The Parallelogram Hypothesis Revisited: Why Language Models Outperform Humans at Analogical Completion

Introduction

The four term word analogy, formalized as A:B::C:D, has served as a cornerstone of cognitive science research for decades. The canonical example, king:queen::man:woman, appears deceptively simple, yet it captures the essence of relational reasoning: the ability to recognize that the relationship between king and queen mirrors that between man and woman, despite the surface differences between the pairs. Since Rumelhart and Abrahamson (1973), the dominant computational account of this phenomenon has been the parallelogram model, which posits that concepts occupy points in a geometric mental space and that analogical relationships correspond to vectors. To solve for D in an analogy, one simply applies the vector difference between A and B to the point C, completing the geometric parallelogram.

However, this elegant geometric account faced serious challenges in recent years. Peterson et al. (2020) demonstrated that when humans generate completions for word analogies, their responses are often better predicted by simple local similarity heuristics than by the parallelogram structure. Specifically, humans frequently select words that are highly similar to the C term alone, regardless of whether the A:B relationship is preserved. These findings suggested that the parallelogram model might be psychologically implausible; perhaps humans rely on associative similarity rather than abstract relational structure when generating analogies.

A new study by Liu, Marjieh, Zhu, Goldberg, and Griffiths, titled "Parallelograms Strike Back: LLMs Generate Better Analogies than People," offers a provocative reframing of this debate. Rather than assuming the parallelogram model is flawed, the authors ask whether humans might simply be inconsistent at generating relation preserving analogies. By comparing human and large language model (LLM) completions on identical analogy problems, they provide evidence that the geometric model captures something important about analogical quality, even if humans frequently fail to instantiate it.

Experimental Design and Core Findings

The study employed a straightforward yet rigorous methodology. The researchers selected analogy problems from Peterson et al. (2020) spanning multiple semantic relation categories from SemEval 2012 Task 2, including taxonomic relations (flower:tulip), part whole relations (car:engine), contrasts (old:young), and causal relations (joke:laughter). They prompted six state of the art LLMs to complete these analogies and collected human judgments rating the quality of both human and LLM generated completions.

The results challenge the narrative that local similarity heuristics provide the best account of analogy. Human judges reliably rated LLM generated analogies as superior to those produced by human participants. Critically, the researchers quantified the geometric alignment of completions using GloVe embeddings (Pennington et al., 2014), a distributional semantic model that represents words as vectors in a high dimensional space. They computed parallelogram alignment by measuring how closely the vector operation C + (B - A) approximated the actual embedding of the generated D term.

The analysis revealed that LLM completions showed significantly greater alignment with the parallelogram structure than human completions. Moreover, this geometric alignment was the primary driver of the quality advantage, not enhanced sensitivity to local similarity. The researchers also examined word accessibility, operationalized as word frequency in natural language corpora. They found that human completions tended to rely on more frequent, accessible words, whereas LLMs produced less common but relationally more appropriate terms. This suggests that human analogy generation is contaminated by retrieval biases and working memory constraints, pushing participants toward easily accessible but relationally weak responses.

The Distribution Problem: Modal Responses vs. the Long Tail

Perhaps the most nuanced finding of the paper concerns the distribution of responses. When the authors compared the full set of human completions against LLM outputs, the LLMs appeared markedly superior. However, when the analysis was restricted to modal responses, defined as the most frequently generated completion for each analogy, the LLM advantage disappeared. Human modal responses were competitive with LLM outputs in terms of judged quality.

This pattern reveals that the LLM advantage stems primarily from consistency rather than uniformly superior relational insight. Human participants generate a "long tail" of weak completions driven by word accessibility and associative priming. While humans occasionally produce excellent analogies that satisfy geometric constraints, they also frequently produce poor completions that share only superficial similarity with the C term. LLMs, by contrast, demonstrate remarkable consistency in generating relationally appropriate completions across the entire distribution.

