引入
我们知道很多豆瓣的用户都会对电影进行打分，根据这个评分，我们可能知道这些用户除了自己评过分之外的电影还偏好哪些电影？这是一个推荐问题，解决这类问题的算法被称作推荐算法。
算法原理
协同过滤
协同过滤简单来说是利用某兴趣相投、拥有共同经验之群体的喜好来推荐用户感兴趣的信息，个人通过合作的机制给予信息相当程度的回应（如评分）并记录下来以达到过滤的目的进而帮助别人筛选信息，回应不一定局限于特别感兴趣的，特别不感兴趣信息的纪录也相当重要。一般来说协同过滤分为三种类型：1.基于用户(user-based)的协同过滤，，通过计算用户和用户的相似度找到跟用户A相似的用户B, C, D…再把这些用户喜欢的内容推荐给A；2.基于物品(item-based)的协同过滤，通过计算物品和物品的相似度找到跟物品1相似的物品2, 3, 4…再把这些物品推荐给看过物品1的用户们；3.基于模型(model-based)的协同过滤。主流方法分为矩阵分解，关联算法，聚类算法，分类算法，回归算法，神经网络。
ALS
本文要介绍的算法ALS是基于矩阵分解，所谓矩阵分解就是将矩阵拆解为多个矩阵的乘积。如果想通过矩阵分解的方法实现基于模型的协同过滤，ALS算法是一个不错的选择，ALS的全称为Alternating Least Square，翻译过来为交替最小二乘法。这里假设用户为a，物品为b，评分矩阵为R(m, n)可以分解为用户矩阵U(k,m)和物品矩阵I(k,n)，其中m,n,k代表矩阵的维度。
- 根据矩阵分解的定义，有
R=UT∗I - 用MSE作为损失函数，为了方便化简，加法符号左侧的常数改为-1/2
- L=−21∑i=0n(Rai−UaT∗Ii)2
- 对损失函数求U_a的一阶偏导数，那么
dUadL=(Ra−UaT∗I)∗IT
dUadL=I∗(RaT−IT∗Ua) - 令一阶偏导数等于0
I∗RaT=I∗IT∗Ua
Ua=(I∗IT)−1∗I∗RaT…(1) - 同理，可求
Ib=(U∗UT)−1∗U∗Rb…(2)
ALS算法的目标是在已知R的情况下，求出用户矩阵U和物品矩阵I。
1.我们先随机生成用户矩阵U，利用(2)，R，U求出I。
2.利用(2)式子，R和I求出U。
如此交替地执行步骤1和2，直到算法收敛或者迭代次数超过了最大限制，最终我们用RMSE来评价模型的表现。
代码实现
#coding=utf-8
from collections import defaultdict
from random import random
from itertools import product, chain
from time import time
def load_movie_ratings():
f = open("boston/movie_ratings.csv")
lines = iter(f)
col_names = ", ".join(next(lines)[:-1].split(",")[:-1])
print("The column names are: %s." % col_names)
data = [[float(x) if i == 2 else int(x)
for i, x in enumerate(line[:-1].split(",")[:-1])]
for line in lines]
f.close()
return data
class Matrix(object):
def __init__(self, data):
self.data = data
self.shape = (len(data), len(data[0]))
def row(self, row_no):
return Matrix([self.data[row_no]])
def col(self, col_no):
m = self.shape[0]
return Matrix([[self.data[i][col_no]] for i in range(m)])
@property
def is_square(self):
return self.shape[0] == self.shape[1]
@property
def transpose(self):
data = list(map(list, zip(*self.data)))
return Matrix(data)
# 生成一个长度为n的单位阵
def _eye(self, n):
return [[0 if i != j else 1 for j in range(n)] for i in range(n)]
@property
def eye(self):
assert self.is_squre, "The matrix has to be squre"
data = self._eye(self.shape[0])
return Matrix(data)
# 高斯消元
def gaussian_elimination(self, aug_matrix):
n = len(aug_matrix)
m = len(aug_matrix[0])
# From top to bottom.
for col_idx in range(n):
# Check if element on the diagonal is zero.
if aug_matrix[col_idx][col_idx] == 0:
row_idx = col_idx
# Find a row whose element has same column index with
# the element on the diagonal is not zero.
while row_idx < n and aug_matrix[row_idx][col_idx] == 0:
row_idx += 1
# Add this row to the row of the element on the diagonal.
for i in range(col_idx, m):
aug_matrix[col_idx][i] += aug_matrix[row_idx][i]
