In "$\,f_w(x) = \sum_{i}\frac{1}{2}\|wx_i + c - Y\|_2^2\,$", what do the stacked "$2$"s at the end mean?

Question

I got an equation and for me it is not clear for what the last stacked 2's is for?

$$f_w(x) = \sum_{i}\frac{1}{2}\|wx_i + c - Y\|_2^2$$

Answer 1

That’s the square of the two-norm (aka the Euclidean norm).

Answer 2

The subscript is relative to the norm being used. The "2" stands for the euclidean norm. The superscript is a power. $$ \|X\|_2^2 = \left(\left(\sum_{i=1}^n X_i^2 \right)^{1/2}\right)^2 = \sum_{i=1}^n X_i^2 $$

Other examples of norms could be

$$ \|X\|_1 = \sum_{i=1}^n |X_i|, \quad \|X\|_{\infty}= \max_{i=1,\cdots, n}|X_i|. $$