# Any four real numbers are on all fours with analogy

arXiv: https://arxiv.org/abs/2407.18770 · HTML: https://mlsanity.com/paper/2407.18770/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 52 references that are in the map.
- First posted on arXiv 26 Jul 2024.

## Builds on (most cited first, 15 of 15)

- [Complex Embeddings for Simple Link Prediction](https://mlsanity.com/paper/1606.06357/) (2016)
- [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)
- [Image-embodied Knowledge Representation Learning](https://mlsanity.com/paper/1609.07028/) (2016)
- [Analogies Explained: Towards Understanding Word Embeddings](https://mlsanity.com/paper/1901.09813/) (2019)
- [BERT is to NLP what AlexNet is to CV: Can Pre-Trained Language Models Identify Analogies?](https://mlsanity.com/paper/2105.04949/) (2021)
- [Issues in evaluating semantic spaces using word analogies](https://mlsanity.com/paper/1606.07736/) (2016)
- [VASR: Visual Analogies of Situation Recognition](https://mlsanity.com/paper/2212.04542/) (2022)
- [VISALOGY: Answering Visual Analogy Questions](https://mlsanity.com/paper/1510.08973/) (2015)
- [Can language models learn analogical reasoning? Investigating training objectives and comparisons to human performance](https://mlsanity.com/paper/2310.05597/) (2023)
- [Towards Analogy-Based Explanations in Machine Learning](https://mlsanity.com/paper/2005.12800/) (2020)
- [Relevant Entity Selection: Knowledge Graph Bootstrapping via Zero-Shot Analogical Pruning](https://mlsanity.com/paper/2306.16296/) (2023)
- [Galois theory for analogical classifiers](https://mlsanity.com/paper/2205.04593/) (2022)
- [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, 2 of 2)

- [Proportional Analogies on Probability Distributions via Bayesian Updating](https://mlsanity.com/paper/2608.11724/) (2026)
- [A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds](https://mlsanity.com/paper/2608.14220/) (2026)

## Abstract (arXiv)

This work presents a formalization of analogy on numbers that relies on generalized means. It is motivated by recent advances in artificial intelligence and applications of machine learning, where the notion of analogy is used to infer results, create data and even as an assessment tool of object representations, or embeddings, that are basically collections of numbers (vectors, matrices, tensors). This extended analogy use asks for mathematical foundations and clear understanding of the notion of analogy between numbers. We propose a unifying view of analogies that relies on generalized means defined in terms of a power parameter. In particular, we show that any four increasing positive real numbers is an analogy in a unique suitable power. In addition, we show that any such analogy can be reduced to an equivalent arithmetic analogy and that any analogical equation has a solution for increasing numbers, which generalizes without restriction to complex numbers. These foundational results provide a better understanding of analogies in areas where representations are numerical.