Foundations of Data Science

less than n, then richness is not reasonable to demand. In the next section, we will see a
possibility result to contrast with this impossibility theorem.
8.12.3 Relaxing the axioms
Given that no clustering algorithm can satisfy scale invariance, richness, and consistency,
one might want to relax the axioms in some way. Then one gets the following results.
1. Single linkage with a distance stopping condition satisfies a relaxed scale-invariance
property that states that for α > 1, then f (αd) is a refinement of f(d).
2. Define refinement consistency to be that shrinking distances within a cluster or
expanding distances between clusters gives a refinement of the clustering. Single linkage
with α stopping condition satisfies scale invariance, refinement consistency and richness
except for the trivial clustering of all singletons.
8.12.4 A Satisfiable Set of Axioms
In this section, we propose a different set of axioms that are reasonable for distances
between points in Euclidean space and show that the clustering measure, the sum of
squared distances between all pairs of points in the same cluster, slightly modified, is consistent
with the new axioms. We assume through the section that points are in Euclidean
d-space. Our three new axioms follow.
We say that a clustering algorithm satisfies the consistency condition if, for the clustering
produced by the algorithm on a set of points, moving a point so that its distance
to any point in its own cluster is not increased and its distance to any point in a different
cluster is not decreased, then the algorithm returns the same clustering after the move.
Remark: Although it is not needed in the sequel, it is easy to see that for an infinitesimal
perturbation dx of x, the perturbation is consistent if and only if each point in the cluster
containing x lies in the half space through x with dx as the normal and each point in a
different cluster lies in the other half space.
An algorithm is scale-invariant if multiplying all distances by a positive constant does
not change the clustering returned.
An algorithm has the richness property if for any set K of k distinct points in the
ambient space, there is some placement of a set S of n points to be clustered so that the
algorithm returns a clustering with the points in K as centers. So there are k clusters,
each cluster consisting of all points of S closest to one particular point of K.
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