Calibrating WEAT Against Anisotropy: ZCA Whitening as a Geometric Pre-Processing Step for Embedding Association Tests
Authors: Seitaro Ono, Senna Ross, Jun Saiki
Organizations: Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, Japan · Brock University, St. Catharines, Ontario, Canada
Abstract
We propose Zero-phase Component Analysis (ZCA) whitening as a geometric pre-processing step for the Word Embedding Association Test (WEAT). WEAT is a bias measurement method widely used in both computational social science and AI fairness research. It relies on cosine similarity as a measure of semantic association, which assumes that the embedding space is approximately isotropic. However, prior work has reported that many widely used language models do not satisfy this assumption, raising concerns about the reliability of bias measurements. ZCA whitening transforms the covariance of the embedding space into the identity matrix while minimizing perturbation to the original vectors. This transformation restores the isotropy condition on which WEAT relies. We evaluate our approach on ten standard WEAT test suites and seven models spanning three architectural families, yielding 70 model-task combinations. The results show that ZCA whitening substantially reduces the anisotropy of the embedding spaces across all models. Particularly for highly anisotropic models, we further observe improvements on standard semantic similarity benchmarks, indicating that the calibrated space better captures semantic associations. After calibration, over 30% of WEAT results change significance status, and effect sizes shift in both directions depending on bias category. These shifts suggest that uncalibrated measurements may both overestimate and underestimate the associations encoded in the embedding space. These findings indicate that previously reported bias measurements in anisotropic embedding spaces should be interpreted with caution and may benefit from re-evaluation with calibrated methods. Our approach contributes to restoring the measurement foundation of WEAT across both computational social science and AI fairness research.
Debiasing methods based on principal component analysis (PCA) are broadly used to reduce gender bias in word embeddings used in LLMs, yet it remains unclear what aspects of bias they actually remove and how destructive this process is. These methods are based on the understanding that bias resides in a low-dimensional subspace, with the assumption that most of it can be captured by a few principal components. In this work, we conduct a systematic geometric analysis of PCA-based gender debiasing and investigate what is actually removed from the embedding space. Our experiments across multiple embeddings show that direct gender bias is primarily concentrated in the first principal component, supporting the low-rank bias hypothesis. However, associative bias measured by WEAT does not align with these principal directions and is instead spread across multiple embedding dimensions. Furthermore, as expected, we demonstrate that removing an increasing number of principal components leads to a consistent degradation of the embedding geometry, affecting semantic structure and vector relationships. These results reveal that PCA-based debiasing operates as a trade-off: while it effectively reduces certain forms of direct bias, it fails to eliminate distributed associations and introduces geometric distortion. Moreover, there is no universal optimal level of debiasing, as the balance between bias reduction and semantic preservation depends on the chosen metric and embedding. Overall, our findings suggest that bias in word embeddings is not purely low-rank and that simple subspace removal methods may be insufficient for comprehensive debiasing.
The standard way to compare two text embeddings is cosine similarity. Scattered studies report that a different metric does better, but never pin down the geometric condition that decides when, or why. We settle both with a comprehensive empirical study: nineteen parameter-free similarity metrics on nineteen encoders, from compact sentence transformers up to seven-billion-parameter large language models, across seven datasets. The answer is geometric. When an encoder spreads its variance evenly across directions, cosine is the best parameter-free choice and no other metric helps by a usable margin. When the variance concentrates into a few dominant directions, a property known as anisotropy, rank-based and L1-type metrics beat cosine by a clear margin. The absolute gain is modest, but because cosine starts low on these encoders it is a sizable relative improvement, around twenty percent on average and largest where cosine is weakest. What decides this is the geometry of the embedding space, not how the model was trained: where the two disagree, the metric follows the geometry. One number, the fraction of variance held by the single most dominant dimension, predicts how much the alternatives help across all nineteen encoders, with a rank correlation of 0.86 and a linear correlation of 0.95. To test this as the cause rather than a correlate, we project out the dominant directions: cosine recovers and the advantage of the other metrics nearly vanishes, but only on the encoders that were anisotropic to begin with. The effect is directional, not magnitude based, since it survives normalizing every vector to unit length. Among parameter-free metrics, then, cosine is the right tool wherever an encoder is well spread, which includes the fine-tuned embedders commonly deployed for retrieval, and we give a one-number diagnostic for when it is not.
The embedding of word sequences into continuous vector spaces has been one of the most important developments in Natural Language Processing in the recent past. Such embeddings have found application in areas such as Automatic Speech Recognition, Machine Translation, Sentiment Analysis and many more. This essay reviews the various mechanisms that have been proposed for the calculation of word embeddings, investigates popular toolkits and embedding matrices that are available in the public domain, and experiments with one or more selected implementations to better understand their characteristics. La représentation vectorielle continue de mots a été l'un des développements les plus importants dans le domaine du traitement automatique du langage naturel au cours des dernières années. Ces représentations ont trouvé application dans des domaines tels que la reconnaissance vocale, la traduction automatique, l'analyse des sentiments, etc. Ce travail passe en revue les différents mécanismes proposés pour le calcul de ces vecteurs de mots, étudie les kits d'outils populaires et les matrices disponibles publiquement en ligne, et expérimente avec une ou plusieurs implémentations sélectionnées pour mieux comprendre leurs caractéristiques.