What does it mean when a variable is defined with a subscript but used without one in equations?

Question

This article defines $x_i ,i=1,\dots,n$ where $x$ has a subscript. However succeeding the definition, the formula contains only $x$.

How am I supposed to interpret this generic notation in terms of math?

Answer 1

Welcome to the world of dummy variables.

In $$\mathbf x_i,i=1,\cdots n$$

$i$ is a dummy index, meaning that you refer to a set of $n$ points, which are numbered. But you could has well have written

$$\mathbf x_k,k=1,\cdots n$$ with exactly the same meaning. You can use the same trick in summation formulas, like

$$\sum_{j=1}^n \mathbf x_j.$$

Finally, the $\mathbf x$ that troubles you is also a dummy variable, used in

$$\sum_{\mathbf x\in \mathbf c}\|\mathbf x-\mu\|_2^2$$ as the range of a summation. They could as well have written

$$\sum_{\mathbf \xi\in \mathbf c}\|\mathbf\xi-\mu\|_2^2.$$