Metricizing the Euclidean Space Toward Desired Distance Relations in Point Clouds

We introduce the concept of an $\varepsilon $ -semimetric that satisfies the same axioms as a topological metric, except for an arbitrarily small allowance to violate the triangle inequality. Under this modification, we demonstrate the possibility of taking arbitrary points in space, assigning arbitrary desired distances between them (independent of their geometric location relative to each other, that is, independent of their “features”), and constructing an $\varepsilon $ -semimetric that measures exactly the desired distances in the point cloud. This results in a threat to fairness and objectiveness in applications of clustering algorithms: suppose that an adversary subjectively classifies people according to its whim or discriminatory preferences. Upon accusations of unethical behavior, the malicious data processor can plausibly deny these as follows: it designs a distance function (an $\varepsilon $ -semimetric) that is (up to a fully controllable numeric “round-off-error” $\varepsilon $ ) equivalent to a standard distance like the Euclidean. However, this crafted distance will exactly reproduce the (malicious) results and thus confirm them while pretending objectivity and transparency, since only standard and explainable artificial intelligence was used. This demonstration works without any data poisoning. We illustrate the method on randomly chosen points with stochastically independent random classifications assigned to them. Then, we apply standard implementations of k-Means and DBSCAN on the data points, which both exactly reproduce the desired (randomly chosen) classes. We also discuss non-adversarial applications of $\varepsilon $ -semimetrics, and corroborate the construction with examples and implementation in Octave.