Is the average squared distance between two clusters equal to the squared distance of the difference in the two centroids?
Simplify \displaystyle \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2} \left\| \mathbf{x}^{(i)}-\mathbf{x}^{(j)} \right\| _2^2where C_1 and C_2 are two different clusters.
For simplicity, use the Euclidean distance for \left\| \mathbf{x}^{(i)}-\mathbf{x}^{(j)} \right\|.
Choose all expressions below that are equal to the above expression.
\displaystyle \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2}\left\| \mathrm{{\boldsymbol \mu }}_1-\mathrm{{\boldsymbol \mu }}_2 \right\| ^2
\displaystyle \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2} \left\| (\mathbf{x}^{(i)}-\mathrm{{\boldsymbol \mu }}_1)+(\mathrm{{\boldsymbol \mu }}_1-\mathrm{{\boldsymbol \mu }}_2)-(\mathbf{x}^{(j)}-\mathrm{{\boldsymbol \mu }}_2) \right\| ^2
\displaystyle \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2}\left( \left\| \mathbf{x}^{(i)}-\mathrm{{\boldsymbol \mu }}_1 \right\| ^2+\left\| \mathrm{{\boldsymbol \mu }}_1-\mathrm{{\boldsymbol \mu }}_2 \right\| ^2+\left\| \mathbf{x}^{(j)}-\mathrm{{\boldsymbol \mu }}_2) \right\| ^2\right)
unanswered
The correct expression that is equal to the given expression is:
\displaystyle \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2}\left\| \mathbf{x}^{(i)}-\mathbf{x}^{(j)} \right\| _2^2
The other expressions provided are not equal to the given expression.
To determine if the average squared distance between two clusters is equal to the squared distance of the difference in the two centroids, let's analyze the given expressions:
\displaystyle \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2} \left\| \mathbf{x}^{(i)}-\mathbf{x}^{(j)} \right\| _2^2
This expression represents the average squared distance between all pairs of points, one from cluster C1 and another from cluster C2.
Now let's evaluate the provided options:
\displaystyle \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2}\left\| \mathrm{{\boldsymbol \mu }}_1-\mathrm{{\boldsymbol \mu }}_2 \right\| ^2
This expression represents the squared distance between the centroids of the two clusters. It does not consider the distances between individual points within the clusters.
\displaystyle \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2} \left\| (\mathbf{x}^{(i)}-\mathrm{{\boldsymbol \mu }}_1)+(\mathrm{{\boldsymbol \mu }}_1-\mathrm{{\boldsymbol \mu }}_2)-(\mathbf{x}^{(j)}-\mathrm{{\boldsymbol \mu }}_2) \right\| ^2
This expression seems to be a manipulation of the original expression, but it does not simplify to the original expression. It adds and subtracts terms involving the cluster centroids.
\displaystyle \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2}\left( \left\| \mathbf{x}^{(i)}-\mathrm{{\boldsymbol \mu }}_1 \right\| ^2+\left\| \mathrm{{\boldsymbol \mu }}_1-\mathrm{{\boldsymbol \mu }}_2 \right\| ^2+\left\| \mathbf{x}^{(j)}-\mathrm{{\boldsymbol \mu }}_2) \right\| ^2\right)
This expression adds the squared distance of each point in C1 from its centroid, the squared distance between the two centroids, and the squared distance of each point in C2 from its centroid. However, this sum is not equivalent to the original expression because it includes additional terms involving the distances between the individual points and their centroids.
Therefore, the only expression that is equal to the original expression is:
\displaystyle \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2} \left\| \mathbf{x}^{(i)}-\mathbf{x}^{(j)} \right\| _2^2
To determine which expressions are equal to the given expression, let's break down the components and simplify them one by one:
Given expression: \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2} \left\| \mathbf{x}^{(i)}-\mathbf{x}^{(j)} \right\| _2^2
First, let's consider the term \left\| \mathbf{x}^{(i)}-\mathbf{x}^{(j)} \right\| _2^2, which represents the squared Euclidean distance between two data points:
\left\| \mathbf{x}^{(i)}-\mathbf{x}^{(j)} \right\| _2^2 = \sum_{k=1}^{d} (x_{k}^{(i)} - x_{k}^{(j)})^2
Where d is the dimensionality of the data and x_{k}^{(i)} and x_{k}^{(j)} are the k-th features of the i-th and j-th data points, respectively.
Now, let's analyze each of the given expressions:
1. \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2}\left\| \mathrm{{\boldsymbol \mu }}_1-\mathrm{{\boldsymbol \mu }}_2 \right\| ^2
This expression involves the squared Euclidean distance between the centroids of the two clusters, \mathrm{{\boldsymbol \mu }}_1 and \mathrm{{\boldsymbol \mu }}_2. It does not consider individual data points within each cluster, so it is not equivalent to the given expression.
2. \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2} \left\| (\mathbf{x}^{(i)}-\mathrm{{\boldsymbol \mu }}_1)+(\mathrm{{\boldsymbol \mu }}_1-\mathrm{{\boldsymbol \mu }}_2)-(\mathbf{x}^{(j)}-\mathrm{{\boldsymbol \mu }}_2) \right\| ^2
This expression involves the squared Euclidean distance between the sum of differences within and between clusters, where \mathrm{{\boldsymbol \mu }}_1 and \mathrm{{\boldsymbol \mu }}_2 represent the centroids of the clusters. While it includes individual data points within each cluster, it also includes additional terms with the centroid differences. Thus, it is not equivalent to the given expression.
3. \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2}\left( \left\| \mathbf{x}^{(i)}-\mathrm{{\boldsymbol \mu }}_1 \right\| ^2+\left\| \mathrm{{\boldsymbol \mu }}_1-\mathrm{{\boldsymbol \mu }}_2 \right\| ^2+\left\| \mathbf{x}^{(j)}-\mathrm{{\boldsymbol \mu }}_2) \right\| ^2\right)
This expression includes both the squared Euclidean distance between data points within each cluster and the squared Euclidean distance between the centroids of the clusters. It is equivalent to the given expression as it captures the same terms (squared distances) for each pair of data points (\mathbf{x}^{(i)} and \mathbf{x}^{(j)}) within the clusters.
So, the only expression that is equal to the given expression is the third one:
\displaystyle \frac{1}{n_1 n_2}\sum _{\mathbf{x}^{(i)}\in C_1}\sum _{\mathbf{x}^{(j)}\in C_2}\left( \left\| \mathbf{x}^{(i)}-\mathrm{{\boldsymbol \mu }}_1 \right\| ^2+\left\| \mathrm{{\boldsymbol \mu }}_1-\mathrm{{\boldsymbol \mu }}_2 \right\| ^2+\left\| \mathbf{x}^{(j)}-\mathrm{{\boldsymbol \mu }}_2) \right\| ^2\right)