Metricizing the Euclidean Space towards Desired Distance Relations in Point Clouds

Given a set of points in the Euclidean space ℝℓ with ℓ>1, the pairwise distances between the points are determined by their spatial location and the metric d that we endow ℝℓ with. Hence, the distance d(mathbf x,mathbf y)=δ between two points is fixed by the choice of mathbf x and mathbf y and d. We study the related problem of fixing the value δ, and the points mathbf x,mathbf y, and ask if there is a topological metric d that computes the desired distance δ. We demonstrate this problem to be solvable by constructing a metric to simultaneously give desired pairwise distances between up to O(√ℓ) many points in ℝℓ. We then introduce the notion of an varepsilon-semimetric tilded to formulate our main result: for all varepsilon>0, for all m≥ 1, for any choice of m points mathbf y1,ldots,mathbf ym∈ℝℓ, and all chosen sets of values δij≥ 0: 1≤ i<j≤ m, there exists an varepsilon-semimetric tildeδ:ℝℓ× ℝℓ→ℝ such that tilded(mathbf yi,mathbf yj)=δij, i.e., the desired distances are accomplished, irrespectively of the topology that the Euclidean or other norms would induce. We showcase our results by using them to attack unsupervised learning algorithms, specifically k-Means and density-based (DBSCAN) clustering algorithms. These have manifold applications in artificial intelligence, and letting them run with externally provided distance measures constructed in the way as shown here, can make clustering algorithms produce results that are pre-determined and hence malleable. This demonstrates that the results of clustering algorithms may not generally be trustworthy, unless there is a standardized and fixed prescription to use a specific distance function.