Clustering ; Single Linkage ; and Pairwise Distance Concentration

But what about high dimensions? What is the density of the points near the mean? And how far away is the average point from it’s component mean? Let us address this questions for a single isotropic Gaussian distribution. First, note that E[‖x‖] = nσ. Hence, on average, we expect a point to be rather far from mean, but let us quantify this. Recall, that the distribution of ‖x‖ is a χn distribution with n degrees of freedom. Hence, the variance of ‖x‖ is 2nσ (so the deviation if √ 2nσ). Hence, we expect the average distance to the mean to be √ nσ ± O(nσ); so not only do we expect the points to be far away, they also will be quite far away with low variance.