Binary classifiers, perceptrons and connectedness in metric spaces and graphs

Abstract

Minsky and Papert's well-known results about the inability of diameter-limited perceptrons to recognise connectedness in the plane are generalised in three important respects: First, to a large class of metric spaces of arbitrary dimension, not necessarily Euclidean; secondly, to the class of all connected graphs; thirdly, in the case of infinite diameter, including Minsky and Papert's case, it is shown that no diameter-limited binary classifier (including neural nets of arbitrary complexity, etc.) can recognise connectedness.