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What is the optimized value of distance $V(x,c)$ when $k=1$ (number of clusters) in k-means? What is the centroid such that it is optimal?

$$V(x,c) = \sum_j \sum_{x_i \rightarrow c_j} D(x_i,c_j)^2$$

which is the sum of square of the distances of each point to the center of the cluster it is assigned to. This cost function depends on the distance $D$ used.

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It just means taking the mean of the data. You can do this by finding the mean of each marginal distribution and putting the marginal means into a vector that is the multivariate mean.

Example: Let $X=(X_1, X_2, X_3)$ have $\bar{X}_1=3$, $\bar{X}_2=-13$, and $\bar{X}_3=-5$. The the multivariate mean is $\bar{X} = (3, -13, -5)$.

(I think this assumes using $L_2$ distance, the usual sense of distance.)

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