微分関連のまとめ(ベクトル@数学) - さっと調べる公式/まとめ

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ベクトルによる微分の基礎

$$ \frac{ \partial a }{ \partial \boldsymbol{x} } = \left( \begin{array}{c} \frac{\partial a}{\partial x_1} \\ \frac{\partial a}{\partial x_2} \\ \vdots \\ \frac{\partial a}{\partial x_n} \end{array} \right) $$ ただし、$a$はスカラー、$\boldsymbol{x}$はベクトル

定数の微分

Eq
導出

$$ \frac{ \partial c }{ \partial \boldsymbol{x} } = \boldsymbol{0} $$ ただし、$c$はスカラー定数、$\boldsymbol{x}$はベクトル

$$ \frac{ \partial c }{ \partial \boldsymbol{x} } = \left( \begin{array}{c} \frac{\partial c}{\partial x_1} \\ \frac{\partial c}{\partial x_2} \\ \vdots \\ \frac{\partial c}{\partial x_n} \end{array} \right)\\ = \left( \begin{array}{c} 0\\ 0\\ \vdots \\ 0 \end{array} \right)\\ =\boldsymbol{0} $$

内積の微分

Eq
導出

$$ \frac{ \partial \boldsymbol{w}^T\boldsymbol{x} }{ \partial \boldsymbol{x} } = \frac{ \partial \boldsymbol{x}^T\boldsymbol{w} }{ \partial \boldsymbol{x} } = w $$

$$ \frac{ \partial \boldsymbol{w}^T\boldsymbol{x} }{ \partial \boldsymbol{x} } = \left( \begin{array}{c} \frac{\partial (w_1x_1 + w_2x_2 + \cdots + w_nx_n)}{\partial x_1} \\ \frac{\partial (w_1x_1 + w_2x_2 + \cdots + w_nx_n)}{\partial x_2} \\ \vdots \\ \frac{\partial (w_1x_1 + w_2x_2 + \cdots + w_nx_n)}{\partial x_n} \end{array} \right)\\ = \left( \begin{array}{c} w_1\\ w_2\\ \vdots \\ w_n \end{array} \right)\\ = \boldsymbol{w} $$

内積の微分(同一ベクトル)

Eq
導出

$$\frac{\partial \boldsymbol{x}^T\boldsymbol{x}}{\partial \boldsymbol{x}} =\frac{\partial |\boldsymbol{x}|^2}{\partial \boldsymbol{x}} = 2\boldsymbol{x}$$

$$\frac{\partial \boldsymbol{x}^T\boldsymbol{x}}{\partial \boldsymbol{x}} = \left( \begin{array}{c} \frac{\partial (x_1^2 + x_2^2 + \cdots + x_n^2)}{\partial x_1} \\ \frac{\partial (x_1^2 + x_2^2 + \cdots + x_n^2)}{\partial x_2} \\ \vdots \\ \frac{\partial (x_1^2 + x_2^2 + \cdots + x_n^2)}{\partial x_n} \end{array} \right)\\ = \left( \begin{array}{c} 2x_1\\ 2x_2\\ \vdots \\ 2x_n \end{array} \right)\\ = 2\boldsymbol{x}$$

内積の微分(同一ベクトル+オフセット)

Eq
導出

$$\frac{\partial (\boldsymbol{x}-\boldsymbol{a})^T(\boldsymbol{x}-\boldsymbol{a})}{\partial \boldsymbol{x}} = 2(\boldsymbol{x}-\boldsymbol{a})$$

$$\frac{\partial (\boldsymbol{x}-\boldsymbol{a})^T(\boldsymbol{x}-\boldsymbol{a})}{\partial \boldsymbol{x}} = \left( \begin{array}{c} \frac{\partial ((x_1-a_1)^2 + (x_2-a_2)^2 + \cdots + (x_n-a_n)^2)}{\partial x_1} \\ \frac{\partial ((x_1-a_1)^2 + (x_2-a_2)^2 + \cdots + (x_n-a_n)^2)}{\partial x_2} \\ \vdots \\ \frac{\partial ((x_1-a_1)^2 + (x_2-a_2)^2 + \cdots + (x_n-a_n)^2)}{\partial x_n} \end{array} \right)\\ = \left( \begin{array}{c} 2(x_1-a_1)\\ 2(x_2-a_2)\\ \vdots \\ 2(x_n-a_n) \end{array} \right)\\ = 2(\boldsymbol{x}-\boldsymbol{a})$$