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1132 lines
59 KiB
1132 lines
59 KiB
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<title>10.3:基于矩阵分解的协同过滤算法原理 · GitBook</title>
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<li class="chapter " data-level="1.1" data-path="../">
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<a href="../">
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前言
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</a>
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</li>
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<li class="chapter " data-level="1.2" data-path="../Chapter1/">
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<a href="../Chapter1/">
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第一章 绪论
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</a>
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<ul class="articles">
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<li class="chapter " data-level="1.2.1" data-path="../Chapter1/为什么要数据挖掘.html">
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<a href="../Chapter1/为什么要数据挖掘.html">
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1.1:为什么要数据挖掘
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</a>
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</li>
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<li class="chapter " data-level="1.2.2" data-path="../Chapter1/什么是数据挖掘.html">
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<a href="../Chapter1/什么是数据挖掘.html">
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1.2: 什么是数据挖掘
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</a>
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</li>
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<li class="chapter " data-level="1.2.3" data-path="../Chapter1/数据挖掘主要任务.html">
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<a href="../Chapter1/数据挖掘主要任务.html">
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1.3:数据挖掘主要任务
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</a>
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</li>
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</ul>
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</li>
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<li class="chapter " data-level="1.3" data-path="../Chapter2/">
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<a href="../Chapter2/">
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第二章 数据探索
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</a>
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<ul class="articles">
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<li class="chapter " data-level="1.3.1" data-path="../Chapter2/数据与属性.html">
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<a href="../Chapter2/数据与属性.html">
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2.1:数据与属性
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</a>
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</li>
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<li class="chapter " data-level="1.3.2" data-path="../Chapter2/数据的基本统计指标.html">
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<a href="../Chapter2/数据的基本统计指标.html">
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2.2:数据的基本统计指标
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</a>
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</li>
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<li class="chapter " data-level="1.3.3" data-path="../Chapter2/数据可视化.html">
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<a href="../Chapter2/数据可视化.html">
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2.3:数据可视化
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</a>
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</li>
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<li class="chapter " data-level="1.3.4" data-path="../Chapter2/相似性度量.html">
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<a href="../Chapter2/相似性度量.html">
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2.4:相似性度量
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</a>
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</li>
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</ul>
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</li>
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<li class="chapter " data-level="1.4" data-path="../Chapter3/">
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<a href="../Chapter3/">
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第三章 数据预处理
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</a>
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<ul class="articles">
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<li class="chapter " data-level="1.4.1" data-path="../Chapter3/为什么要数据预处理.html">
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<a href="../Chapter3/为什么要数据预处理.html">
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3.1:为什么要数据预处理
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</a>
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</li>
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<li class="chapter " data-level="1.4.2" data-path="../Chapter3/标准化.html">
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<a href="../Chapter3/标准化.html">
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3.2:标准化
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</a>
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</li>
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<li class="chapter " data-level="1.4.3" data-path="../Chapter3/非线性变换.html">
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<a href="../Chapter3/非线性变换.html">
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3.3:非线性变换
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</a>
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</li>
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<li class="chapter " data-level="1.4.4" data-path="../Chapter3/归一化.html">
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<a href="../Chapter3/归一化.html">
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3.4:归一化
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</a>
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</li>
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<li class="chapter " data-level="1.4.5" data-path="../Chapter3/离散值编码.html">
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<a href="../Chapter3/离散值编码.html">
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3.5:离散值编码
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</a>
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</li>
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<li class="chapter " data-level="1.4.6" data-path="../Chapter3/生成多项式特征.html">
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<a href="../Chapter3/生成多项式特征.html">
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3.6:生成多项式特征
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</a>
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</li>
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<li class="chapter " data-level="1.4.7" data-path="../Chapter3/估算缺失值.html">
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<a href="../Chapter3/估算缺失值.html">
