矩阵求导与最小二乘法

2019-09-28  本文已影响0人  热爱生活的大川

一、矩阵求导

  1. 矩阵求导就是对内部每一项求导
    F \in R^{a*b}, X \in R^{m*n}
    \frac{\partial{F}}{\partial{X}} = \left[\frac{\partial{F}}{\partial{x_{ij}}}\right]_{m*n} ， \frac{\partial{F}}{\partial{x}} = \left[\frac{\partial{f_{ij}}}{\partial{x}}\right]_{a*b}

  2. 矩阵的迹有如下性质：

    • tr(AB)=tr(BA)
    • tr(A^T)=tr(A)

因而可推出如下性质：设x=(x_{ij})_{m*1}

  1. \frac{\partial{x^TA}}{\partial{x}} = A
  2. \frac{\partial{tr(AB)}}{\partial{A}} = B^T
  3. \frac{\partial{tr(ABA^TC)}}{\partial{A}} = C^TAB^T+CAB，相当于分别对A和A^T取偏导后相加
  4. \frac{\partial{x^TAy}}{\partial{A}} = \frac{\partial{tr(x^TAy)}}{\partial{A}} = xy^T，分子为标量可看做矩阵的迹

二、最小二乘法

已知X \in R^{m*n}为参数矩阵，对应标签值为y \in R^{m*1}
引入参数\theta \in R^{n*1}，构造\hat{y}=X\theta，令最小化目标函数为L=\frac{1}{2}(y-X\theta)^T(y-X\theta)，可求出\theta=(X^TX)^{-1}X^Ty.
推导方法：
\begin{align} \frac{\partial{L}}{\partial\theta} & = \frac{\partial{tr((y-X\theta)^T(y-X\theta))}}{2\partial{\theta}} \\ & = \frac{\partial{tr(\theta^TX^TX\theta)}-\partial{tr(2\theta^TX^Ty)}}{2\partial{\theta}} \\ & = X^TX\theta - X^Ty \\ let & = 0 \\ \theta & =(X^TX)^{-1}X^Ty \end{align}

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