Revisiting Cosine Similarity via Normalized ICA-transformed Embeddings

TL;DR

This paper introduces a novel interpretation of cosine similarity using ICA-transformed embeddings, emphasizing interpretability and statistical significance of axes, supported by numerical experiments and probability analysis.

Contribution

It proposes a new interpretation of cosine similarity based on ICA axes, enhancing interpretability and statistical analysis of embedding similarities.

Findings

ICA-transformed embeddings are sparse and interpretable.

Semantic similarity can be represented by the product of ICA components.

Method for selecting statistically significant axes based on probability distributions.

Abstract

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. The normalized ICA-transformed embeddings exhibit sparsity, enhancing the interpretability of each axis, and the semantic similarity defined by the product of the components represents the shared meaning between the two embeddings along each axis. The effectiveness of this approach is demonstrated through intuitive numerical examples and thorough numerical experiments. By deriving the probability distributions that govern each component and the product of components, we propose a method for selecting statistically significant axes.