Learning Weakly Convex Sets in Metric Spaces

We introduce the notion of weak convexity in metric spaces, a generalization ordinary commonly used machine learning. It is shown that weakly convex sets can be characterized by closure operator and have unique decomposition into set pairwise disjoint connected blocks. give two generic efficient algorithms, an extensional intensional one for learning concepts study their formal properties. Our experimental results concerning vertex classification clearly demonstrate excellent predictive performance algorithm. Two non-trivial applications algorithm to polynomial PAC-learnability are presented. The first deals with k-convex Boolean functions, which already known efficiently PAC-learnable. how derive this positive result fairly easy way second concerned Euclidean space equipped Manhattan distance. For space, form union axis-aligned hyperrectangles. show consistent examples contains minimum number hyperrectangles found time. In contrast, problem NP-complete if may overlapping.