Notation of min with i

Question

Ive seen an unfamiliar notation of min. Min Notation:

$$\min_{i=1,\dots,N}\left|\langle w,x_i\rangle\,+\,b\right|\,=\,1.$$

What do they mean with it? Does it mean: Choose $x_i$ that $\left|\langle w,x_i\rangle\,+\,b\right|\,=\,1?$

They want me to take my $x_i$'s in a way that $|\quad|$ of the function is $1$, is that right?

That wouldn't make much sense because $x_i$ are my training examples for an algorithm. So i cant minimize them. They are fixed.

You can read this as "the minimum of $|\langle w,x_i\rangle+b|$ with respect to $i\in{1,2,3,\ldots,N}$ is $1$". In other words, for at least one of the $i\in{1,2,3,\ldots,N}$, the minimum of your quantity is $1$. — user170231, Aug 20 '21 at 01:27
I can't speak to why some author chose to do one thing or another... — user170231, Aug 20 '21 at 02:37

score 2 · Accepted Answer · answered Aug 20 '21 at 01:58

2

Most likely your decision variables are $w$ and $b$.

Choose $w$ and $b$ such that when they satisfy the condition that the smallest element of

$$\{|wx_i+b|: i = 1, \ldots, n\}$$ is equal to $1$.

answered Aug 20 '21 at 01:58

Siong Thye Goh

1 Answers1