Cheaper Spaces

Minds and Machines (2025) 35:6 https://doi.org/10.1007/s11023-024-09704-x Cheaper Spaces Matthieu Moullec1 · Igor Douven2 Received: 27 March 2024 / Accepted: 30 October 2024 © The Author(s) 2024 Abstract Similarity spaces are standardly constructed by collecting pairwise similarity judgments and subjecting those to a dimension-reduction technique such as multidimensional scaling or principal component analysis. While this approach can be effective, it has some known downsides, most notably, it tends to be costly and has limited generalizability. Recently, a number of authors have attempted to mitigate these issues through machine learning techniques. For instance, neural networks have been trained on human similarity judgments to infer the spatial representation of unseen stimuli. However, these newer methods are still costly and fail to generalize widely beyond their initial training sets. This paper proposes leveraging prebuilt semantic vector spaces as a cheap alternative to collecting similarity judgments. Our results suggest that some of those spaces can be used to approximate human similarity judgments at low cost and high speed. Keywords Conceptual spaces · Deep learning · Multidimensional scaling · Psychological representations · Similarity judgments 1 Introduction Over the past two decades, the conceptual spaces framework (CSF) has been gaining popularity in cognitive science and beyond (Gärdenfors, 2000, 2014; Nosofsky, 1986, 1987, 1992; Shepard, 1964, 1987). This is in large part because it offers researchers a mathematical framework for modeling concepts, concept learning, and the use of concepts in categorization and induction (see, among many other publications, Douven, 2016, 2023, 2024a, b; Douven & Gärdenfors, 2020; Douven et al., 2023; Gärdenfors, 2000; Gärdenfors & Osta-Vélez, 2023; Gärdenfors & Warglien, * Matthieu Moullec Igor Douven 1 IHPST, Panthéon–Sorbonne University, Paris, France 2 IHPST, CNRS, Panthéon–Sorbonne University, Paris, France Vol.:(0123456789) 6 Page 2 of 21 M. Moullec, I. Douven 2012; Gärdenfors & Williams, 2001; Osta-Vélez & Gärdenfors, 2020, 2022). According to the CSF, concepts can be represented geometrically, as regions in similarity spaces, which are one- or multidimensional structures with a metric defined on them (for an overview, see Gärdenfors, 2000). The metric measures dissimilarity between items, in that the farther apart two items are in the space, the more dissimilar they are in the respect represented by the space (e.g., the more dissimilar their colors are, if the relevant space is color space; see below). The dimensions of a similarity space aim at representing measurable properties items may have, so that items can be mapped onto points in the space according to the values they assume on these properties. The conceptual spaces approach has been applied across diverse domains of varying complexity. Examples of simple conceptual spaces are temporal space, which is represented by a singular dimension (time), and auditory space, which is characterized by the dimensions of pitch and loudness. Somewhat more complex are color spaces like CIELAB and CIELUV, which are three-dimensional, with hue, luminosity, and saturation as their dimensions. Still more complex conceptual spaces to be found in the literature are ones for actions, events, faces, tastes, scents, moral values, socio-economic status, pain, and much else (see, e.g., Bendifallah et al., 2023; Bourdieu, 1989; Castro et al., 2013; Churchland, 2012; Deauvieau et al., 2014; Douven, 2016; Gärdenfors & Warglien, 2012; Petitot, 1988; Valentine et al., 2016). As intimated, conceptual spaces are built on top of similarity spaces. Similarity spaces can be constructed in a number of different ways. The most common approach entails gathering pairwise similarity judgments for a set of items and then using these judgments as input for a dimension-reduction technique. Multidimensional Scaling (MDS) is the most commonly used technique for this purpose (Borg & Groenen, 1999), while others such as Principal Component Analysis (PCA) and Non-Negative Matrix Factorization (NMF) are employed less frequently (Abdi & Williams, 2010; Castro et al., 2013). Other types of input data, such as confusion probabilities (indicating the likelihood of different items being mistaken for each other) and correlation coefficients (depicting the strength of correlation between items), are also occasionally used. A more recent alternative technique used for building similarity spaces is the spatial arrangement method (SpAM). This method requires participants to position items on a surface such that the distances among the items reflect the participant’s similarity judgments, providing an intuitive and visual representation of perceived similarities (Goldstone, 1994). A common procedure for transforming a similarity space into a conceptual space involves locating the prototypes of the concepts one wishes to represent within the relevant similarity space (Gärdenfors, 2000; Gärdenfors & Williams, 2001). For instance, color prototypes would be situated in CIELAB or CIELUV space. Following the identification and placement of prototypes, the mathematical technique of Voronoi tessellations (Okabe et al., 2000) is applied to segment the similarity space into distinct regions, by associating with each prototype all points in the space that are at least as close to it as they are to any of the other prototypes (Douven, 2016). Each of these regions represents a different concept (e.g., a different color concept if the underlying similarity space was CIELAB or CIELUV space). Cheaper Spaces Page 3 of 21 6 Part of the appeal of the CSF stems from the fact that it is typically fairly straightforward to derive empirical predictions from a conceptual space (e.g., about issues like concept acquisition, or category-based induction, or graded membership, or a variety of other issues), thereby giving theories about those issues clear empirical content. That requires, of course, that the conceptual space is described in some detail. Ideally, one can load it onto one’s computer and use modern software to interact with it (e.g., to measure distances in it, or to measure volumes of regions in the space). In practice, however, there is still only a limited number of conceptual spaces that are easily accessible for researchers, or even accessible at all. The root problem really concerns similarity spaces. For once we have a similarity space, its conversion into a conceptual spaces is generally rather straightforward. However, the construction of the underlying similarity space can be both very time-consuming and very expensive. In addition to this, many of the similarity spaces that are available are not readily generalizable to items that were not used in the process of generating the space. Recently, a number of authors have attempted to mitigate these issues through machine learning techniques. Most (...truncated)