What Does Debiasing Really Remove? A Geometric Study of PCA-Based Gender Debiasing in Word Embeddings

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.