(itemSum): if self.ratingMatrix[user,item]>0:# 未评分的项目不参与计算 eui = self.ratingMatrix[user, item] - self.predict(user, item,ru) # 真实值减去预测的值 self.bu[user] += self.alpha * (eui - self.λ * self.bu[user]) # 更新bu self.bi[item] += self.alpha * (eui - self.λ * self.bi[item]) # 更新bi for f in range(self.F): sum[f]+=self.I[item,f]*eui*ru #Y更新中的部分eui*ru*I self.U[user, f] += self.alpha * (self.I[item, f] * eui - self.λ * self.U[user, f]) # 更新U self.I[item,f]+=self.alpha*(eui*(self.U[user,f]+Z[user,f])-self.λ*self.I[item,f])#更新I #更新Y for item in range(itemSum): for f in range(self.F): self.Y[item,f]+=self.alpha*(sum[f]-self.λ*self.Y[item,f]) #self.alpha*=0.9 predictRating = []#存放全部预测数据 for user in range(userSum): userItemRating = [] ru = 1 / math.sqrt(len(self.U[user][self.U[user, :] > 0])) # 1/sqrt(Usum>0)用户评过分的项目个数 for item in range(itemSum): pui = self.predict(user, item,ru) userItemRating.append(pui) predictRating.append(userItemRating) return np.round(np.array(predictRating), 0) #预测打分，用户的行与项目的列 def predict(self,user,item,ru): z=np.zeros(self.F) for i in range(len(self.ratingMatrix[user])): if self.ratingMatrix[user,i]>0: for f in range(self.F): z[f]+=self.Y[i,f] pui =0. for f in range(self.F): pui+=(self.U[user,f]+z[f]/ru)*self.I[item,f] pui+=self.meanV+self.bu[user]+self.bi[item] return pui # 预测误差训练，convergence:误差收敛，小于这个误差,则终止训练 def convergence_train(self, convergence): userSum = len(self.ratingMatrix) # 用户个数 itemSum = len(self.ratingMatrix[0]) # 项目个数 self.__initPQ(userSum, itemSum) # 初始化U,I,Y,bu,bi Z = np.zeros((userSum, self.F)) flag=True while flag: for user in range(userSum): Z[user] = self.U[user] ru = 1 / math.sqrt(len(self.U[user][self.U[user, :] > 0])) # 1/sqrt(Usum>0)用户评过分的项目个数 for item in range(itemSum): if self.ratingMatrix[user, item] > 0: # 未评分的项目不参与计算 for f in range(self.F): Z[user, f] += self.Y[item, f] * ru # I更新中的部分(1/sqrt(Usum))*self.Y sum = np.zeros(self.F) for item in range(itemSum): if self.ratingMatrix[user, item] > 0: # 未评分的项目不参与计算 eui = self.ratingMatrix[user, item] - self.predict(user, item, ru) # 真实值减去预测的值 self.bu[user] += self.alpha * (eui - self.λ * self.bu[user]) # 更新bu self.bi[item] += self.alpha * (eui - self.λ * self.bi[item]) # 更新bi for f in range(self.F): sum[f] += self.I[item, f] * eui * ru # Y更新中的部分eui*ru*I self.U[user, f] += self.alpha * (self.I[item, f] * eui - self.λ * self.U[user, f]) # 更新U self.I[item, f] += self.alpha * ( eui * (self.U[user, f] + Z[user, f]) - self.λ * self.I[item, f]) # 更新I # 更新Y for item in range(itemSum): for f in range(self.F): self.Y[item, f] += self.alpha * (sum[f] - self.λ * self.Y[item, f]) cost = 0 # 误差 for user in range(userSum): ru = 1 / math.sqrt(len(self.U[user][self.U[user, :] > 0])) # 1/sqrt(Usum>0)用户评过分的项目个数 for item in range(itemSum): if self.ratingMatrix[user, item] > 0: cost += (1 / 2) * math.pow(self.ratingMatrix[user, item] - self.predict(user, item,ru), 2) cost += (1 / 2) * self.λ * (math.pow(self.bu[user], 2) + math.pow(self.bi[item], 2)) for f in range(self.F): cost += (1 / 2) * self.λ * (math.pow(self.U[user, f], 2) + math.pow(self.I[item, f], 2)+math.pow(self.Y[item,f],2)) if cost < convergence: flag = False predictRating = [] # 存放全部预测数据 for user in range(userSum): userItemRating = [] ru = 1 / math.sqrt(len(self.U[user][self.U[user, :] > 0])) # 1/sqrt(Usum>0)用户评过分的项目个数 for item in range(itemSum): pui = self.predict(user, item, ru) userItemRating.append(pui) predictRating.append(userItemRating) return np.round(np.array(predictRating), 0)
