The Size of a Hyperball in a Conceptual Space
This work provides a foundational mathematical tool for the cognitive framework of conceptual spaces, aiding in knowledge representation and similarity analysis, though it is incremental as it builds on existing metric definitions.
The paper tackles the problem of computing the size of a hyperball in a conceptual space, which represents the set of points with minimal similarity to a center, and derives a formula for this size under the combined Euclidean-Manhattan metric used in such spaces.
The cognitive framework of conceptual spaces [3] provides geometric means for representing knowledge. A conceptual space is a high-dimensional space whose dimensions are partitioned into so-called domains. Within each domain, the Euclidean metric is used to compute distances. Distances in the overall space are computed by applying the Manhattan metric to the intra-domain distances. Instances are represented as points in this space and concepts are represented by regions. In this paper, we derive a formula for the size of a hyperball under the combined metric of a conceptual space. One can think of such a hyperball as the set of all points having a certain minimal similarity to the hyperball's center.