Revisiting Cosine Similarity via Normalized ICA-transformed Embeddings

Cosine similarity is widely used to measure the similarity between two embeddings, while interpretations based on angle and correlation coefficient are common. In this study, we focus on the interpretable axes of embeddings transformed by Independent Component Analysis (ICA), and propose a novel interpretation of cosine similarity as the sum of semantic similarities over axes. To investigate this, we first show experimentally that unnormalized embeddings contain norm-derived artifacts. We then demonstrate that normalized ICA-transformed embeddings exhibit sparsity, with a few large values in each axis and across embeddings, thereby enhancing interpretability by delineating clear semantic contributions. Finally, to validate our interpretation, we perform retrieval experiments using ideal embeddings with and without specific semantic components.