# What is the definition of a “Clustering Feature” in BIRCH algorithm?

The paper for BIRCH (a clustering algorithm) contains definitions of a Clustering Feature (CF) where the notation is unclear (cf. PDF page 3 / section 4).

A cluster contains N d-dimensional entries $\{ \vec{X}_1, \vec{X}_2, \dots, \vec{X}_N \}$ and its CF is composed of three values.

1. N the number of elements (straight forward)
2. $\vec{LS}$ "linear sum of the elements" $\sum_{i=1}^N \vec{X}_i$
3. $SS$ "square sum of the N data points" $\sum_{i=1}^N \vec{X}_i^2$

The third value seems to cause confusion and is throwing me off. Let's start with the paper's definition.

I am parsing the notation as $\sum_{i=1}^N \vec{X}_i^2 = \vec{X}_1^2 + \vec{X}_2^2 + \dots + \vec{X}_N^2$. Now I am left with $\vec{X}^2$ and to my current understanding a dot/scalar product has to be applied, e.g. $\vec{X}^2 = \vec{X} \cdot \vec{X} = \sum_{j=1}^d x_j^2$ where $x_j$ is an Element in $\vec{X} = ( x_1, x_2, \dots , x_d )$.

On the other hand, there are plenty of lectures/slides providing examples in which the value is not a scalar, but the element-wise squared sum, while the notation in the respective definition does not reflect this. The German Wikipedia article even states $SS$ is a "d-dimensional vector".

Basically they are doing it like this, using the earlier notation: $\vec{X}_i = ( x_{i,1}, x_{i,2}, \dots , x_{i,d} )$ $\vec{SS} = (\sum_{i=1}^N x_{i,1}, \sum_{i=1}^N x_{i,2}, \dots, \sum_{i=1}^N x_{i,d}) = (x_{1,1}^2 + x_{2,1}^2 + \dots + x_{N,1}^2, ... )$.

In the end the sum of the components would be the same.

Why is are those examples doing it differently? Am I missing something?

Examples describing SS as a scalar

Examples describing SS as a d-dimensional vector