学习来源：日撸 Java 三百行（61-70天，决策树与集成学习）_闵帆的博客——CSDN博客
矩阵分解 1.奇异值分解奇异值分解（SVD）是一种矩阵分解算法，用于将矩阵简化为其组成部分，从而简化后续矩阵计算。SVD的表达式为：
其中： 为以 或 的 特征值的平方根为主对角线元素的对角阵； 是由 的特征向量组成的矩阵； 是由 的特征向量组成的矩阵。
例：矩阵 ， 则 ，特征值为50和0，特征向量为 和； ，特征向量为 和 ；因此得到SVD分解：
2.矩阵分解矩阵分解算法是由奇异值分解算法（SVD）演变而来，传统的SVD只能对数据稠密的矩阵进行分解。而在推荐系统中，评分矩阵是极度稀疏的，因此对SVD进行改进来对用户对项目的评分进行预测。
3.BasicSVDBasicSVD是最简单的矩阵分解方法，它将评分矩阵分解为两个低阶矩阵的乘积，在实际推荐计算时不再使用原矩阵，而是使用分解得到的两个低阶矩阵。
图1.评分矩阵例子
图1描述了5个用户（）对4个物品（）的评分（1-5分） ， “ - ”代表没有评分。通过矩阵分解可以对缺失的评分进行预测。对于矩阵分解有公式：
对应图1的评分矩阵， 为 的矩阵， 为 的矩阵， （）为预测评分矩阵。
4.代码部分1）衡量矩阵分解优劣的指标采用的是MAE。即预测评分与实际评分差值的绝对值之和与预测评分数之比。MAE的值越小越好。
2）由学习因子 和预测评分误差共同作用于每轮训练子矩阵的更新。
3）代码如下：
package JavaDay20;
import java.io.*;
import java.util.Random;
public class MatrixFactorization {
Random rand = new Random();
int numUsers;
int numItems;
int numRatings;
Triple[] dataset;
double alpha;
double lambda;
int rank;
double[][] userSubspace;
double[][] itemSubspace;
double ratingLowerBound;
double ratingUpperBound;
public MatrixFactorization(String paraFilename, int paraNumUsers, int paraNumItems,
int paraNumRatings, double paraRatingLowerBound, double paraRatingUpperBound) {
numUsers = paraNumUsers;
numItems = paraNumItems;
numRatings = paraNumRatings;
ratingLowerBound = paraRatingLowerBound;
ratingUpperBound = paraRatingUpperBound;
try {
readData(paraFilename, paraNumUsers, paraNumItems, paraNumRatings);
// adjustUsingMeanRating();
} catch (Exception ee) {
System.out.println("File " + paraFilename + " cannot be read! " + ee);
System.exit(0);
} // Of try
}// Of the first constructor
public void setParameters(int paraRank, double paraAlpha, double paraLambda) {
rank = paraRank;
alpha = paraAlpha;
lambda = paraLambda;
}// Of setParameters
public void readData(String paraFilename, int paraNumUsers, int paraNumItems,
int paraNumRatings) throws IOException {
File tempFile = new File(paraFilename);
if (!tempFile.exists()) {
System.out.println("File " + paraFilename + " does not exists.");
System.exit(0);
} // Of if
BufferedReader tempBufferReader = new BufferedReader(new FileReader(tempFile));
// Allocate space.
