Distance-driven adaptive trees in biological metric spaces: uninformed accretion does not prevent convergence

We present several variants of a stochastic algorithm which all evolve tree-structured sets adapted to the geometry of general target subsets in metric spaces, and we briefly discuss their relevance to biological modelling. In all variants, one repeatedly draws random points from the target (step 1), each time selecting from the tree to be grown the point which is closest to the point just randomly drawn (step 2), then adding to the tree a new point in the vicinity of that closest point (step 3 or accretion step). The algorithms differ in their accretion rule, which can use the position of the target point drawn, or not. The informed case relates to the early behaviour of self-organizing maps that mimic somatotopy. It is simple enough to be studied analytically near its branching points, which generally follow some unsuccessful bifurcations. Further modifying step 2 leads to a fast version of the algorithm that builds oblique binary search trees, and we show how to use it in high-dimensional spaces to address a problem relevant to interventional medical imaging and artificial vision. In the case of an uninformed accretion rule, some adaptation also takes place, the behaviour near branching points is computationally very similar to the informed case, and we discuss its interpretations within the Darwinian paradigm.