# A Thorough Formalization of Conceptual Spaces

arXiv: https://arxiv.org/abs/1706.06366 · HTML: https://mlsanity.com/paper/1706.06366/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jun 2017.

## Built on by (most cited first, 5 of 5)

- [A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces](https://mlsanity.com/paper/2110.05985/) (2021)
- [The Conceptual VAE](https://mlsanity.com/paper/2203.11216/) (2022)
- [From Conceptual Spaces to Quantum Concepts: Formalising and Learning Structured Conceptual Models](https://mlsanity.com/paper/2401.08585/) (2023)
- [Flexible and Inherently Comprehensible Knowledge Representation for Data-Efficient Learning and Trustworthy Human-Machine Teaming in Manufacturing Environments](https://mlsanity.com/paper/2305.11597/) (2023)
- [Formalising and Learning a Quantum Model of Concepts](https://mlsanity.com/paper/2302.14822/) (2023)

## Abstract (arXiv)

The highly influential framework of conceptual spaces provides a geometric way of representing knowledge. Instances are represented by points in a high-dimensional space and concepts are represented by convex regions in this space. After pointing out a problem with the convexity requirement, we propose a formalization of conceptual spaces based on fuzzy star-shaped sets. Our formalization uses a parametric definition of concepts and extends the original framework by adding means to represent correlations between different domains in a geometric way. Moreover, we define computationally efficient operations on concepts (intersection, union, and projection onto a subspace) and show that these operations can support both learning and reasoning processes.