05_LFM算法实现

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 P_{11} P11​表示用户1对隐含特征1的权重值

• 矩阵值 Q 11 Q_{11} Q11​表示隐含特征1在物品1上的权重值

• 矩阵值 R 11 ​ R_{11}​ R11​​就表示预测的用户1对物品1的评分，且 R 11 = P 1 , k ⃗ ⋅ Q k , 1 ⃗ ​ R_{11}=\vec{P_{1,k}}\cdot \vec{Q_{k,1}}​ R11​=P1,k​ ​⋅Qk,1​ ​​

利用LFM预测用户对物品的评分，$k$表示隐含特征数量：
r ^ u i = p u k ⃗ ⋅ q i k ⃗ = ∑ k = 1 k p u k q i k \begin{split} \hat {r}_{ui} &=\vec {p_{uk}}\cdot \vec {q_{ik}} \\&={\sum_{k=1}}^k p_{uk}q_{ik} \end{split} r^ui​​=puk​ ​⋅qik​ ​=k=1∑​kpuk​qik​​
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：
C o s t = ∑ u , i ∈ R ( r u i − r ^ u i ) 2 = ∑ u , i ∈ R ( r u i − ∑ k = 1 k p u k q i k ) 2 \begin{split} Cost &= \sum_{u,i\in R} (r_{ui}-\hat{r}_{ui})^2 \\&=\sum_{u,i\in R} (r_{ui}-{\sum_{k=1}}^k p_{uk}q_{ik})^2 \end{split} Cost​=u,i∈R∑​(rui​−r^ui​)2=u,i∈R∑​(rui​−k=1∑​kpuk​qik​)2​
加入L2正则化：
C o s t = ∑ u , i ∈ R ( r u i − ∑ k = 1 k p u k q i k ) 2 + λ ( ∑ U p u k 2 + ∑ I q i k 2 ) 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) Cost=u,i∈R∑​(rui​−k=1∑​kpuk​qik​)2+λ(U∑​puk​2+I∑​qik​2)
对损失函数求偏导：
∂ ∂ p u k C o s t = ∂ ∂ p u k [ ∑ u , i ∈ R ( r u i − ∑ k = 1 k p u k q i k ) 2 + λ ( ∑ U p u k 2 + ∑ I q i k 2 ) ] = 2 ∑ u , i ∈ R ( r u i − ∑ k = 1 k p u k q i k ) ( − q i k ) + 2 λ p u k ∂ ∂ q i k C o s t = ∂ ∂ q i k [ ∑ u , i ∈ R ( r u i − ∑ k = 1 k p u k q i k ) 2 + λ ( ∑ U p u k 2 + ∑ I q i k 2 ) ] = 2 ∑ u , i ∈ R ( r u i − ∑ k = 1 k p u k q i k ) ( − p u k ) + 2 λ q i k \begin{split} \cfrac {\partial}{\partial p_{uk}}Cost &= \cfrac {\partial}{\partial p_{uk}}[\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)] \\&=2\sum_{u,i\in R} (r_{ui}-{\sum_{k=1}}^k p_{uk}q_{ik})(-q_{ik}) + 2\lambda p_{uk} \\\\ \cfrac {\partial}{\partial q_{ik}}Cost &= \cfrac {\partial}{\partial q_{ik}}[\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)] \\&=2\sum_{u,i\in R} (r_{ui}-{\sum_{k=1}}^k p_{uk}q_{ik})(-p_{uk}) + 2\lambda q_{ik} \end{split} ∂puk​∂​Cost∂qik​∂​Cost​=∂puk​∂​[u,i∈R∑​(rui​−k=1∑​kpuk​qik​)2+λ(U∑​puk​2+I∑​qik​2)]=2u,i∈R∑​(rui​−k=1∑​kpuk​qik​)(−qik​)+2λpuk​=∂qik​∂​[u,i∈R∑​(rui​−k=1∑​kpuk​qik​)2+λ(U∑​puk​2+I∑​qik​2)]=2u,i∈R∑​(rui​−k=1∑​kpuk​qik​)(−puk​)+2λqik​​

随机梯度下降法优化

梯度下降更新参数 p u k ​ p_{uk}​ puk​​：
p u k : = p u k − α ∂ ∂ p u k C o s t : = p u k − α [ 2 ∑ u , i ∈ R ( r u i − ∑ k = 1 k p u k q i k ) ( − q i k ) + 2 λ p u k ] : = p u k + α [ ∑ u , i ∈ R ( r u i − ∑ k = 1 k p u k q i k ) q i k − λ p u k ] \begin{split} p_{uk}&:=p_{uk} - \alpha\cfrac {\partial}{\partial p_{uk}}Cost \\&:=p_{uk}-\alpha [2\sum_{u,i\in R} (r_{ui}-{\sum_{k=1}}^k p_{uk}q_{ik})(-q_{ik}) + 2\lambda p_{uk}] \\&:=p_{uk}+\alpha [\sum_{u,i\in R} (r_{ui}-{\sum_{k=1}}^k p_{uk}q_{ik})q_{ik} - \lambda p_{uk}] \end{split} puk​​:=puk​−α∂puk​∂​Cost:=puk​−α[2u,i∈R∑​(rui​−k=1∑​kpuk​qik​)(−qik​)+2λpuk​]:=puk​+α[u,i∈R∑​(rui​−k=1∑​kpuk​qik​)qik​−λpuk​]​
同理：
q i k : = q i k + α [ ∑ u , i ∈ R ( r u i − ∑ k = 1 k p u k q i k ) p u k − λ q i k ] \begin{split} q_{ik}&:=q_{ik} + \alpha[\sum_{u,i\in R} (r_{ui}-{\sum_{k=1}}^k p_{uk}q_{ik})p_{uk} - \lambda q_{ik}] \end{split} qik​​:=qik​+α[u,i∈R∑​(rui​−k=1∑​kpuk​qik​)puk​−λqik​]​
随机梯度下降： 向量乘法 每一个分量相乘 求和
p u k : = p u k + α [ ( r u i − ∑ k = 1 k p u k q i k ) q i k − λ 1 p u k ] q i k : = q i k + α [ ( r u i − ∑ k = 1 k p u k q i k ) p u k − λ 2 q i k ] \begin{split} &p_{uk}:=p_{uk}+\alpha [(r_{ui}-{\sum_{k=1}}^k p_{uk}q_{ik})q_{ik} - \lambda_1 p_{uk}] \\&q_{ik}:=q_{ik} + \alpha[(r_{ui}-{\sum_{k=1}}^k p_{uk}q_{ik})p_{uk} - \lambda_2 q_{ik}] \end{split} ​puk​:=puk​+α[(rui​−k=1∑​kpuk​qik​)qik​−λ1​puk​]qik​:=qik​+α[(rui​−k=1∑​kpuk​qik​)puk​−λ2​qik​]​
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 λ 1 ​ \lambda_1​ λ1​​和 λ 2 ​ \lambda_2​ λ2​​

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)))

0, 100, [“userId”, “movieId”, “rating”])
lfm.fit(dataset)

while True:uid = input("uid: ")iid = input("iid: ")print(lfm.predict(int(uid), int(iid)))

本文来自互联网用户投稿，文章观点仅代表作者本人，不代表本站立场，不承担相关法律责任。如若转载，请注明出处。 如若内容造成侵权/违法违规/事实不符，请点击【内容举报】进行投诉反馈！