I don't understand where this formula for Mean Squared Error is coming from.
How do we arrive at:
$$MSE = \frac{1}{m}||y' - y||_2^2$$
from:
$$MSE = \frac{1}{m}\cdot\sum_i(y'_{i} - y_{i})^2$$
(The source is deeplearningbook)
$\begingroup$
$\endgroup$
asked Jun 10, 2019 at 4:02
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
3
We have $$\|x\|_2=\sqrt{\sum_{i=1}^n x_i^2}$$
Hence $$\|x\|_2^2=\sum_{i=1}^n x_i^2$$
Now let $x=y'-y$ and you obtain your formula.
answered Jun 10, 2019 at 5:16
-
$\begingroup$ Thank you for your response. What is this $||x||_2$ called? $\endgroup$ Commented Jun 10, 2019 at 5:56
-
2
-
2
$\begingroup$
$\endgroup$
- Single bars refers to a vector norm: http://mathworld.wolfram.com/VectorNorm.html
- Double bars refers to a matrix norm: http://mathworld.wolfram.com/MatrixNorm.html
- sub-scripted number refers the power that $x$ is being raised too
The Euclidean norm from above falls into this class and is the 2-norm, and the 1-norm is the norm that corresponds to the rectilinear distance.
Check out $L_p$ space (https://en.wikipedia.org/wiki/Lp_space) for info on the relations between norm spaces and a generalization of the p-norm for finite-dimensional vector spaces.
