singular value decomposition tutorial

来源：互联网发布：gentoo安装软件编辑：程序博客网时间：2024/05/20 10:24

原文地址:

http://www.puffinwarellc.com/index.php/news-and-articles/articles/30-singular-value-decomposition-tutorial.html

为什么svd能够解析矩阵背后的结构?

假设这是phil, tiger, vijar 三个人去打高尔夫, 打了9个洞, 每个人都打出了par

HoleParPhilTigerVijay144442555533333444445444464444744448333395555

加入两个参数: 打球人能力参数和每洞难度系数

根据以上两个参数, 我们就可以计算每洞的预测得分 = 打球人能力参数 * 每洞难度系数

PhilTigerVijay444555333444444444444333555=HoleDifficulty453444435*PlayerAbilityPhilTigerVijay111

根据这两个新添参数, 我们可以还原原先计分板(矩阵)

数据家比较惹人烦, 所以他们要做 rescale

PhilTigerVijay444555333444444444444333555=
HoleDifficulty0.330.410.250.330.330.330.330.250.41*
ScaleFactor21.07*
PlayerAbilityPhilTigerVijay0.580.580.58因为rescaling, 导致又新添了个参数 scale factor, 但实际上是物理意义最小的一个参数

ps: holedifficulty = 左奇异向量 playerability = 右奇异向量

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升级版

刚才的例子中, 我们用了一组参数, 但是实际中有无法用一组参数就能重构的矩阵存在

真实的score board

HoleParPhilTigerVijay144452545533332444545444464354744438324495555

SVD 生成的hole difficulty & player ability, 这是在拥有一组参数情况下最佳的拟合

生成过程: 有一组 hole difficulty 参数后, 问player ability应该怎么改针对预测分值, ->> 有一组 player ability 参数后, 问hole difficulty 应该怎么改针对预测分值 ->> 周而复始

PhilTigerVijay3.954.644.344.275.024.692.422.852.663.974.674.363.644.284.003.694.334.053.333.923.663.083.633.394.555.355.00=HoleDifficulty4.344.692.664.364.004.053.663.395.00*PlayerAbilityPhilTigerVijay0.911.071.00

现在将预测分值减去真实得分, 然后用第2组参数去拟合这个区别, 通过第2组参数对第一组参数所拟合出的预测结果与真实结果的差距进行弥补

PhilTigerVijay0.05-0.640.66-0.28-0.020.310.580.15-0.660.030.33-0.360.36-0.280.00-0.690.67-0.050.670.08-0.66-1.080.370.610.45-0.350.00=HoleDifficulty2-0.18-0.380.800.150.35-0.670.89-1.290.44*PlayerAbility2PhilTigerVijay0.82-0.20-0.53

最后 , 原 M*N 矩阵, 当 M>= N时, 可生成N*N的矩阵

PhilTigerVijay445455332454444354443244555=
HoleDifficulty 1-34.34-0.18-0.904.69-0.38-0.152.660.800.404.360.150.474.000.35-0.294.05-0.670.683.660.890.333.39-1.290.145.000.44-0.36*
PlayerAbility 1-3PhilTigerVijay0.911.071.000.82-0.20-0.53-0.210.76-0.62

1-3 指 3组holeDifficulty & playerAbility 参数

rescaling 之后, 我们得到PhilTigerVijay445455332454444354443244555=
HoleDifficulty 1-30.350.09-0.640.380.19-0.100.22-0.400.280.36-0.080.330.33-0.18-0.200.330.330.480.30-0.440.230.280.640.100.41-0.22-0.25*
ScaleFactor 1-321.070002.010001.42*
PlayerAbility 1-3PhilTigerVijay0.530.620.58-0.820.200.53-0.210.76-0.62

总结, 说白了 svd 干了件降维的工作, 并且在生成的方阵中保存原矩阵的物理特性

PlayerAbility 1-3PhilTigerVijay0.530.620.58-0.820.200.53-0.210.76-0.62 phil 33 杆

tiger 39 杆

vijay 36 杆

另外不难看出方阵中第一条vector 被赋予的原矩阵物理特性最大, 接下来的是为了弥补第一组player ability 未能完全覆盖的部分

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