# Elimiate the non-zero element.
for i in range(col_idx + 1, n):
# Skip the zero element.
if aug_matrix[i][col_idx] == 0:
continue
# Elimiate the non-zero element.
k = aug_matrix[i][col_idx] / aug_matrix[col_idx][col_idx]
for j in range(col_idx, m):
aug_matrix[i][j] -= k * aug_matrix[col_idx][j]
# From bottom to top.
for col_idx in range(n - 1, -1, -1):
# Elimiate the non-zero element.
for i in range(col_idx):
# Skip the zero element.
if aug_matrix[i][col_idx] == 0:
continue
# Elimiate the non-zero element.
k = aug_matrix[i][col_idx] / aug_matrix[col_idx][col_idx]
for j in chain(range(i, col_idx + 1), range(n, m)):
aug_matrix[i][j] -= k * aug_matrix[col_idx][j]
# Iterate the element on the diagonal.
for i in range(n):
k = 1 / aug_matrix[i][i]
aug_matrix[i][i] *= k
for j in range(n, m):
aug_matrix[i][j] *= k
return aug_matrix
# 矩阵求逆
def _inverse(self, data):
n = len(data)
unit_matrix = self._eye(n)
aug_matrix = [a + b for a, b in zip(self.data, unit_matrix)]
ret = self.gaussian_elimination(aug_matrix)
return list(map(lambda x: x[n:], ret))
# 矩阵求逆，原理：https://baike.baidu.com/item/%E9%AB%98%E6%96%AF%E6%B6%88%E5%85%83%E6%B3%95/619561?fr=aladdin
@property
def inverse(self):
assert self.is_square, "The matrix has to be square!"
data = self._inverse(self.data)
return Matrix(data)
def row_mul(self, row_A, row_B):
return sum(x[0] * x[1] for x in zip(row_A, row_B))
def _mat_mul(self, row_A, B):
row_pairs = product([row_A], B.transpose.data)
return [self.row_mul(*row_pair) for row_pair in row_pairs]
def mat_mul(self, B):
assert self.shape[1] == B.shape[0], "A's column count does not match B's row count!"
return Matrix([self._mat_mul(row_A, B) for row_A in self.data])
def _mean(self, data):
m = len(data)
n = len(data[0])
ret = [0 for _ in range(n)]
for row in data:
for j in range(n):
ret[j] += row[j] / m
return ret
def mean(self, data):
return Matrix(self._mean(self.data))
# 统计程序运行时间函数
# fn代表运行的函数
def run_time(fn):
def fun():
start = time()
fn()
ret = time() - start
if ret < 1e-6:
unit = "ns"
ret *= 1e9
elif ret < 1e-3:
unit = "us"
ret *= 1e6
elif ret < 1:
unit = "ms"
ret *= 1e3
else:
unit = "s"
print("Total run time is %.1f %s\n" % (ret, unit))
return fun()
class ALS(object):
# 初始化，存储用户ID、物品ID、用户ID与用户矩阵列号的对应关系、物品ID
# 与物品矩阵列号的对应关系、用户已经看过哪些物品、评分矩阵的Shape以及RMSE
def __init__(self):
self.user_ids = None
self.item_ids = None
self.user_ids_dict = None
self.item_ids_dict = None
self.user_matrix = None
self.item_matrix = None
self.user_items = None
self.shape = None
self.rmse = None
# 对训练数据进行处理，得到用户ID、物品ID、用户ID与用户矩阵列号的对应关系、物
# 品ID与物品矩阵列号的对应关系、评分矩阵的Shape、评分矩阵及评分矩阵的转置。
def process_data(self, X):
self.user_ids = tuple((set(map(lambda x: x[0], X))))
self.user_ids_dict = dict(map(lambda x: x[::-1], enumerate(self.user_ids)))
self.item_ids = tuple((set(map(lambda x: x[1], X))))
self.item_ids_dict = dict(map(lambda x: x[::-1], enumerate(self.item_ids)))
self.shape = (len(self.user_ids), len(self.item_ids))
ratings = defaultdict(lambda : defaultdict(int))
ratings_T = defaultdict(lambda : defaultdict(int))
for row in X:
user_id, item_id, rating = row
ratings[user_id][item_id] = rating
ratings_T[item_id][user_id] = rating
err_msg = "Length of user_ids %d and ratings %d not match!" % (
len(self.user_ids), len(ratings))
assert len(self.user_ids) == len(ratings), err_msg
err_msg = "Length of item_ids %d and ratings_T %d not match!" % (
len(self.item_ids), len(ratings_T))
assert len(self.item_ids) == len(ratings_T), err_msg
return ratings, ratings_T
