In this article we present a new modelling framework for structured concepts using a category-theoretic generalisation of conceptual spaces, and show how the conceptual representations can be learned automatically from data, using two very different instantiations: one classical and one quantum. A contribution of the work is a thorough category-theoretic formalisation of our framework. We claim that the use of category theory, and in particular the use of string diagrams to describe quantum processes, helps elucidate some of the most important features of our approach. We build upon Gardenfors' classical framework of conceptual spaces, in which cognition is modelled geometrically through the use of convex spaces, which in turn factorise in terms of simpler spaces called domains. We show how concepts from the domains of shape, colour, size and position can be learned from images of simple shapes, where concepts are represented as Gaussians in the classical implementation, and quantum effects in the quantum one. In the classical case we develop a new model which is inspired by the Beta-VAE model of concepts, but is designed to be more closely connected with language, so that the names of concepts form part of the graphical model. In the quantum case, concepts are learned by a hybrid classical-quantum network trained to perform concept classification, where the classical image processing is carried out by a convolutional neural network and the quantum representations are produced by a parameterised quantum circuit. Finally, we consider the question of whether our quantum models of concepts can be considered conceptual spaces in the Gardenfors sense.
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