The Size of a Hyperball in a Conceptual Space

The cognitive framework of conceptual spaces [3] provides geometric means for representing knowledge. A conceptual space is a highdimensional 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. 1 Conceptual Spaces This section presents the cognitive framework of conceptual spaces as defined by Gärdenfors [3] and introduces our formalization of dimensions, domains, and distances as described in [1]. A conceptual space is a high-dimensional space spanned by a set D of socalled “quality dimensions”. Each of these dimensions d ∈ D represents a way in which two stimuli can be judged to be similar or different. Examples for quality dimensions include temperature, weight, time, pitch, and hue. We denote the distance between two points x and y with respect to a dimension d as |xd− yd|. A domain δ ⊆ D is a set of dimensions that inherently belong together. Different perceptual modalities (like color, shape, or taste) are represented by different domains. The color domain for instance consists of the three dimensions hue, saturation, and brightness. Gärdenfors argues based on psychological evidence that distance within a domain δ should be measured by the weighted Euclidean metric: dE(x, y,Wδ) = √∑ d∈δ wd · |xd − yd| ∗ORCID: 0000-0002-1962-1777 1 ar X iv :1 70 8. 05 26 3v 3 [ cs .A I] 1 8 Se p 20 17 The parameter Wδ contains positive weights wd for all dimensions d ∈ δ representing their relative importance. We assume that ∑ d∈δ wd = 1. The overall conceptual space CS is defined as the product space of all dimensions. Again, based on psychological evidence, Gärdenfors argues that distance within the overall conceptual space should be measured by the weighted Manhattan metric dM of the intra-domain distances. Let ∆ be the set of all domains in CS. We define the distance within a conceptual space as follows: dC (x, y,W ) = ∑ δ∈∆ wδ · dE(x, y,Wδ) = ∑ δ∈∆ wδ · √∑ d∈δ wd · |xd − yd| The parameter W=〈W∆, {Wδ}δ∈∆〉 contains W∆, the set of positive domain weights wδ. We require that ∑ δ∈∆ wδ = |∆|. Moreover, W contains for each domain δ ∈ ∆ a set Wδ of dimension weights as defined above. The weights in W are not globally constant, but depend on the current context. One can easily show that dC (x, y,W ) with a given W is a metric. The similarity of two points in a conceptual space is inversely related to their distance. Gärdenfors expresses this as follows : Sim(x, y) = e−c·d(x,y) with a constant c > 0 and a given metric d Properties (like red, round, and sweet) and concepts (like apple, dog, and chair) can be represented by regions in this space: Properties are defined within a single domain and concepts are defined on the overall space. In [1], we have developed a mathematical formalization of concepts and properties. 2 Hyperballs under the Unweighted Metric In general, a hyperball of radius r around a point p can be defined as the set of all points with a distance of at most r to p: H = {x ∈ CS | d(x, p) ≤ r} If the Euclidean distance dE is used, this corresponds to our intuitive notion of a ball – a round shape centered at p. However, under the Manhattan distance dM , hyperballs have the shape of diamonds. Under the combined distance d ∆ C , a hyperball in three dimensions has the shape of a double cone (cf. Figure 1). As similarity is inversely related to distance, one can interpret a hyperball in a conceptual space as the set of all points that have a minimal similarity α to the central point p, where α depends on the radius of the hyperball. In this section, we assume an unweighted version of dC : dC (x, y) = ∑ δ∈∆ √∑ d∈δ |xd − yd|