  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.
