NMF(非负矩阵分解)的SGD（随机梯度下降）实现

NMF把一个矩阵分解为两个矩阵的乘积，可以用来解决很多问题，例如：用户聚类、item聚类、预测（补全）用户对item的评分、个性化推荐等问题。NMF的过程可以转化为最小化损失函数（即误差函数）的过程，其实整个问题也就是一个最优化的问题。详细实现过程如下：（其中，输入矩阵很多时候会比较稀疏，即很多元素都是缺失项，故数据存储采用的是libsvm的格式，这个类在此忽略）

package NMF_danji;

import java.io.File;
import java.util.ArrayList;

/**
 * @author 玉心sober： http://weibo.com/karensober
 * @date 2013-05-19
 * 
 * */
public class NMF {
	private Dataset dataset = null;
	private int M = -1; // 行数
	private int V = -1; // 列数
	private int K = -1; // 隐含主题数
	double[][] P;
	double[][] Q;

	public NMF(String datafileName, int topics) {
		File datafile = new File(datafileName);
		if (datafile.exists()) {
			if ((this.dataset = new Dataset(datafile)) == null) {
				System.out.println(datafileName + " is null");
			}
			this.M = this.dataset.size();
			this.V = this.dataset.getFeatureNum();
			this.K = topics;
		} else {
			System.out.println(datafileName + " doesn't exist");
		}
	}

	public void initPQ() {
		P = new double[this.M][this.K];
		Q = new double[this.K][this.V];

		for (int k = 0; k < K; k++) {
			for (int i = 0; i < M; i++) {
				P[i][k] = Math.random();
			}
			for (int j = 0; j < V; j++) {
				Q[k][j] = Math.random();
			}
		}
	}

	// 随机梯度下降，更新参数
	public void updatePQ(double alpha, double beta) {
		for (int i = 0; i < M; i++) {
			ArrayList<Feature> Ri = this.dataset.getDataAt(i).getAllFeature();
			for (Feature Rij : Ri) {
				// eij=Rij.weight-PQ for updating P and Q
				double PQ = 0;
				for (int k = 0; k < K; k++) {
					PQ += P[i][k] * Q[k][Rij.dim];
				}
				double eij = Rij.weight - PQ;

				// update Pik and Qkj
				for (int k = 0; k < K; k++) {
					double oldPik = P[i][k];
					P[i][k] += alpha
							* (2 * eij * Q[k][Rij.dim] - beta * P[i][k]);
					Q[k][Rij.dim] += alpha
							* (2 * eij * oldPik - beta * Q[k][Rij.dim]);
				}
			}
		}
	}

	// 每步迭代后计算SSE
	public double getSSE(double beta) {
		double sse = 0;
		for (int i = 0; i < M; i++) {
			ArrayList<Feature> Ri = this.dataset.getDataAt(i).getAllFeature();
			for (Feature Rij : Ri) {
				double PQ = 0;
				for (int k = 0; k < K; k++) {
					PQ += P[i][k] * Q[k][Rij.dim];
				}
				sse += Math.pow((Rij.weight - PQ), 2);
			}
		}

		for (int i = 0; i < M; i++) {
			for (int k = 0; k < K; k++) {
				sse += ((beta / 2) * (Math.pow(P[i][k], 2)));
			}
		}

		for (int i = 0; i < V; i++) {
			for (int k = 0; k < K; k++) {
				sse += ((beta / 2) * (Math.pow(Q[k][i], 2)));
			}
		}

		return sse;
	}

	// 采用随机梯度下降方法迭代求解参数，即求解最终分解后的矩阵
	public boolean doNMF(int iters, double alpha, double beta) {
		for (int step = 0; step < iters; step++) {
			updatePQ(alpha, beta);
			double sse = getSSE(beta);
			if (step % 100 == 0)
				System.out.println("step " + step + " SSE = " + sse);
		}
		return true;
	}

	public void printMatrix() {
		System.out.println("===========原始矩阵==============");
		for (int i = 0; i < this.dataset.size(); i++) {
			for (Feature feature : this.dataset.getDataAt(i).getAllFeature()) {
				System.out.print(feature.dim + ":" + feature.weight + " ");
			}
			System.out.println();
		}
	}

	public void printFacMatrxi() {
		System.out.println("===========分解矩阵==============");
		for (int i = 0; i < P.length; i++) {
			for (int j = 0; j < Q[0].length; j++) {
				double cell = 0;
				for (int k = 0; k < K; k++) {
					cell += P[i][k] * Q[k][j];
				}
				System.out.print(baoliu(cell, 3) + " ");
			}
			System.out.println();
		}
	}

	// 为double类型变量保留有效数字
	public static double baoliu(double d, int n) {
		double p = Math.pow(10, n);
		return Math.round(d * p) / p;
	}

	public static void main(String[] args) {
		double alpha = 0.002;
		double beta = 0.02;

		NMF nmf = new NMF("D:\\myEclipse\\graphModel\\data\\nmfinput.txt", 10);
		nmf.initPQ();
		nmf.doNMF(3000, alpha, beta);

		// 输出原始矩阵
		nmf.printMatrix();

		// 输出分解后矩阵
		nmf.printFacMatrxi();
	}
}

结果：
...

step 2900 SSE = 0.5878774074369989
===========原始矩阵==============
0:9.0 1:2.0 2:1.0 3:1.0 4:1.0
0:8.0 1:3.0 2:2.0 3:1.0
0:3.0 3:1.0 4:2.0 5:8.0
1:1.0 3:2.0 4:4.0 5:7.0
0:2.0 1:1.0 2:1.0 4:1.0 5:3.0
===========分解矩阵==============
8.959 2.007 1.007 0.996 1.007 6.293
7.981 2.972 1.989 1.005 2.046 7.076
3.01 1.601 1.773 1.003 2.005 7.968
4.821 1.009 2.209 1.984 3.968 6.988
2.0 0.991 0.984 0.51 1.0 2.994