Abstract
The purpose of this chapter is to show how it is possible to efficiently extract the structure of a set of objects by use of the notion of proportional analogy. As a proportional analogy involves four objects, the very naive approach to the problem, has basically a complexity of \(O(n^4)\) for a given set of \(n\) objects. We show, under some conditions on proportional analogy, how to reduce this complexity to \(O(n^2)\) by considering an equivalent problem, that of enumerating analogical clusters that are informative and not redundant. We further show how some improvements make the task tractable. We illustrate our technique with a task related with natural language processing, that of clustering Chinese characters. In this way, we re-discover the graphical structure of these characters.
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