From Black Box to Bijection: Interpreting Machine Learning to Build a Zeta Map Algorithm
This addresses the challenge of discovering combinatorial bijections when patterns are not visible at human scale, offering a novel approach for researchers in algebraic combinatorics, though it is incremental as it builds on existing concepts with a new method.
The paper tackles the problem of constructing explicit combinatorial bijections in algebraic combinatorics by proposing a machine learning workflow, and as a result, it derives a new algorithmic description of the zeta map called the Scaffolding Map using a transformer trained on paired Dyck paths.
There is a large class of problems in algebraic combinatorics which can be distilled into the same challenge: construct an explicit combinatorial bijection. Traditionally, researchers have solved challenges like these by visually inspecting the data for patterns, formulating conjectures, and then proving them. But what is to be done if patterns fail to emerge until the data grows beyond human scale? In this paper, we propose a new workflow for discovering combinatorial bijections via machine learning. As a proof of concept, we train a transformer on paired Dyck paths and use its learned attention patterns to derive a new algorithmic description of the zeta map, which we call the \textit{Scaffolding Map}.