From conceptual spaces to quantum concepts: formalising and learning structured conceptual models

Abstract

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 Gärdenfors’ 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 Gärdenfors sense.

Appendices

A The shapes dataset

The parameters used in the Spriteworld software to generate the Shapes dataset:

Additional parameters for the colour domain:

A.1 The extended colour dataset

The parameters used in the Spriteworld software to generate the Shapes dataset with more (rainbow) colours:

B Neural architectures and hyper-parameters

image width	64
image height	64
image channels	3
CNN kernel size	$4\times 4$
CNN stride	$2\times 2$
CNN layers	4
CNN filters	64
CNN dense layers	2
CNN dense layer size	256
dimensions of latent space	6
initialization interval	$[-1.0, 1.0]$
for means of priors
initialization interval	$[-7.0, 0.0]$
for log-variances of priors
batch size	32
Adam learning rate	$10^{-3}$
Adam $\beta _1$	0.9
Adam $\beta _2$	0.999
Adam $\epsilon$	$10^{-7}$