# L_q Norm

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8/11/2021, 2:17:00 PM
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## Norm의 종류

L_q norm에 대해 확 와닿는 표현이 있어서 기록용으로 남깁니다.
Do you count how many non-zero elements it has ? $L_0$
Do you add up all the elements ? $L_1$
Do you measure how "long" the "arrow" is ? $L_2$
Do you use the size of the biggest element ? $L_\infty$
Do you measure the total length of ellipsoid's axes by a matrix ? $\| X \|_*$ (Nuclear norm)

#### Nuclear norm : $\| X \|_*$

Definition
$\| A \| = \text{trace}(\sqrt{A^TA}) = \sum_{i=1}^{\min (m, n)} \sigma_i (A)$
where $\sigma_i (A)$ is the $i$-th largest singular value of $A$.