A Group-Theoretic Approach to Computational Abstraction: Symmetry-Driven Hierarchical Clustering
Authors: Haizi Yu, Igor Mineyev, Lav R. Varshney; 24(47):1−61, 2023.
Abstract
This theory paper presents a mathematical formulation for computationally emulating human-like abstractions, known as computational abstraction. It explores the abstraction processes that are developed hierarchically from innate priors like symmetries. The nature of abstraction is studied using a group-theoretic approach, which formalizes and practically computes abstractions as symmetry-driven hierarchical clustering. Unlike data-driven clustering methods such as k-means or agglomerative clustering, our abstraction model is data-free, feature-free, similarity-free, and globally hierarchical. It is designed as a lattice rather than a chain. This paper also generalizes several existing works, including Shannon’s information lattice and specialized algorithms for certain symmetry-induced clusterings. Furthermore, it formalizes knowledge discovery applications such as learning music theory from scores and chemistry laws from molecules. Computational abstraction is considered as a first step towards achieving human-level concept learning and knowledge discovery in a principled and cognitive manner.
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