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3.7:估算缺失值
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</a>
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</li>
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</ul>
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</li>
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<li class="chapter " data-level="1.5" data-path="../Chapter4/">
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<a href="../Chapter4/">
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第四章 k-近邻
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</a>
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<ul class="articles">
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<li class="chapter " data-level="1.5.1" data-path="../Chapter4/k-近邻算法思想.html">
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<a href="../Chapter4/k-近邻算法思想.html">
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4.1:k-近邻算法思想
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</a>
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</li>
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<li class="chapter " data-level="1.5.2" data-path="../Chapter4/k-近邻算法原理.html">
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<a href="../Chapter4/k-近邻算法原理.html">
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4.2:k-近邻算法原理
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</a>
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</li>
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<li class="chapter " data-level="1.5.3" data-path="../Chapter4/k-近邻算法流程.html">
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<a href="../Chapter4/k-近邻算法流程.html">
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4.3:k-近邻算法流程
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</a>
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</li>
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<li class="chapter " data-level="1.5.4" data-path="../Chapter4/动手实现k-近邻.html">
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<a href="../Chapter4/动手实现k-近邻.html">
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4.4:动手实现k-近邻
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</a>
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</li>
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<li class="chapter " data-level="1.5.5" data-path="../Chapter4/实战案例.html">
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<a href="../Chapter4/实战案例.html">
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4.5:实战案例
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</a>
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</li>
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</ul>
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</li>
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<li class="chapter " data-level="1.6" data-path="../Chapter5/">
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<a href="../Chapter5/">
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第五章 线性回归
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</a>
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<ul class="articles">
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<li class="chapter " data-level="1.6.1" data-path="../Chapter5/线性回归算法思想.html">
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<a href="../Chapter5/线性回归算法思想.html">
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5.1:线性回归算法思想
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</a>
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</li>
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<li class="chapter " data-level="1.6.2" data-path="../Chapter5/线性回归算法原理.html">
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<a href="../Chapter5/线性回归算法原理.html">
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5.2:线性回归算法原理
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</a>
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</li>
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<li class="chapter " data-level="1.6.3" data-path="../Chapter5/线性回归算法流程.html">
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<a href="../Chapter5/线性回归算法流程.html">
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5.3:线性回归算法流程
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</a>
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</li>
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<li class="chapter " data-level="1.6.4" data-path="../Chapter5/动手实现线性回归.html">
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<a href="../Chapter5/动手实现线性回归.html">
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5.4:动手实现线性回归
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</a>
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</li>
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<li class="chapter " data-level="1.6.5" data-path="../Chapter5/实战案例.html">
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<a href="../Chapter5/实战案例.html">
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5.5:实战案例
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</a>
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</li>
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</ul>
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</li>
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<li class="chapter " data-level="1.7" data-path="../Chapter6/">
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<a href="../Chapter6/">
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第六章 决策树
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</a>
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<ul class="articles">
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<li class="chapter " data-level="1.7.1" data-path="../Chapter6/决策树算法思想.html">
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<a href="../Chapter6/决策树算法思想.html">
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6.1:决策树算法思想
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</a>
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</li>
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<li class="chapter " data-level="1.7.2" data-path="../Chapter6/决策树算法原理.html">
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<a href="../Chapter6/决策树算法原理.html">
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6.2:决策树算法原理
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</a>
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</li>
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<li class="chapter " data-level="1.7.3" data-path="../Chapter6/决策树算法流程.html">
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<a href="../Chapter6/决策树算法流程.html">
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6.3:决策树算法流程
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</a>
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</li>
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<li class="chapter " data-level="1.7.4" data-path="../Chapter6/动手实现决策树.html">
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<a href="../Chapter6/动手实现决策树.html">
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6.4:动手实现决策树
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</a>
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</li>
