[Paper Review] A Group-Theoretic Approach to Computational Abstraction: Symmetry-Driven Hierarchical Clustering

Haizi Yu, Igor Mineyev+1 more|arXiv (Cornell University)|Jul 30, 2018

Complex Network Analysis Techniques38 references4 citations

TL;DR

This paper proposes a group-theoretic framework for computational abstraction that generates hierarchical clusterings based on symmetries rather than data, enabling data-free, feature-free, similarity-free, and globally hierarchical concept formation. The method uses group actions to induce partitions of mathematical spaces, and when coupled with statistical inference, it realizes Shannon’s information lattice, enabling transparent, cognition-inspired concept learning as demonstrated in music composition.

ABSTRACT

Abstraction plays a key role in concept learning and knowledge discovery; this paper is concerned with computational abstraction. In particular, we study the nature of abstraction through a group-theoretic approach, formalizing it as symmetry-driven---as opposed to data-driven---hierarchical clustering. Thus, the resulting clustering framework is data-free, feature-free, similarity-free, and globally hierarchical---the four key features that distinguish it from common data clustering models such as $k$-means. Beyond a theoretical foundation for abstraction, we also present a top-down and a bottom-up approach to establish an algorithmic foundation for practical abstraction-generating methods. Lastly, via both a theoretical explanation and a real-world application, we illustrate that further coupling of our abstraction framework with statistics realizes Shannon's information lattice and even further, brings learning into the picture. This not only presents one use case of our proposed computational abstraction, but also gives a first step towards a principled and cognitive way of automatic concept learning and knowledge discovery.

Motivation & Objective

• To develop a universal, principled approach to computational abstraction that does not rely on data, features, or similarity metrics.
• To formalize abstraction as a consequence of symmetry operations, using group theory as a universal prior across domains.
• To establish a theoretical and algorithmic foundation for generating hierarchical abstractions in a top-down or bottom-up manner.
• To demonstrate how coupling the abstraction framework with statistical learning yields an information lattice, enabling transparent, cognitive-inspired concept learning.
• To provide a middle ground between rule-based and data-driven AI by integrating innate-like symmetries with empirical learning.

Proposed method

• Formalizes abstraction as symmetry-driven clustering, where group actions on a mathematical space (e.g., vector space or manifold) generate partitions.
• Uses group actions to induce equivalence relations, resulting in clusterings that treat symmetric elements as equivalent—forgetting within-cluster variations.
• Introduces a top-down method to build abstractions from high-level symmetries and a bottom-up method to recover symmetries from data.
• Employs Tsallis entropy maximization under linear constraints (representing learned rules) to realize abstractions in a probabilistic, information-theoretic framework.
• Represents abstractions as information elements in a lattice, with constraints $ H(\bm{y}^k|\bm{x}) = 0 $ ensuring that higher-level abstractions $ \bm{y}^k $ are consistent with the base information $ \bm{x} $.
• Uses partition matrices $ A^k $ and probability vectors $ y^k $ to encode rules as linear constraints, enabling optimization for maximum randomness under abstraction fidelity.

Experimental results

Research questions

• RQ1 How can abstraction be formalized as a consequence of symmetry rather than data or similarity?
• RQ2 Can a universal, domain-agnostic prior for abstraction be constructed using group-theoretic principles?
• RQ3 How can a hierarchical abstraction structure be algorithmically generated without relying on features or data?
• RQ4 What is the role of statistical inference in connecting abstract symmetries to real-world learning, and how does it yield a cognitive-like learning process?
• RQ5 Can the resulting abstraction framework be integrated with information theory to form a transparent, interpretable learning system?

Key findings

• The proposed framework achieves data-free, feature-free, similarity-free, and globally hierarchical clustering by deriving partitions from group actions on mathematical spaces.
• The method enables a top-down and bottom-up algorithmic construction of abstractions, allowing both conceptual design and recovery from data.
• When combined with statistical learning, the abstraction framework realizes Shannon’s information lattice, with abstractions represented as information elements satisfying $ H(\bm{y}^k|\bm{x}) = 0 $.
• In the MUS-ROVER application, the student agent maximizes Tsallis entropy under rule constraints, generating novel, rule-compliant music while memorizing high-level principles.
• The learning process in MUS-ROVER visually mimics Bach’s compositional mind, with the information lattice reflecting hierarchical abstraction and rule activation.
• The framework provides a transparent, interpretable alternative to black-box models, resembling human-like learning through innate-like symmetries and experiential data.