dataset = new Triple[paraNumRatings];
String tempString;
String[] tempStringArray;
for (int i = 0; i < paraNumRatings; i++) {
tempString = tempBufferReader.readLine();
tempStringArray = tempString.split(",");
dataset[i] = new Triple(Integer.parseInt(tempStringArray[0]),
Integer.parseInt(tempStringArray[1]), Double.parseDouble(tempStringArray[2]));
} // Of for i
tempBufferReader.close();
}// Of readData
void initializeSubspaces() {
userSubspace = new double[numUsers][rank];
for (int i = 0; i < numUsers; i++) {
for (int j = 0; j < rank; j++) {
userSubspace[i][j] = rand.nextDouble();
} // Of for j
} // Of for i
itemSubspace = new double[numItems][rank];
for (int i = 0; i < numItems; i++) {
for (int j = 0; j < rank; j++) {
itemSubspace[i][j] = rand.nextDouble();
} // Of for j
} // Of for i
}// Of initializeSubspaces
public double predict(int paraUser, int paraItem) {
double resultValue = 0;
for (int i = 0; i < rank; i++) {
// The row vector of a user and the column vector of an item
resultValue += userSubspace[paraUser][i] * itemSubspace[paraItem][i];
} // Of for i
return resultValue;
}// Of predict
public void train(int paraRounds) {
initializeSubspaces();
for (int i = 0; i < paraRounds; i++) {
updateNoRegular();
if (i % 50 == 0) {
// Show the process
System.out.println("Round " + i);
System.out.println("MAE: " + mae());
} // Of if
} // Of for i
}// Of train
public void updateNoRegular() {
for (int i = 0; i < numRatings; i++) {
int tempUserId = dataset[i].user;
int tempItemId = dataset[i].item;
double tempRate = dataset[i].rating;
double tempResidual = tempRate - predict(tempUserId, tempItemId); // Residual
// Update user subspace
double tempValue = 0;
for (int j = 0; j < rank; j++) {
tempValue = 2 * tempResidual * itemSubspace[tempItemId][j];
userSubspace[tempUserId][j] += alpha * tempValue;
} // Of for j
// Update item subspace
for (int j = 0; j < rank; j++) {
tempValue = 2 * tempResidual * userSubspace[tempUserId][j];
itemSubspace[tempItemId][j] += alpha * tempValue;
} // Of for j
} // Of for i
}// Of updateNoRegular
public double rsme() {
double resultRsme = 0;
int tempTestCount = 0;
for (int i = 0; i < numRatings; i++) {
int tempUserIndex = dataset[i].user;
int tempItemIndex = dataset[i].item;
double tempRate = dataset[i].rating;
double tempPrediction = predict(tempUserIndex, tempItemIndex);// +
// DataInfo.mean_rating;
if (tempPrediction < ratingLowerBound) {
tempPrediction = ratingLowerBound;
} else if (tempPrediction > ratingUpperBound) {
tempPrediction = ratingUpperBound;
} // Of if
double tempError = tempRate - tempPrediction;
resultRsme += tempError * tempError;
tempTestCount++;
} // Of for i
return Math.sqrt(resultRsme / tempTestCount);
}// Of rsme
public double mae() {
double resultMae = 0;
int tempTestCount = 0;
for (int i = 0; i < numRatings; i++) {
int tempUserIndex = dataset[i].user;
int tempItemIndex = dataset[i].item;
double tempRate = dataset[i].rating;
double tempPrediction = predict(tempUserIndex, tempItemIndex);
if (tempPrediction < ratingLowerBound) {
tempPrediction = ratingLowerBound;
} // Of if
if (tempPrediction > ratingUpperBound) {
tempPrediction = ratingUpperBound;
} // Of if
double tempError = tempRate - tempPrediction;
resultMae += Math.abs(tempError);
// System.out.println("resultMae: " + resultMae);
tempTestCount++;
} // Of for i
return (resultMae / tempTestCount);
}// Of mae
public static void testTrainingTesting(String paraFilename, int paraNumUsers, int paraNumItems,
int paraNumRatings, double paraRatingLowerBound, double paraRatingUpperBound,
int paraRounds) {
try {
// Step 1. read the training and testing data
MatrixFactorization tempMF = new MatrixFactorization(paraFilename, paraNumUsers,
paraNumItems, paraNumRatings, paraRatingLowerBound, paraRatingUpperBound);
tempMF.setParameters(5, 0.0001, 0.005);
// Step 3. update and predict
System.out.println("Begin Training ! ! !");
tempMF.train(paraRounds);
double tempMAE = tempMF.mae();
double tempRSME = tempMF.rsme();
System.out.println("Finally, MAE = " + tempMAE + ", RSME = " + tempRSME);
} catch (Exception e) {
e.printStackTrace();
} // Of try
}// Of testTrainingTesting
public static void main(String args[]) {
testTrainingTesting("D:/data/movielens-943u1682m.txt", 943, 1682, 10000, 1, 5, 2000);
}// Of main
public class Triple {
public int user;
public int item;
public double rating;
public Triple() {
user = -1;
item = -1;
rating = -1;
}// Of the first constructor
public Triple(int paraUser, int paraItem, double paraRating) {
user = paraUser;
item = paraItem;
rating = paraRating;
}// Of the first constructor
public String toString() {
return "" + user + ", " + item + ", " + rating;
}// Of toString
}// Of class Triple
}// Of class MatrixFactorization
运行结果
