Is Cosine-Similarity of Embeddings Really About Similarity?
This addresses a foundational issue for researchers and practitioners in machine learning who rely on embeddings for similarity tasks, highlighting an incremental but critical insight into model interpretability.
The paper tackles the problem of cosine-similarity in embeddings potentially yielding arbitrary or meaningless results, showing analytically that for linear models, similarities can be non-unique or controlled by regularization, and cautioning against its blind use.
Cosine-similarity is the cosine of the angle between two vectors, or equivalently the dot product between their normalizations. A popular application is to quantify semantic similarity between high-dimensional objects by applying cosine-similarity to a learned low-dimensional feature embedding. This can work better but sometimes also worse than the unnormalized dot-product between embedded vectors in practice. To gain insight into this empirical observation, we study embeddings derived from regularized linear models, where closed-form solutions facilitate analytical insights. We derive analytically how cosine-similarity can yield arbitrary and therefore meaningless `similarities.' For some linear models the similarities are not even unique, while for others they are implicitly controlled by the regularization. We discuss implications beyond linear models: a combination of different regularizations are employed when learning deep models; these have implicit and unintended effects when taking cosine-similarities of the resulting embeddings, rendering results opaque and possibly arbitrary. Based on these insights, we caution against blindly using cosine-similarity and outline alternatives.