基于矩阵分解的CF算法实现（一）：LFM

LFM也就是前面提到的Funk SVD矩阵分解

LFM原理解析

LFM(latent factor model)隐语义模型核心思想是通过隐含特征联系用户和物品，如下图：

• P矩阵是User-LF矩阵，即用户和隐含特征矩阵。LF有三个，表示共总有三个隐含特征。
• Q矩阵是LF-Item矩阵，即隐含特征和物品的矩阵
• R矩阵是User-Item矩阵，有P*Q得来
• 能处理稀疏评分矩阵

利用矩阵分解技术，将原始User-Item的评分矩阵（稠密/稀疏）分解为P和Q矩阵，然后利用$P\ast Q$还原出User-Item评分矩阵$R$。整个过程相当于降维处理，其中：

• 矩阵值$P_{11}$表示用户1对隐含特征1的权重值

• 矩阵值$Q_{11}$表示隐含特征1在物品1上的权重值

• 矩阵值$R_{11}$就表示预测的用户1对物品1的评分，且$R_{11}=\overrightarrow{P_{1,k}}\cdot \overrightarrow{Q_{k,1}}$

利用LFM预测用户对物品的评分，$k$表示隐含特征数量：

因此最终，我们的目标也就是要求出P矩阵和Q矩阵及其当中的每一个值，然后再对用户-物品的评分进行预测。

损失函数

同样对于评分预测我们利用平方差来构建损失函数：

加入L2正则化：
$$Cost=\sum _{u,i\in R}(r_{ui}-{\sum _{k=1}}^k{p}_{uk}{q}_{ik}{)}^2+\lambda (\sum _U{p_{uk}}^2+\sum _I{q_{ik}}^2)$$
对损失函数求偏导：

随机梯度下降法优化

梯度下降更新参数$p_{uk}$：

同理：

随机梯度下降： 向量乘法 每一个分量相乘 求和

由于P矩阵和Q矩阵是两个不同的矩阵，通常分别采取不同的正则参数，如${\lambda }_1$和${\lambda }_2$

算法实现

'''
LFM Model
'''
import pandas as pd
import numpy as np# 评分预测    1-5
class LFM(object):def __init__(self, alpha, reg_p, reg_q, number_LatentFactors=10, number_epochs=10, columns=["uid", "iid", "rating"]):self.alpha = alpha # 学习率self.reg_p = reg_p    # P矩阵正则self.reg_q = reg_q    # Q矩阵正则self.number_LatentFactors = number_LatentFactors  # 隐式类别数量self.number_epochs = number_epochs    # 最大迭代次数self.columns = columnsdef fit(self, dataset):'''fit dataset:param dataset: uid, iid, rating:return:'''self.dataset = pd.DataFrame(dataset)self.users_ratings = dataset.groupby(self.columns[0]).agg([list])[[self.columns[1], self.columns[2]]]self.items_ratings = dataset.groupby(self.columns[1]).agg([list])[[self.columns[0], self.columns[2]]]self.globalMean = self.dataset[self.columns[2]].mean()self.P, self.Q = self.sgd()def _init_matrix(self):'''初始化P和Q矩阵，同时为设置0，1之间的随机值作为初始值:return:'''# User-LFP = dict(zip(self.users_ratings.index,np.random.rand(len(self.users_ratings), self.number_LatentFactors).astype(np.float32)))# Item-LFQ = dict(zip(self.items_ratings.index,np.random.rand(len(self.items_ratings), self.number_LatentFactors).astype(np.float32)))return P, Qdef sgd(self):'''使用随机梯度下降，优化结果:return:'''P, Q = self._init_matrix()for i in range(self.number_epochs):print("iter%d"%i)error_list = []for uid, iid, r_ui in self.dataset.itertuples(index=False):# User-LF P## Item-LF Qv_pu = P[uid] #用户向量v_qi = Q[iid] #物品向量err = np.float32(r_ui - np.dot(v_pu, v_qi))v_pu += self.alpha * (err * v_qi - self.reg_p * v_pu)v_qi += self.alpha * (err * v_pu - self.reg_q * v_qi)P[uid] = v_pu Q[iid] = v_qi# for k in range(self.number_of_LatentFactors):#     v_pu[k] += self.alpha*(err*v_qi[k] - self.reg_p*v_pu[k])#     v_qi[k] += self.alpha*(err*v_pu[k] - self.reg_q*v_qi[k])error_list.append(err ** 2)print(np.sqrt(np.mean(error_list)))return P, Qdef predict(self, uid, iid):# 如果uid或iid不在，我们使用全剧平均分作为预测结果返回if uid not in self.users_ratings.index or iid not in self.items_ratings.index:return self.globalMeanp_u = self.P[uid]q_i = self.Q[iid]return np.dot(p_u, q_i)def test(self,testset):'''预测测试集数据'''for uid, iid, real_rating in testset.itertuples(index=False):try:pred_rating = self.predict(uid, iid)except Exception as e:print(e)else:yield uid, iid, real_rating, pred_ratingif __name__ == '__main__':dtype = [("userId", np.int32), ("movieId", np.int32), ("rating", np.float32)]dataset = pd.read_csv("datasets/ml-latest-small/ratings.csv", usecols=range(3), dtype=dict(dtype))lfm = LFM(0.02, 0.01, 0.01, 10, 100, ["userId", "movieId", "rating"])lfm.fit(dataset)while True:uid = input("uid: ")iid = input("iid: ")print(lfm.predict(int(uid), int(iid)))

