Euclidean distance

$$\Vert \mathbf{x}{\Vert }_2:=\sqrt{\sum _{i=1}^n{x}_i^2}=\sqrt{{\mathbf{x}}^{\top }\mathbf{x}}$$

• $L^2$ norm
  • Limitation
    • In many contexts, the squared $L^2$ norm may be undesirable because it increases very slowly near the origin.
    • In several machine learning applications, it is important to discriminate between elements that are exactly zero and elements that are small but nonzero.
      • 이런 경우, 모든 location 에서 동일한 growth rate 를 가지는 $L^1$ norm 을 쓰면 좋다.
      • The $L^1$ norm is commonly used in machine learning when the difference between zero and nonzero elements is very important.