Galois theory for analogical classifiers
This work provides a theoretical foundation for analogical inference in classification, which could benefit researchers in machine learning and reasoning, though it appears incremental as it builds on existing analogical proportion concepts.
The paper tackles the problem of formalizing analogical classifiers by establishing a Galois theory to relate analogical models and analogy-preserving functions, with results including explicit determination of closed sets for Boolean domains.
Analogical proportions are 4-ary relations that read "A is to B as C is to D". Recent works have highlighted the fact that such relations can support a specific form of inference, called analogical inference. This inference mechanism was empirically proved to be efficient in several reasoning and classification tasks. In the latter case, it relies on the notion of analogy preservation. In this paper, we explore this relation between formal models of analogy and the corresponding classes of analogy preserving functions, and we establish a Galois theory of analogical classifiers. We illustrate the usefulness of this Galois framework over Boolean domains, and we explicitly determine the closed sets of analogical classifiers, i.e., classifiers that are compatible with the analogical inference, for each pair of Boolean analogies.