基于矩阵分解的CF算法实现（二）：BiasSvd

BiasSvd其实就是前面提到的Funk SVD矩阵分解基础上加上了偏置项。

BiasSvd

利用BiasSvd预测用户对物品的评分，$k$表示隐含特征数量：

损失函数

同样对于评分预测我们利用平方差来构建损失函数：

加入L2正则化：
$$Cost=\sum _{u,i\in R}(r_{ui}-\mu -b_u-b_i-{\sum _{k=1}}^k{p}_{uk}{q}_{ik}{)}^2+\lambda (\sum _U{b_u}^2+\sum _I{b_i}^2+\sum _U{p_{uk}}^2+\sum _I{q_{ik}}^2)$$
对损失函数求偏导：

随机梯度下降法优化

梯度下降更新参数$p_{uk}$：

同理：

$$b_u:=b_u+\alpha [\sum _{u,i\in R}(r_{ui}-\mu -b_u-b_i-{\sum _{k=1}}^k{p}_{uk}{q}_{ik})-\lambda b_u]$$

$$b_i:=b_i+\alpha [\sum _{u,i\in R}(r_{ui}-\mu -b_u-b_i-{\sum _{k=1}}^k{p}_{uk}{q}_{ik})-\lambda b_i]$$

随机梯度下降：

$$b_u:=b_u+\alpha [(r_{ui}-\mu -b_u-b_i-{\sum _{k=1}}^k{p}_{uk}{q}_{ik})-{\lambda }_3{b}_u]$$

$$b_i:=b_i+\alpha [(r_{ui}-\mu -b_u-b_i-{\sum _{k=1}}^k{p}_{uk}{q}_{ik})-{\lambda }_4{b}_i]$$

由于P矩阵和Q矩阵是两个不同的矩阵，通常分别采取不同的正则参数，如${\lambda }_1$和${\lambda }_2$

算法实现

'''
BiasSvd Model
'''
import math
import random
import pandas as pd
import numpy as npclass BiasSvd(object):def __init__(self, alpha, reg_p, reg_q, reg_bu, reg_bi, number_LatentFactors=10, number_epochs=10, columns=["uid", "iid", "rating"]):self.alpha = alpha # 学习率self.reg_p = reg_pself.reg_q = reg_qself.reg_bu = reg_buself.reg_bi = reg_biself.number_LatentFactors = number_LatentFactors  # 隐式类别数量self.number_epochs = number_epochsself.columns = columnsdef fit(self, dataset):'''fit dataset:param dataset: uid, iid, rating:return:'''self.dataset = pd.DataFrame(dataset)self.users_ratings = dataset.groupby(self.columns[0]).agg([list])[[self.columns[1], self.columns[2]]]self.items_ratings = dataset.groupby(self.columns[1]).agg([list])[[self.columns[0], self.columns[2]]]self.globalMean = self.dataset[self.columns[2]].mean()self.P, self.Q, self.bu, self.bi = self.sgd()def _init_matrix(self):'''初始化P和Q矩阵，同时为设置0，1之间的随机值作为初始值:return:'''# User-LFP = dict(zip(self.users_ratings.index,np.random.rand(len(self.users_ratings), self.number_LatentFactors).astype(np.float32)))# Item-LFQ = dict(zip(self.items_ratings.index,np.random.rand(len(self.items_ratings), self.number_LatentFactors).astype(np.float32)))return P, Qdef sgd(self):'''使用随机梯度下降，优化结果:return:'''P, Q = self._init_matrix()# 初始化bu、bi的值，全部设为0bu = dict(zip(self.users_ratings.index, np.zeros(len(self.users_ratings))))bi = dict(zip(self.items_ratings.index, np.zeros(len(self.items_ratings))))for i in range(self.number_epochs):print("iter%d"%i)error_list = []for uid, iid, r_ui in self.dataset.itertuples(index=False):v_pu = P[uid]v_qi = Q[iid]err = np.float32(r_ui - self.globalMean - bu[uid] - bi[iid] - np.dot(v_pu, v_qi))v_pu += self.alpha * (err * v_qi - self.reg_p * v_pu)v_qi += self.alpha * (err * v_pu - self.reg_q * v_qi)P[uid] = v_pu Q[iid] = v_qibu[uid] += self.alpha * (err - self.reg_bu * bu[uid])bi[iid] += self.alpha * (err - self.reg_bi * bi[iid])error_list.append(err ** 2)print(np.sqrt(np.mean(error_list)))return P, Q, bu, bidef predict(self, uid, iid):if uid not in self.users_ratings.index or iid not in self.items_ratings.index:return self.globalMeanp_u = self.P[uid]q_i = self.Q[iid]return self.globalMean + self.bu[uid] + self.bi[iid] + np.dot(p_u, q_i)if __name__ == '__main__':dtype = [("userId", np.int32), ("movieId", np.int32), ("rating", np.float32)]dataset = pd.read_csv("datasets/ml-latest-small/ratings.csv", usecols=range(3), dtype=dict(dtype))bsvd = BiasSvd(0.02, 0.01, 0.01, 0.01, 0.01, 10, 20)bsvd.fit(dataset)while True:uid = input("uid: ")iid = input("iid: ")print(bsvd.predict(int(uid), int(iid)))

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