When Efficient Communication Explains Convexity

Much recent work has argued that the variation in the languages of the world can be explained from the perspective of efficient communication; in particular, languages can be seen as optimally balancing competing pressures to be simple and to be informative. Focusing on the expression of meaning – semantic typology – the present paper asks what factors are responsible for successful explanations in terms of efficient communication. Using the Information Bottleneck (IB) approach to formalizing this trade-off, we first demonstrate and analyze a correlation between optimality in the IB sense and a novel generalization of convexity to this setting. In a second experiment, we manipulate various modeling parameters in the IB framework to determine which factors drive the correlation between convexity and optimality. We find that the convexity of the communicative need distribution plays an especially important role. These results move beyond showing that efficient communication can explain aspects of semantic typology into explanations for why that is the case by identifying which underlying factors are responsible.

💡 Research Summary

The paper investigates why languages tend to encode word meanings as convex regions and how the efficient communication framework, specifically the Information Bottleneck (IB) model, can account for this phenomenon. The authors first formalize a “quasi‑convexity” measure that extends traditional binary convexity to probabilistic meaning representations. For a probability distribution p(x), they define level‑sets ls(p, t) = {x | p(x) ≥ t}, compute the convexity of each level‑set, and integrate over t to obtain dcon(p). The overall convexity of a conditional distribution (e.g., the IB encoder q(m | w)) is then a weighted average of dcon across all conditioning variables. An algorithm is provided to approximate this measure efficiently.

Using the IB framework, the speaker’s meaning m (a distribution over referents) is compressed into a word w, balancing complexity I(M; W) and accuracy I(W; U) via a trade‑off parameter β. Optimal encoders minimize Fβ = I(M; W) − β I(W; U). The authors compute optimal frontiers with reverse deterministic annealing and generate sub‑optimal encoders by shuffling portions of the encoder’s mapping.

Two experiments are conducted. Experiment 1 applies the methodology to the World Color Survey (WCS). Meanings are Gaussian distributions over CIELab chips; natural language data are turned into frequency‑based encoders. Sub‑optimal encoders are created by shuffling 10‑100 % of the meaning‑word mappings. Quasi‑convexity of q(m | w) and q(u | w) is measured, and optimality is quantified as the negative Euclidean distance to the nearest optimal encoder. Results show a modest but significant positive correlation between quasi‑convexity and optimality, and stronger correlations with complexity and accuracy. Mixed‑effects regression indicates that optimal encoders have slightly lower convexity scores than natural language encoders, while sub‑optimal encoders derived from optimal ones retain higher convexity than those derived from natural encoders. A full linear model including interactions among optimality, complexity, and accuracy explains 93 % of the variance in convexity, revealing that each factor contributes positively but their interactions attenuate the effects.

Experiment 2 explores the causal factors behind the observed correlation by constructing synthetic meaning spaces. Four base environments are defined: convex priors with unique meanings (CPUM), non‑convex priors with unique meanings (NPUM), convex priors with duplicate meanings (CPDM), and non‑convex priors with duplicate meanings (NPDM). Additional variants manipulate the referent set, introduce dual peaks, cluster duplicate meanings, or use Manhattan distance as the referent‑meaning mapping. For each environment, optimal IB frontiers are computed across β values, and the quasi‑convexity of the resulting optimal encoders is evaluated. The key finding is that environments with convex prior distributions over meanings consistently yield higher quasi‑convexity scores, whereas non‑convex priors lead to markedly lower scores, regardless of whether meanings are unique or duplicated. This demonstrates that the shape of the communicative‑need distribution (the prior over meanings) is the primary driver of the convexity‑optimality relationship.

The paper’s contributions are threefold: (1) a novel, mathematically grounded quasi‑convexity metric for probabilistic semantic representations; (2) empirical evidence linking IB optimality to convexity in both real‑world color naming and controlled synthetic settings; (3) identification of the prior’s convexity as the pivotal factor that makes efficient communication predict convex semantic organization. Limitations include reliance on shuffling to generate sub‑optimal encoders (which may not capture natural language evolution) and the focus on color naming as the sole empirical domain. Future work is suggested to test other semantic domains, incorporate learning or evolutionary simulations, and investigate how communicative‑need distributions become convex in real languages. Overall, the study deepens our understanding of why efficient communication not only aligns with linguistic simplicity and informativeness but also gives rise to the geometric regularities observed in semantic typology.