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<li class="chapter " data-level="1.7.5" data-path="../Chapter6/实战案例.html">
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<a href="../Chapter6/实战案例.html">
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6.5:实战案例
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</a>
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</li>
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</ul>
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</li>
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<li class="chapter " data-level="1.8" data-path="../Chapter7/">
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<a href="../Chapter7/">
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第七章 k-均值
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</a>
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<ul class="articles">
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<li class="chapter " data-level="1.8.1" data-path="../Chapter7/k-均值算法思想.html">
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<a href="../Chapter7/k-均值算法思想.html">
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7.1:k-均值算法思想
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</a>
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</li>
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<li class="chapter " data-level="1.8.2" data-path="../Chapter7/k-均值算法原理.html">
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<a href="../Chapter7/k-均值算法原理.html">
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7.2:k-均值算法原理
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</a>
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</li>
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<li class="chapter " data-level="1.8.3" data-path="../Chapter7/k-均值算法流程.html">
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<a href="../Chapter7/k-均值算法流程.html">
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7.3:k-均值算法流程
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</a>
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</li>
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<li class="chapter " data-level="1.8.4" data-path="../Chapter7/动手实现k-均值.html">
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<a href="../Chapter7/动手实现k-均值.html">
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7.4:动手实现k-均值
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</a>
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</li>
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<li class="chapter " data-level="1.8.5" data-path="../Chapter7/实战案例.html">
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<a href="../Chapter7/实战案例.html">
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7.5:实战案例
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</a>
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</li>
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</ul>
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</li>
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<li class="chapter " data-level="1.9" data-path="../Chapter8/">
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<a href="../Chapter8/">
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第八章 Apriori
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</a>
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<ul class="articles">
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<li class="chapter " data-level="1.9.1" data-path="../Chapter8/Apriori算法思想.html">
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<a href="../Chapter8/Apriori算法思想.html">
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8.1:Apriori算法思想
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</a>
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<li class="chapter " data-level="1.9.2" data-path="../Chapter8/Apriori算法原理.html">
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<a href="../Chapter8/Apriori算法原理.html">
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8.2:Apriori算法原理
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</a>
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<li class="chapter " data-level="1.9.3" data-path="../Chapter8/Apriori算法流程.html">
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<a href="../Chapter8/Apriori算法流程.html">
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8.3:Apriori算法流程
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<li class="chapter " data-level="1.9.4" data-path="../Chapter8/动手实现Apriori.html">
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<a href="../Chapter8/动手实现Apriori.html">
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8.4:动手实现Apriori
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</a>
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</li>
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<li class="chapter " data-level="1.9.5" data-path="../Chapter8/实战案例.html">
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<a href="../Chapter8/实战案例.html">
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8.5:实战案例
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<li class="chapter " data-level="1.10" data-path="../Chapter9/">
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<a href="../Chapter9/">
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第九章 PageRank
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<ul class="articles">
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<li class="chapter " data-level="1.10.1" data-path="../Chapter9/PageRank算法思想.html">
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<a href="../Chapter9/PageRank算法思想.html">
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9.1:PageRank算法思想
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<li class="chapter " data-level="1.10.2" data-path="../Chapter9/PageRank算法原理.html">
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<a href="../Chapter9/PageRank算法原理.html">
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9.2:PageRank算法原理
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<li class="chapter " data-level="1.10.3" data-path="../Chapter9/PageRank算法流程.html">
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<a href="../Chapter9/PageRank算法流程.html">
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9.3:PageRank算法流程
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</a>
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</li>
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<li class="chapter " data-level="1.10.4" data-path="../Chapter9/动手实现PageRank.html">
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<a href="../Chapter9/动手实现PageRank.html">
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9.4:动手实现PageRank
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</a>
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</li>
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<li class="chapter " data-level="1.10.5" data-path="../Chapter9/实战案例.html">
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<a href="../Chapter9/实战案例.html">
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9.5:实战案例
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</a>
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</li>
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</ul>
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<li class="chapter " data-level="1.11" data-path="./">
