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On convexity and efficiency in semantic systems

Nathaniel Imel, Noga Zaslavasky

TL;DR

This paper investigates the relationship between convexity and efficiency in human semantic category systems, using color naming as a focal domain. It combines analytical proofs within the Information Bottleneck (IB) framework and empirical analyses to show that convexity and efficiency are distinct properties, though IB-optimal color naming is largely convex; the IB objective is $\mathcal{F}_{\beta}[q] = I_q(M;W) - \beta I_q(W;U)$ and the deviation from optimality is $\varepsilon[q]$. Efficiency proves to be a stronger predictor than convexity for distinguishing attested color naming from rotated or hypothetical variants. Beyond empirical fit, efficiency captures phenomena that convexity cannot, such as adaptation to environmental communicative needs and language evolution. Overall, efficiency provides a more comprehensive account of semantic typology, with convexity offering only partial explanatory power and not a universal constraint.

Abstract

There are two widely held characterizations of human semantic category systems: (1) they form convex partitions of conceptual spaces, and (2) they are efficient for communication. While prior work observed that convexity and efficiency co-occur in color naming, the analytical relation between them and why they co-occur have not been well understood. We address this gap by combining analytical and empirical analyses that build on the Information Bottleneck (IB) framework for semantic efficiency. First, we show that convexity and efficiency are distinct in the sense that neither entails the other: there are convex systems which are inefficient, and optimally-efficient systems that are non-convex. Crucially, however, the IB-optimal systems are mostly convex in the domain of color naming, explaining the main empirical basis for the convexity approach. Second, we show that efficiency is a stronger predictor for discriminating attested color naming systems from hypothetical variants, with convexity adding negligible improvement on top of that. Finally, we discuss a range of empirical phenomena that convexity cannot account for but efficiency can. Taken together, our work suggests that while convexity and efficiency can yield similar structural observations, they are fundamentally distinct, with efficiency providing a more comprehensive account of semantic typology.

On convexity and efficiency in semantic systems

TL;DR

This paper investigates the relationship between convexity and efficiency in human semantic category systems, using color naming as a focal domain. It combines analytical proofs within the Information Bottleneck (IB) framework and empirical analyses to show that convexity and efficiency are distinct properties, though IB-optimal color naming is largely convex; the IB objective is and the deviation from optimality is . Efficiency proves to be a stronger predictor than convexity for distinguishing attested color naming from rotated or hypothetical variants. Beyond empirical fit, efficiency captures phenomena that convexity cannot, such as adaptation to environmental communicative needs and language evolution. Overall, efficiency provides a more comprehensive account of semantic typology, with convexity offering only partial explanatory power and not a universal constraint.

Abstract

There are two widely held characterizations of human semantic category systems: (1) they form convex partitions of conceptual spaces, and (2) they are efficient for communication. While prior work observed that convexity and efficiency co-occur in color naming, the analytical relation between them and why they co-occur have not been well understood. We address this gap by combining analytical and empirical analyses that build on the Information Bottleneck (IB) framework for semantic efficiency. First, we show that convexity and efficiency are distinct in the sense that neither entails the other: there are convex systems which are inefficient, and optimally-efficient systems that are non-convex. Crucially, however, the IB-optimal systems are mostly convex in the domain of color naming, explaining the main empirical basis for the convexity approach. Second, we show that efficiency is a stronger predictor for discriminating attested color naming systems from hypothetical variants, with convexity adding negligible improvement on top of that. Finally, we discuss a range of empirical phenomena that convexity cannot account for but efficiency can. Taken together, our work suggests that while convexity and efficiency can yield similar structural observations, they are fundamentally distinct, with efficiency providing a more comprehensive account of semantic typology.
Paper Structure (14 sections, 2 theorems, 9 equations, 5 figures, 1 table, 2 algorithms)

This paper contains 14 sections, 2 theorems, 9 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Convexity
  4. Efficiency
  5. Revisiting Gärdenfors' conjecture
  6. Convexity and efficiency are distinct: Neither is necessary for the other
  7. Efficiency explains convexity and beyond
  8. Phenomena beyond convexity
  9. Discussion
  10. Acknowledgments
  11. Theoretical results: convexity and efficiency do not entail each other
  12. Preliminaries
  13. Proofs
  14. Algorithms for finding inefficient convex category systems

Key Result

Theorem 1

Optimal IB systems are not necessarily convex. That is, there exists a non-convex system that is IB-optimal. Our second theorem shows that convexity does not entail efficiency by showing that for any convex $k$-category system there is an environment where that system is highly inefficient. It also shows that for the same environment, there is a different convex system that is maximally efficient

Figures (5)

  • Figure 1: (a) The WCS color naming grid, standard in color naming research. (b) The WCS color stimuli visualized in the 3D CIELAB perceptual color space (see our \SMurlFigures for exploring the structure of this 3D space). (c-e) Illustration of convex and non-convex sets. (d-f) Examples of convex and non-convex color category systems in CIELAB. (g) The IB communication model (see main text). (h) The IB theoretical bound (black curve) and empirical results (blue dots) in the domain of color naming, reproduced from Zaslavsky2018Efficient.
  • Figure 2: Convexity consistency (\ref{['eq:convexity_consistency']}) of attested color naming systems (blue), rotated variants (gray) and optimal IB systems (black curve), as a function of their IB-complexity.
  • Figure 3: (a) IB trade-off for human color naming systems (blue circles) from the WCS and a large sample of artificial convex systems (gray circles). Black circles indicate costly convex systems, visualized in (e-g). (b) Same systems in (a), with $k$ (number of major terms) vs. communicative cost. (c,d) Example human color naming system (Buglere, Panama) and its fitted IB system. (e-g) Three example convex, inefficient category systems as partitions in the 2D WCS grid (top row) and convex hulls in the 3D CIELAB color space (bottom row). Interactive versions of these figures and additional systems can be viewed in our \SMurlFigures.
  • Figure 4: (a) Advantage in efficiency (blue) and convexity consistency (orange) scores for rotations of attested systems along the hue dimension of the WCS grid. Shaded regions denote $95\%$ CIs. (b) Mean ROC AUC scores for classifiers trained on efficiency advantage and convexity advantage, evaluated over $5$-fold CV.
  • Figure 5: Example of a non-convex IB optimal system ($\beta \approx 9.11$) with $k=2$ categories. The categories are periodic with crossing boundaries.

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1: Efficiency does not entail convexity
  • proof
  • Theorem 2: Convexity does not entail efficiency
  • proof