This distinction between modal performance and distributional performance has profound implications for cognitive modeling. Previous work, such as Peterson et al. (2020), interpreted the poor fit of the parallelogram model to human data as evidence against geometric representations of analogy. However, Liu et al. demonstrate that the parallelogram structure remains a valid predictor of which analogies humans judge to be good, even if humans frequently fail to generate such analogies themselves. The model was not wrong about the structure of analogy; it was wrong as a descriptive model of human generation, which is noisy and subject to accessibility biases.

Cognitive Implications and the Nature of Relational Reasoning

These findings force us to reconsider the relationship between competence and performance in analogical reasoning. Humans appear to possess the competence to recognize high quality relational mappings, as evidenced by their ratings of LLM completions, yet they lack the consistent performance to generate such mappings under typical experimental conditions. Time pressure, memory retrieval failures, and the cognitive demands of mapping abstract relations may all contribute to this performance gap.

LLMs, conversely, demonstrate consistent performance that aligns with the parallelogram model, but this consistency does not necessarily imply that these models possess human like competence in relational reasoning. The attention mechanisms and next token prediction objectives that underlie LLM behavior may simply extract statistical regularities from training data that happen to correspond to geometric relational structure. The fact that LLM completions align with GloVe parallelograms suggests that the geometric structure of analogy is deeply embedded in linguistic distributions, even if individual humans access this structure inconsistently.

This also raises questions about the role of word frequency in analogical reasoning. The finding that lower frequency words predict higher analogy ratings suggests that the most relationally precise terms are often less cognitively accessible. Humans appear to settle for "good enough" completions that come easily to mind, a phenomenon reminiscent of satisficing behavior in decision making. LLMs, unburdened by working memory limitations or retrieval dynamics, can search the solution space more thoroughly to find terms that satisfy the relational constraint.

Original Insights and Limitations

This work illuminates a methodological pitfall in cognitive science: conflating the noise of human performance with the structure of the underlying competence. When behavioral data contains high variance due to accessibility effects or response biases, models that account for this noise (such as local similarity heuristics) may outperform models that capture the true relational structure (such as the parallelogram model). By using LLMs as a "consistency filter," Liu et al. demonstrate that the geometric model captures normative standards of analogical quality that humans endorse but do not always achieve.

However, several limitations warrant consideration. The study relies on GloVe embeddings, which are static, word level representations trained on co occurrence statistics. Modern contextual embeddings from transformer models might reveal different geometric properties, and the parallelogram structure might manifest differently in these higher dimensional, context sensitive spaces. Additionally, the four term word analogy task, while classic, represents only a narrow slice of analogical reasoning. Complex analogies involving visual, causal, or scientific relations may not conform to simple vector arithmetic, and the consistency advantage of LLMs might diminish as analogical complexity increases.

Furthermore, the study does not establish whether LLMs are explicitly computing parallelogram like operations internally or whether their outputs merely covary with geometric structure in embedding space. The alignment between LLM completions and GloVe parallelograms could reflect shared training objectives rather than explicit geometric reasoning.

Conclusion

The parallelogram model of analogy has survived its apparent empirical refutation, but in an unexpected form. Rather than describing how humans generate analogies, it appears to prescribe what humans consider good analogies. Large language models outperform human participants not because they access deeper cognitive primitives, but because they avoid the accessibility driven errors that plague human generation. They serve as consistency engines that approximate the geometric ideal with regularity that biological cognition cannot match under standard conditions.

Looking forward, this research suggests that LLMs may serve as valuable tools for denoising human behavioral data, helping researchers distinguish between variance due to task demands and variance that reflects genuine cognitive structure. Future work should examine whether the consistency advantage extends to more complex relational reasoning tasks, and whether training humans to avoid accessibility biases can close the gap between human and machine performance. The geometry of analogy was right all along; we simply needed to look beyond the noise of human inconsistency to see it clearly.