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I am given the following algorithm for agglomerative hierarchical clustering:

Consider $\mathbb{X} = [ \mathbf{X}_1 \dots \mathbf{X}_n ]$. Fix $\Delta$ and $\mathcal{L}$ (where $\Delta$ and $\mathcal{L}$ are the distance function and linkage, respectively). Allocate $\mathbb{X}$ to $K$ clusters. Suppose $\mathbb{X}$ are in $\kappa$ clusters, $\kappa \le n$. To assign $\mathbb{X}$ to $K < \kappa$ clusters, put $\nu = \kappa$ and consider the $\nu$ clusters $\mathcal{C}_1 \dots \mathcal{C}_\nu$.
1. Calculate pairwise linkages: For $\mathcal{C}_k, \mathcal{C}_\mathscr{l}$, calculate $\mathcal{L}_{k, \mathscr{l}}$.
2. Find the smallest linkage: Call it $\mathcal{L}_{(1)}$.
3. Marge the closest clusters: If $\mathcal{C}_\alpha$ and $\mathcal{C}_\beta$ such that $\mathcal{L}_{\alpha, \beta} = \mathcal{L}_{(1)}$, then merge $\mathcal{C}_\alpha$ and $\mathcal{C}_\beta$ into one new cluster.
4. Rename the remaining clusters: Consider the collection $\mathcal{C}_1 \dots \mathcal{C}_{\nu - 1}$ that is derived from replacing $\mathcal{C}_\alpha$ and $\mathcal{C}_\beta$ by their merged cluster. Now put $\nu = \nu - 1$.
5. Update $\nu$: If $\nu > K$ steps, then repeat steps 1 to 4.

I don't understand this part:

Allocate $\mathbb{X}$ to $K$ clusters. Suppose $\mathbb{X}$ are in $\kappa$ clusters, $\kappa \le n$.

If we allocate $\mathbb{X}$ to $K$ clusters, then how does it then make sense to say that $\mathbb{X}$ are in $\kappa$ clusters? What is meant here by "allocate"?

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  • $\begingroup$ I think this just means that $K$ is the target number of clusters where the procedure stops. $\endgroup$ Oct 10 '20 at 9:02
  • $\begingroup$ @Lewian So "allocate" means that it is the number of clusters that we are setting the algorithm to cluster the data in? $\endgroup$ Oct 10 '20 at 9:04
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    $\begingroup$ I'd think so. That would make sense. I also think the bit you are having trouble with is badly written, and ultimately I cannot guarantee what is meant by something that is badly written. $\endgroup$ Oct 10 '20 at 10:16
  • $\begingroup$ @Lewian I agree. Anyway, thank you for the help. Feel free to post an answer, and I will accept it. $\endgroup$ Oct 10 '20 at 10:17
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I think this just means that K is the target number of clusters where the procedure stops.

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