# Galois theory for analogical classifiers

arXiv: https://arxiv.org/abs/2205.04593 · HTML: https://mlsanity.com/paper/2205.04593/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 60 references that are in the map.
- First posted on arXiv 9 May 2022.

## Builds on (most cited first, 9 of 9)

- [Abstraction and Analogy-Making in Artificial Intelligence](https://mlsanity.com/paper/2102.10717/) (2021)
- [Good Counterfactuals and Where to Find Them: A Case-Based Technique for Generating Counterfactuals for Explainable AI (XAI)](https://mlsanity.com/paper/2005.13997/) (2020)
- [VISALOGY: Answering Visual Analogy Questions](https://mlsanity.com/paper/1510.08973/) (2015)
- [Analogical Dissimilarity: Definition, Algorithms and Two Experiments in Machine Learning](https://mlsanity.com/paper/1401.3427/) (2014)
- [Analogical proportions](https://mlsanity.com/paper/2006.02854/) (2020)
- [Towards Analogy-Based Explanations in Machine Learning](https://mlsanity.com/paper/2005.12800/) (2020)
- [A Neural Approach for Detecting Morphological Analogies](https://mlsanity.com/paper/2108.03945/) (2021)
- [Learning to Rank based on Analogical Reasoning](https://mlsanity.com/paper/1711.10207/) (2017)
- [On the Transferability of Neural Models of Morphological Analogies](https://mlsanity.com/paper/2108.03938/) (2021)

## Built on by (most cited first, 6 of 6)

- [Relevant Entity Selection: Knowledge Graph Bootstrapping via Zero-Shot Analogical Pruning](https://mlsanity.com/paper/2306.16296/) (2023)
- [Proportoids](https://mlsanity.com/paper/2210.01751/) (2022)
- [Solving morphological analogies: from retrieval to generation](https://mlsanity.com/paper/2303.18062/) (2023)
- [Any four real numbers are on all fours with analogy](https://mlsanity.com/paper/2407.18770/) (2024)
- [Analogical proportions II](https://mlsanity.com/paper/2405.13461/) (2024)
- [Similarity-based analogical proportions](https://mlsanity.com/paper/2402.18360/) (2024)

## Abstract (arXiv)

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.