\[{{\bf{X}}_2} = \left( {\frac{{{x_1}}}{{{{\left\| {\bf{x}} \right\|}_2}}},\frac{{{x_2}}}{{{{\left\| {\bf{x}} \right\|}_2}}}, \cdots ,\frac{{{x_n}}}{{{{\left\| {\bf{x}} \right\|}_2}}}} \right) = \left( {\frac{{{x_1}}}{{\sqrt {x_1^2 + x_2^2 + \cdots + x_n^2} }},\frac{{{x_2}}}{{\sqrt {x_1^2 + x_2^2 + \cdots + x_n^2} }}, \cdots ,\frac{{{x_n}}}{{\sqrt {x_1^2 + x_2^2 + \cdots + x_n^2} }}} \right)\]
\[{\left\| {\bf{A}} \right\|_2} = \sqrt {{2^2} + {3^2} + {6^2}} = \sqrt {4 + 9 + 36} = \sqrt {49} = 7\]
\[{{\bf{A}}_2} = \left( {\frac{2}{7},\frac{3}{7},\frac{6}{7}} \right)\]
图1 L2范数可以看作是向量的长度
L2范数有一大优势：经过L2范数归一化后，一组向量的欧式距离和它们的余弦相似度可以等价
\[\begin{array}{l} D\left( {{{\bf{X}}_{\rm{2}}},{{\bf{Y}}_{\rm{2}}}} \right) = \sqrt {{{\left( {\frac{{{x_1}}}{{{{\left\| {\bf{X}} \right\|}_2}}} - \frac{{{y_1}}}{{{{\left\| {\bf{Y}} \right\|}_2}}}} \right)}^2} + {{\left( {\frac{{{x_2}}}{{{{\left\| {\bf{X}} \right\|}_2}}} - \frac{{{y_2}}}{{{{\left\| {\bf{Y}} \right\|}_2}}}} \right)}^2} + \cdots + {{\left( {\frac{{{x_n}}}{{{{\left\| {\bf{X}} \right\|}_2}}} - \frac{{{y_n}}}{{{{\left\| {\bf{Y}} \right\|}_2}}}} \right)}^2}} \\ \quad \quad \quad \quad \quad \;\;\; = \sqrt {\left( {\frac{{\bf{X}}}{{{{\left\| {\bf{X}} \right\|}_2}}} - \frac{{\bf{Y}}}{{{{\left\| {\bf{Y}} \right\|}_2}}}} \right){{\left( {\frac{{\bf{X}}}{{{{\left\| {\bf{X}} \right\|}_2}}} - \frac{{\bf{Y}}}{{{{\left\| {\bf{Y}} \right\|}_2}}}} \right)}^T}} \\ \quad \quad \quad \quad \quad \;\;\; = \sqrt {\frac{{{\bf{X}}{{\bf{X}}^T}}}{{\left\| {\bf{X}} \right\|_2^2}} - \frac{{{\bf{X}}{{\bf{Y}}^T}}}{{{{\left\| {\bf{X}} \right\|}_2}{{\left\| {\bf{Y}} \right\|}_2}}} - \frac{{{\bf{Y}}{{\bf{X}}^T}}}{{{{\left\| {\bf{X}} \right\|}_2}{{\left\| {\bf{Y}} \right\|}_2}}} + \frac{{{\bf{Y}}{{\bf{Y}}^T}}}{{\left\| {\bf{Y}} \right\|_2^2}}} \\ \quad \quad \quad \quad \quad \;\;\; = \sqrt {\frac{{{\bf{X}}{{\bf{X}}^T}}}{{{\bf{X}}{{\bf{X}}^T}}} - \frac{{2{\bf{X}}{{\bf{Y}}^T}}}{{{{\left\| {\bf{X}} \right\|}_2}{{\left\| {\bf{Y}} \right\|}_2}}} + \frac{{{\bf{Y}}{{\bf{Y}}^T}}}{{{\bf{Y}}{{\bf{Y}}^T}}}} \\ \quad \quad \quad \quad \quad \;\;\; = \sqrt {2 - 2\frac{{{\bf{X}}{{\bf{Y}}^T}}}{{{{\left\| {\bf{X}} \right\|}_2}{{\left\| {\bf{Y}} \right\|}_2}}}} \\ \end{array}\]