# 用户矩阵乘以评分矩阵，实现稠密矩阵与稀疏矩阵的矩阵乘法，得到用户矩阵与评分矩阵的乘积。
def users_mul_ratings(self, users, ratings_T):
def f(users_row, item_id):
user_ids = iter(ratings_T[item_id].keys())
scores = iter(ratings_T[item_id].values())
col_nos = map(lambda x: self.user_ids_dict[x], user_ids)
_users_row = map(lambda x: users_row[x], col_nos)
return sum(a * b for a, b in zip(_users_row, scores))
ret = [[f(users_row, item_id) for item_id in self.item_ids]
for users_row in users.data]
return Matrix(ret)
# 物品矩阵乘以评分矩阵，实现稠密矩阵与稀疏矩阵的矩阵乘法，得到物品矩阵与评分矩阵的乘积。
def items_mul_ratings(self, items, ratings):
def f(items_row, user_id):
item_ids = iter(ratings[user_id].keys())
scores = iter(ratings[user_id].values())
col_nos = map(lambda x: self.item_ids_dict[x], item_ids)
_items_row = map(lambda x: items_row[x], col_nos)
return sum(a * b for a, b in zip(_items_row, scores))
ret = [[f(items_row, user_id) for user_id in self.user_ids]
for items_row in items.data]
return Matrix(ret)
# 生成随机矩阵
def gen_random_matrix(self, n_rows, n_colums):
data = [[random() for _ in range(n_colums)] for _ in range(n_rows)]
return Matrix(data)
# 计算RMSE
def get_rmse(self, ratings):
m, n = self.shape
mse = 0.0
n_elements = sum(map(len, ratings.values()))
for i in range(m):
for j in range(n):
user_id = self.user_ids[i]
item_id = self.item_ids[j]
rating = ratings[user_id][item_id]
if rating > 0:
user_row = self.user_matrix.col(i).transpose
item_col = self.item_matrix.col(j)
rating_hat = user_row.mat_mul(item_col).data[0][0]
square_error = (rating - rating_hat) ** 2
mse += square_error / n_elements
return mse ** 0.5
# 训练模型
# 1.数据预处理
# 2.变量k合法性检查
# 3.生成随机矩阵U
# 4.交替计算矩阵U和矩阵I，并打印RMSE信息，直到迭代次数达到max_iter
# 5.保存最终的RMSE
def fit(self, X, k, max_iter=10):
ratings, ratings_T = self.process_data(X)
self.user_items = {k: set(v.keys()) for k,v in ratings.items()}
m, n = self.shape
error_msg = "Parameter k must be less than the rank of original matrix"
assert k < min(m, n), error_msg
self.user_matrix = self.gen_random_matrix(k, m)
for i in range(max_iter):
if i % 2:
items = self.item_matrix
self.user_matrix = self.items_mul_ratings(
items.mat_mul(items.transpose).inverse.mat_mul(items),
ratings
)
else:
users = self.user_matrix
self.item_matrix = self.users_mul_ratings(
users.mat_mul(users.transpose).inverse.mat_mul(users),
ratings_T
)
rmse = self.get_rmse(ratings)
print("Iterations: %d, RMSE: %.6f" % (i + 1, rmse))
self.rmse = rmse
# 预测一个用户
def _predict(self, user_id, n_items):
users_col = self.user_matrix.col(self.user_ids_dict[user_id])
users_col = users_col.transpose
items_col = enumerate(users_col.mat_mul(self.item_matrix).data[0])
items_scores = map(lambda x: (self.item_ids[x[0]], x[1]), items_col)
viewed_items = self.user_items[user_id]
items_scores = filter(lambda x: x[0] not in viewed_items, items_scores)
return sorted(items_scores, key=lambda x: x[1], reverse=True)[:n_items]
# 预测多个用户
def predict(self, user_ids, n_items=10):
return [self._predict(user_id, n_items) for user_id in user_ids]
def format_prediction(item_id, score):
return "item_id:%d score:%.2f" % (item_id, score)
@run_time
def main():
print("Tesing the accuracy of ALS...")
X = load_movie_ratings()
model = ALS()
model.fit(X, k=3, max_iter=5)
print("Showing the predictions of users...")
user_ids = range(1, 5)
predictions = model.predict(user_ids, n_items=2)
for user_id, prediction in zip(user_ids, predictions):
_prediction = [format_prediction(item_id, score)
for item_id, score in prediction]
print("User id:%d recommedation: %s" % (user_id, _prediction))
京公网安备 11010502036488号