A concept geometry for conceptual spaces

Abstract

This paper generalizes and extends the theory of conceptual spaces as originally proposed by Gardenförs (Conceptual spaces. Cambridge, MA: MIT Press) to provide further geometric representations of both concepts and object observations within a multi-dimensional fuzzy space corresponding to a subset of a unit hypercube. With these representations, we are able directly to calculate normalized scalar measures both of the similarity of two different concepts and of the degree to which an observation satisfies a concept description, and thus to perform inferences with respect to situational assessments and predictions. This capability is directly relevant to the implementation of Levels 2 and 3 data fusion functions using our approach.

Transferring embodied concepts between perceptually heterogeneous robots

“…In our work, we take into account the degree of membership for a property, as well as the degree of membership for a concept, allowing the robot to take this in consideration during the communication process and when deciding how to act. In order to do this, we use a modified extension of conceptual spaces to allow fuzzy memberships proposed by Rickard [15]. A concept is represented as a graph of nodes consisting of properties, with salience weights for the concept.…”

“…In this paper, all saliency weights are set to one (since all domains are equally important). Nodes for pairs of properties p j and p k are connected with directional edges, with weight C ( j, k), corresponding to the conditional probability that the concept will have property p k given that it has property p j [15]. If the two properties are disjoint (non-overlapping regions) and are from the same domain then C (j,k) = 0.…”

“…Once the properties have stabilized (in that their associations and parameters do not change greatly) concepts can also be learned via supervised learning [15]. Instances again take the form of sensory readings, with the robot attaching a random label, this time to a concept.…”

“…Here, membership in both properties is represented as the minimum of the two. Note that as described in [15], the identity portion of the matrix all have a value of one. However, we modified this to contain the mean value of the property for all instances, in order to preserve the strength of a membership of a property in a concept (e.g.…”

“…In order to compare concepts and categorize instances, the concept matrix described above can be projected into a hypercube graph representation [15]. Specifically, the matrix C is converted into a vector c by concatenating subsequent rows together, so that values from row j and column k correspond to element r = (j − 1)N + k in the vector, where N is the dimensionality of the matrix (corresponding to the number of properties).…”