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<a href="./">
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第十章 推荐系统
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</a>
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<ul class="articles">
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<li class="chapter " data-level="1.11.1" data-path="推荐系统概述.html">
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<a href="推荐系统概述.html">
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10.1:推荐系统概述
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</a>
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</li>
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<li class="chapter " data-level="1.11.2" data-path="基于矩阵分解的协同过滤算法思想.html">
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<a href="基于矩阵分解的协同过滤算法思想.html">
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10.2:基于矩阵分解的协同过滤算法思想
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</a>
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<li class="chapter active" data-level="1.11.3" data-path="基于矩阵分解的协同过滤算法原理.html">
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<a href="基于矩阵分解的协同过滤算法原理.html">
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10.3:基于矩阵分解的协同过滤算法原理
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</a>
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</li>
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<li class="chapter " data-level="1.11.4" data-path="基于矩阵分解的协同过滤算法流程.html">
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<a href="基于矩阵分解的协同过滤算法流程.html">
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10.4:基于矩阵分解的协同过滤算法流程
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</li>
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<li class="chapter " data-level="1.11.5" data-path="动手实现基于矩阵分解的协同过滤.html">
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<a href="动手实现基于矩阵分解的协同过滤.html">
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10.5:动手实现基于矩阵分解的协同过滤
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</a>
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<li class="chapter " data-level="1.11.6" data-path="实战案例.html">
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<a href="实战案例.html">
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10.6:实战案例
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</a>
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<h1 id="103基于矩阵分解的协同过滤算法原理">10.3:基于矩阵分解的协同过滤算法原理</h1>
|
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<p>将用户喜好矩阵与内容矩阵进行矩阵乘法就能得到用户对物品的预测结果，而我们的目的是预测结果与真实情况越接近越好。所以，我们将预测值与评分表中已评分部分的值构造平方差损失函数：</p>
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>l</mi><mi>o</mi><mi>s</mi><mi>s</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mo>∑</mo><mrow><mo>(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>)</mo><mo>∈</mo><mi>r</mi><mo>(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>)</mo><mo>=</mo><mn>1</mn></mrow></msub><mo>(</mo><msubsup><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></msubsup><msub><mi>x</mi><mrow><mi>i</mi><mi>l</mi></mrow></msub><msub><mi>w</mi><mrow><mi>l</mi><mi>j</mi></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mo>)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">
|
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loss = \frac{1}{2}\sum\limits_{(i,j)\in r(i,j)=1}(\sum\limits_{l=1}^dx_{il}w_{lj}-y_{ij})^2
|
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.836113em;"></span><span class="strut bottom" style="height:3.352118em;vertical-align:-1.516005em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord mathit" style="margin-right:0.01968em;">l</span><span class="mord mathit">o</span><span class="mord mathit">s</span><span class="mord mathit">s</span><span class="mrel">=</span><span class="mord reset-textstyle displaystyle textstyle uncramped"><span class="mopen sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.686em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle cramped"><span class="mord textstyle cramped"><span class="mord mathrm">2</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mop op-limits"><span class="vlist"><span style="top:1.241005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mopen mtight">(</span><span class="mord mathit mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span><span class="mrel mtight">∈</span><span class="mord mathit mtight" style="margin-right:0.02778em;">r</span><span class="mopen mtight">(</span><span class="mord mathit mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span><span class="mrel mtight">=</span><span class="mord mathrm mtight">1</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="mop op-symbol large-op">∑</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mop op-limits"><span class="vlist"><span style="top:1.202113em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span><span class="mrel mtight">=</span><span class="mord mathrm mtight">1</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-1.250005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord mathit mtight">d</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02691em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span></span></p>
|
|
<p>其中：</p>
|
|
<ul>
|
|
<li>i:第i个用户</li>
|
|
<li>j:第j个物品</li>
|
|
<li>d:第d种因素</li>
|
|
<li>x:用户喜好矩阵</li>
|
|
<li>w:内容矩阵</li>
|
|
<li>y:评分矩阵</li>
|
|
<li>r:评分记录矩阵，无评分记为0，有评分记为1。r(i,j)=1代表用户i对物品j进行过评分，r(i,j)=0代表用户i对物品j未进行过评分</li>
|
|
</ul>
|
|
<p>损失函数<code>python</code>实现代码如下：</p>
|
|
<pre><code class="lang-python"><span class="hljs-keyword">import</span> numpy <span class="hljs-keyword">as</span> np
|
|
loss = np.mean(np.multiply((y-np.dot(x,w))**<span class="hljs-number">2</span>,record))
|
|
</code></pre>
|
|
<p>其中，<code>record</code>为评分记录矩阵。</p>
|
|
<p>我们的目的就是最小化平方差损失函数，通常机器学习都是使用梯度下降的方法来最小化损失函数得到正确的参数。</p>
|
|
<p>对每个参数求得偏导如下：</p>
|
|
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>l</mi><mi>o</mi><mi>s</mi><mi>s</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow></mfrac><mo>=</mo><msub><mo>∑</mo><mrow><mi>j</mi><mo>∈</mo><mi>r</mi><mo>(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>)</mo><mo>=</mo><mn>1</mn></mrow></msub><mo>(</mo><msubsup><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></msubsup><msub><mi>x</mi><mrow><mi>i</mi><mi>l</mi></mrow></msub><msub><mi>w</mi><mrow><mi>l</mi><mi>j</mi></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo><msub><mi>w</mi><mrow><mi>k</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">
|
|
\frac{\partial loss}{\partial x_{ik}} = \sum\limits_{j\in r(i,j)=1}(\sum\limits_{l=1}^dx_{il}w_{lj}-y_{ij})w_{kj}
|
|
