Matrix factorization

导语:承载上集的矩阵代数入门,今天来聊聊进阶版,矩阵分解。其他集数可在[线性代数]标籤文章找到。有空再弄目录什麽的。

Matrix factorization is quite like an application of invertible matrices, where L is an invertible matrix in LU factorization.

As you may have seen, that solving Ax=b for x can be tedious with all the row-reduction algorithm. Here, we are going to explore another efficient algorithm for find x in matrix equation, which is LU Factorization. Suppose we are given L and U in the following form which reconstruct A. L is an invertible unit lower triangular mxm matrix, while U is the mxn echelon form of A. Recall a way to solve for x is by x=A-1b and A-1 need to be invertible. Since L is invertible, LU is also invertible as proved in previous article in this series. The motivation here is that if we are to compute x for different b, we need to compute A-1bi for every single b. That's not desirable and we should look for ways to circumvent this…

Suppose LU are already given, expressing A=LU is just the first step in LU factorization. Remember our goal of using matrix factorization is to solve for x in matrix equation. So we rely on the following:

Above suggests by row-reducing the following, we can get x. So we introduce y as the intermediate results along our way to get b. Noted that we still need to calculate each b individually for Ax=b, just that with the assistance of LU, less steps are involved.

As we know L as an lower unit triangular matrix, columns must be linearly independent. Since it's mxm, L is also invertible. This means the following:

Indeed, when you get a lower triangular unit matrix L, it's trivial to get Imxm from it. As U is the echelon form of A and is of size mxn, so identity matrix is not guaranteed as the reduced echelon form may not be of square matrix.

The LU factorization algorithm

The prerequisite for using this algorithm is that, given any matrix A in Ax=b, A must be reduceable to echelon form, U, using row replacements of rows in a TOP-DOWN manner. However, this is always a hard requirement to meet and people sometimes relax this restriction into allowing row interchanges before performing top-down sequential row replacement in A. If the requirements are satisfied, it's guaranteed we can get the lower triangular unit matrix L, and the proof of which is shown below:

And if we apply the same sequence of elementary matrices onto L, we restore the identity matrix I as follows:

But now it sounds a bit abstract. What exactly does give us btw? And how is it utilized to find L? The following example shows how. During the row-reduction of A into U, entries below pivot position in each pivot column is zeroed-out. The reverse of elementary row operations just require us to gather all pivot columns before their transformation and pack them into a nxn matrix.

When all pivot columns before row replacement gathered, L is easily available.

Examples

例子不定期更新

[线性代数] 矩阵代数進階:矩阵分解 Matrix factorization的更多相关文章

  1. 矩阵分解(Matrix Factorization)与推荐系统

    转自:http://www.tuicool.com/articles/RV3m6n 对于矩阵分解的梯度下降推导参考如下:

  2. 【Math for ML】矩阵分解(Matrix Decompositions) (下)

    [Math for ML]矩阵分解(Matrix Decompositions) (上) I. 奇异值分解(Singular Value Decomposition) 1. 定义 Singular V ...

  3. 【Math for ML】矩阵分解(Matrix Decompositions) (上)

    I. 行列式(Determinants)和迹(Trace) 1. 行列式(Determinants) 为避免和绝对值符号混淆,本文一般使用\(det(A)\)来表示矩阵\(A\)的行列式.另外这里的\ ...

  4. 推荐系统之矩阵分解及C++实现

    1.引言 矩阵分解(Matrix Factorization, MF)是传统推荐系统最为经典的算法,思想来源于数学中的奇异值分解(SVD), 但是与SVD 还是有些不同,形式就可以看出SVD将原始的评 ...

  5. Mahout分布式运行实例:基于矩阵分解的协同过滤评分系统(一个命令实现文件格式的转换)

     Apr 08, 2014  Categories in tutorial tagged with Mahout hadoop 协同过滤  Joe Jiang 前言:之前配置Mahout时测试过一个简 ...

  6. matlab之矩阵分解

    矩阵分解 矩阵分解 (decomposition, factorization)是将矩阵拆解为数个矩阵的乘积. 1.三角分解法: 要求原矩阵为方阵,将之分解成一个上三角形矩阵(或是排列(permute ...

  7. Matrix Factorization SVD 矩阵分解

    Today we have learned the Matrix Factorization, and I want to record my study notes. Some kownledge ...

  8. 【RS】Sparse Probabilistic Matrix Factorization by Laplace Distribution for Collaborative Filtering - 基于拉普拉斯分布的稀疏概率矩阵分解协同过滤

    [论文标题]Sparse Probabilistic Matrix Factorization by Laplace Distribution for Collaborative Filtering  ...

  9. 【RS】List-wise learning to rank with matrix factorization for collaborative filtering - 结合列表启发排序和矩阵分解的协同过滤

    [论文标题]List-wise learning to rank with matrix factorization for collaborative filtering   (RecSys '10 ...

随机推荐

  1. SQL 多并发 多人取号,防止重复取号SQL 办法

    BEGIN TRAN         SELECT  *  from 表明 WITH(HOLDLOCK)        UPDATE 表名 SET 值=1 WHERE 字段=@carrierNo;   ...

  2. 字节数组(byte[])与16进制字符串转换

    /// <summary> /// 转换扩展类 /// </summary> public static class ConvertExtend { /// <summa ...

  3. kubernets 证书过期的问题

    .问题起源 kubeadm 是 kubernetes 提供的一个初始化集群的工具,使用起来非常方便.但是它创建的apiserver.controller-manager等证书默认只有一年的有效期,同时 ...

  4. Jerry Wang从2017年到2019年的自由泳学习笔记

    打腿 把注意力调整到脚部,尽量不要让他打出水面,因为在空气中大腿完全是无用功,但是如果只是脚跟出水一点,倒也没什么关心,但是主观上,要控制一下,如果你听到你的打腿是"咚咚咚咚"这样 ...

  5. 前台.cshtml得到cookie值方法

    function Cookie_() { $.ajax({ url: "/Login_/do_cookie",//请求地址 dataType: "json",/ ...

  6. Computer Vision_33_SIFT:An efficient SIFT-based mode-seeking algorithm for sub-pixel registration of remotely sensed images——2015

    此部分是计算机视觉部分,主要侧重在底层特征提取,视频分析,跟踪,目标检测和识别方面等方面.对于自己不太熟悉的领域比如摄像机标定和立体视觉,仅仅列出上google上引用次数比较多的文献.有一些刚刚出版的 ...

  7. 基于numpy实现矩阵计算器

    要求 制作一个Python的矩阵计算器: ① 程序提供任意两矩阵的加.乘法运算:方阵的行列式计算.逆矩阵计算.特征分解:任意矩阵的转置等计算功能,可自行添加功能 ② 从控制台通过键盘获取数据并完成以上 ...

  8. Nginx的平滑升级记录---适用于编译安装的Nginx

    一.查看自己的Nginx的版本号 [root@localhost sbin]# cd /usr/local/nginx/sbin/ [root@localhost sbin]# ls nginx [r ...

  9. nginx基础概述

    为什么选择nginx     nginx非常轻量     互联网公司都选择nginx nginx技术成熟,具备的功能时企业最常用使用而且最需要的 适合当前主流架构趋势,微服务.云架构.中间层 统一技术 ...

  10. 18计科学期总成绩明细 && 小黄衫团队预选

    1.个人所有作业+团队作业总得分排名 2.千帆竞发图 3.各团队每次作业得分汇总 根据团队成绩排名,众志陈成队得分最高,预选为小黄衫团队. 对团队成绩有异议的团队,请在成绩预选发布后的两天内私信助教解 ...