Bongard Problems: A Topological Data Analysis Approach

Bongard problems are a set of 100 visual puzzles posed by M. M. Bongard, where each puzzle consists of twelve images separated into two groups of six images. The task is to find the unique rule separating the two classes in each given problem. The problems were first posed as a challenge for the AI community to test machines ability to imitate complex, context-depending thinking processes using only minimal information. Although some work was done to solve these problems, none of the previous approaches could automatically solve all of them. The present paper is a contribution to attack these problems with a different approach, combining the tools of persistent homology alongside with machine learning methods. In this work, we present an algorithm and show that it is able to solve problems involving differences in connectivity and size as examples, we also show that it can solve problems involving a much larger set of differences provided the right G-equivariant operators