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?
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2Most probably, $x = (x_1, \ldots, x_n)$ is a vector with the given components. – Martin R Jun 10 '20 at 07:15
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1@MartinR: no, have a look at the paper. $\mathbf x$ is a dummy vector variable taking all values in a cluster. The $\mathbf x_i$ are also vectors. – Jun 10 '20 at 07:17
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That may be. The question should contain all relevant information in itself, and not refer to an external link only (which may become unreachable or go offline in the future). – Martin R Jun 10 '20 at 07:21
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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
$$\sum_{\mathbf \xi\in \mathbf c}\|\mathbf\xi-\mu\|_2^2.$$
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@ArtTatum: not at all. This defines $\mathbf x$ to be a set of points, not a dummy variable. – Jun 11 '20 at 06:36
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Hi! Can't we just continue to use $\mathbf x_i$ instead of $\mathbf x$? I.e. $\sum_{\mathbf x_i\in \mathbf c_i}|\mathbf x_i-\mu_i|_2^2$? – JDoeDoe Jun 21 '20 at 19:34
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Hi! But does it matter if we use $\mathbf x_i$ or $\mathbf x$? I.e. aren't $\sum_{\mathbf x\in \mathbf c}|\mathbf x-\mu|2^2$ equivalent to $\sum{\mathbf x_i\in \mathbf c}|\mathbf x_i-\mu|_2^2$? – JDoeDoe Jun 22 '20 at 10:06
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