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.836113em;"></span><span class="strut bottom" style="height:3.352118em;vertical-align:-1.516005em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord reset-textstyle displaystyle textstyle uncramped"><span class="mopen sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.686em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle cramped"><span class="mord textstyle cramped"><span class="mord mathrm" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord mathrm" style="margin-right:0.05556em;">∂</span><span class="mord mathit" style="margin-right:0.01968em;">l</span><span class="mord mathit">o</span><span class="mord mathit">s</span><span class="mord mathit">s</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mrel">=</span><span class="mop op-limits"><span class="vlist"><span style="top:1.241005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">∈</span><span class="mord mathit mtight" style="margin-right:0.02778em;">r</span><span class="mopen mtight">(</span><span class="mord mathit mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span><span class="mrel mtight">=</span><span class="mord mathrm mtight">1</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="mop op-symbol large-op">∑</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mop op-limits"><span class="vlist"><span style="top:1.202113em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span><span class="mrel mtight">=</span><span class="mord mathrm mtight">1</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-1.250005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord mathit mtight">d</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02691em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02691em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span></span></p>
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>l</mi><mi>o</mi><mi>s</mi><mi>s</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>w</mi><mrow><mi>k</mi><mi>j</mi></mrow></msub></mrow></mfrac><mo>=</mo><msub><mo>∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>r</mi><mo>(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>)</mo><mo>=</mo><mn>1</mn></mrow></msub><mo>(</mo><msubsup><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></msubsup><msub><mi>x</mi><mrow><mi>i</mi><mi>l</mi></mrow></msub><msub><mi>w</mi><mrow><mi>l</mi><mi>j</mi></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo><msub><mi>x</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">
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\frac{\partial loss}{\partial w_{kj}} = \sum\limits_{i\in r(i,j)=1}(\sum\limits_{l=1}^dx_{il}w_{lj}-y_{ij})x_{ik}
|
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.836113em;"></span><span class="strut bottom" style="height:3.352118em;vertical-align:-1.516005em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord reset-textstyle displaystyle textstyle uncramped"><span class="mopen sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.6859999999999999em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle cramped"><span class="mord textstyle cramped"><span class="mord mathrm" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02691em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord mathrm" style="margin-right:0.05556em;">∂</span><span class="mord mathit" style="margin-right:0.01968em;">l</span><span class="mord mathit">o</span><span class="mord mathit">s</span><span class="mord mathit">s</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mrel">=</span><span class="mop op-limits"><span class="vlist"><span style="top:1.241005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span><span class="mrel mtight">∈</span><span class="mord mathit mtight" style="margin-right:0.02778em;">r</span><span class="mopen mtight">(</span><span class="mord mathit mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mclose mtight">)</span><span class="mrel mtight">=</span><span class="mord mathrm mtight">1</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="mop op-symbol large-op">∑</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mopen">(</span><span class="mop op-limits"><span class="vlist"><span style="top:1.202113em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span><span class="mrel mtight">=</span><span class="mord mathrm mtight">1</span></span></span></span><span style="top:-0.000005000000000032756em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-1.250005em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord mathit mtight">d</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.02691em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mbin">−</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span></span></p>
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<p>则梯度为：</p>
|
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>=</mo><mi>r</mi><mi mathvariant="normal">.</mi><mo>(</mo><mi>x</mi><mi>w</mi><mo>−</mo><mi>y</mi><mo>)</mo><msup><mi>w</mi><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">
|
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\Delta x = r.(xw-y)w^T
|
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8913309999999999em;"></span><span class="strut bottom" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord mathrm">Δ</span><span class="mord mathit">x</span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mord mathrm">.</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="mbin">−</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord"><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord mathit mtight" style="margin-right:0.13889em;">T</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span></span></p>
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>w</mi><mo>=</mo><msup><mi>x</mi><mi>T</mi></msup><mo>[</mo><mo>(</mo><mi>x</mi><mi>w</mi><mo>−</mo><mi>y</mi><mo>)</mo><mi mathvariant="normal">.</mi><mi>r</mi><mo>]</mo></mrow><annotation encoding="application/x-tex">
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\Delta w = x^T[(xw-y).r]
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.8913309999999999em;"></span><span class="strut bottom" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord mathrm">Δ</span><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="mrel">=</span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist"><span style="top:-0.413em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord mathit mtight" style="margin-right:0.13889em;">T</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mopen">[</span><span class="mopen">(</span><span class="mord mathit">x</span><span class="mord mathit" style="margin-right:0.02691em;">w</span><span class="mbin">−</span><span class="mord mathit" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathrm">.</span><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mclose">]</span></span></span></span></span></p>
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<p>其中：</p>
|
|
<pre><code>.表示点乘法，无则表示矩阵相乘
|
|
上标T表示矩阵转置
|
|
</code></pre><p>梯度<code>python</code>代码如下：</p>
|
|
<pre><code class="lang-python">x_grads = np.dot(np.multiply(record,np.dot(x,w)-y),w.T)
|
|
w_grads = np.dot(x.T,np.multiply(record,np.dot(x,w)-y))
|
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</code></pre>
|
|
<p>然后再进行梯度下降:</p>
|
|
<pre><code class="lang-python"><span class="hljs-comment">#梯度下降，更新参数</span>
|
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<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(n_iter):
|
|
x_grads = np.dot(np.multiply(record,np.dot(x,w)-y),w.T)
|
|
w_grads = np.dot(x.T,np.multiply(record,np.dot(x,w)-y))
|
|
x = alpha*x - lr*x_grads
|
|
w = alpha*w - lr*w_grads
|
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</code></pre>
|
|
<p>其中：</p>
|
|
<pre><code>n_iter:训练轮数
|
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lr:学习率
|
|
alpha：权重衰减系数，用来防止过拟合
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</code></pre>
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