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Given a set of points in the Euclidean space $\mathbb{R}^\ell$ with $\ell>1$,the pairwise distances between the points are determined by their spatiallocation and the metric $d$ that we endow $\mathbb{R}^\ell$ with. Hence, thedistance $d(\mathbf x,\mathbf y)=\delta$ between two points is fixed by thechoice of $\mathbf x$ and $\mathbf y$ and $d$. We study the related problem offixing the value $\delta$, and the points $\mathbf x,\mathbf y$, and ask ifthere is a topological metric $d$ that computes the desired distance $\delta$.We demonstrate this problem to be solvable by constructing a metric tosimultaneously give desired pairwise distances between up to $O(\sqrt\ell)$many points in $\mathbb{R}^\ell$. We then introduce the notion of an$\varepsilon$-semimetric $\tilde{d}$ to formulate our main result: for all$\varepsilon>0$, for all $m\geq 1$, for any choice of $m$ points $\mathbfy_1,\ldots,\mathbf y_m\in\mathbb{R}^\ell$, and all chosen sets of values$\{\delta_{ij}\geq 0: 1\leq i<j\leq m\}$, there exists an$\varepsilon$-semimetric $\tilde{\delta}:\mathbb{R}^\ell\times\mathbb{R}^\ell\to\mathbb{R}$ such that $\tilde{d}(\mathbf y_i,\mathbfy_j)=\delta_{ij}$, i.e., the desired distances are accomplished, irrespectivelyof the topology that the Euclidean or other norms would induce. We showcase ourresults by using them to attack unsupervised learning algorithms, specifically$k$-Means and density-based (DBSCAN) clustering algorithms. These have manifoldapplications in artificial intelligence, and letting them run with externallyprovided distance measures constructed in the way as shown here, can makeclustering algorithms produce results that are pre-determined and hencemalleable. This demonstrates that the results of clustering algorithms may notgenerally be trustworthy, unless there is a standardized and fixed prescriptionto use a specific distance function.

Metricizing the Euclidean Space towards Desired Distance Relations in Point Clouds