\[\begin{array}{l} Sim\left( {{{\bf{X}}_{\rm{2}}},{{\bf{Y}}_{\rm{2}}}} \right) = \frac{{\frac{{{x_1}}}{{{{\left\| {\bf{X}} \right\|}_2}}} \cdot \frac{{{y_1}}}{{{{\left\| {\bf{Y}} \right\|}_2}}}{\rm{ + }}\frac{{{x_{\rm{2}}}}}{{{{\left\| {\bf{X}} \right\|}_2}}} \cdot \frac{{{y_{\rm{2}}}}}{{{{\left\| {\bf{Y}} \right\|}_2}}}{\rm{ + }} \cdots {\rm{ + }}\frac{{{x_n}}}{{{{\left\| {\bf{X}} \right\|}_2}}} \cdot \frac{{{y_n}}}{{{{\left\| {\bf{Y}} \right\|}_2}}}}}{{\sqrt {{{\left( {\frac{{{x_1}}}{{{{\left\| {\bf{X}} \right\|}_2}}}} \right)}^{\rm{2}}}{\rm{ + }}{{\left( {\frac{{{x_{\rm{2}}}}}{{{{\left\| {\bf{X}} \right\|}_2}}}} \right)}^{\rm{2}}}{\rm{ + }} \cdots {{\left( {\frac{{{x_{\rm{n}}}}}{{{{\left\| {\bf{X}} \right\|}_2}}}} \right)}^{\rm{2}}}} \cdot \sqrt {{{\left( {\frac{{{y_1}}}{{{{\left\| {\bf{Y}} \right\|}_2}}}} \right)}^{\rm{2}}}{\rm{ + }}{{\left( {\frac{{{y_{\rm{2}}}}}{{{{\left\| {\bf{Y}} \right\|}_2}}}} \right)}^{\rm{2}}}{\rm{ + }} \cdots {\rm{ + }}{{\left( {\frac{{{y_n}}}{{{{\left\| {\bf{Y}} \right\|}_2}}}} \right)}^{\rm{2}}}} }} \\ \quad \quad \quad \quad \quad \;\;\; = \frac{{\frac{{{x_1}{y_1} + {x_2}{y_2} + \cdots + {x_n}{y_n}}}{{{{\left\| {\bf{X}} \right\|}_2}{{\left\| {\bf{Y}} \right\|}_2}}}}}{{\sqrt {\frac{{x_1^2 + x_2^2 + \cdots + x_n^2}}{{\left\| {\bf{X}} \right\|_2^2}}} \cdot \sqrt {\frac{{y_1^2 + y_2^2 + \cdots y_n^2}}{{\left\| {\bf{Y}} \right\|_2^2}}} }} \\ \quad \quad \quad \quad \quad \;\;\; = \frac{{\frac{{{x_1}{y_1} + {x_2}{y_2} + \cdots + {x_n}{y_n}}}{{{{\left\| {\bf{X}} \right\|}_2}{{\left\| {\bf{Y}} \right\|}_2}}}}}{{\sqrt {\frac{{x_1^2 + x_2^2 + \cdots + x_n^2}}{{x_1^2 + x_2^2 + \cdots + x_n^2}}} \cdot \sqrt {\frac{{y_1^2 + y_2^2 + \cdots y_n^2}}{{y_1^2 + y_2^2 + \cdots y_n^2}}} }} \\ \quad \quad \quad \quad \quad \;\;\; = \frac{{{x_1}{y_1} + {x_2}{y_2} + \cdots + {x_n}{y_n}}}{{{{\left\| {\bf{X}} \right\|}_2}{{\left\| {\bf{Y}} \right\|}_2}}} \\ \quad \quad \quad \quad \quad \;\;\; = \frac{{{\bf{X}}{{\bf{Y}}^T}}}{{{{\left\| {\bf{X}} \right\|}_2}{{\left\| {\bf{Y}} \right\|}_2}}} \\ \end{array}\]
\[D\left( {{{\bf{X}}_{\rm{2}}},{{\bf{Y}}_{\rm{2}}}} \right) = \sqrt {2 - 2sim\left( {{{\bf{X}}_{\rm{2}}},{{\bf{Y}}_{\rm{2}}}} \right)} \]
\[\cos \angle POQ = \frac{{O{P^2} + O{Q^2} - P{Q^2}}}{{2 \cdot OP \cdot OQ}}\]
图2 L2范数归一化后向量对应的点都在单位圆上
\[\cos \angle POQ = \frac{{{1^2} + {1^2} - P{Q^2}}}{2} = \frac{{2 - P{Q^2}}}{2}\]
\[sim\left( {{{\bf{X}}_{\rm{2}}},{{\bf{Y}}_{\rm{2}}}} \right) = \frac{{2 - D{{\left( {{{\bf{X}}_{\rm{2}}},{{\bf{Y}}_{\rm{2}}}} \right)}^2}}}{2} \Rightarrow D\left( {{{\bf{X}}_{\rm{2}}},{{\bf{Y}}_{\rm{2}}}} \right) = \sqrt {2 - 2sim\left( {{{\bf{X}}_{\rm{2}}},{{\bf{Y}}_{\rm{2}}}} \right)} \]