What Cosine Similarity of Label Representations Can and Cannot Tell us

Cosine similarity is often used to measure the similarity of vector representations of neural network models. However, the cosine similarity of representations is not guaranteed to tell us anything about model probabilities. In this paper we show that for a softmax classifier, be it an image classifier or an autoregressive language model, the cosine similarity between label representations (called unembeddings in the paper) does not give any information on the probabilities assigned by the model. Specifically, we prove that given two unembeddings, it is possible to create another model which assigns the same probabilities for all inputs, but where the cosine similarity between the representations is now either $1$ or $-1$. We also show that for a sigmoid classifier (where each input can be assigned multiple labels), all pairwise cosine similarities between the unembeddings define the set of possible label combinations. However, for softmax classifiers (where each input is assigned a ranking of the labels from most to least likely), we need all pairwise cosine similarities between all \textit{differences} of unembeddings to know which rankings the model can predict. We conclude that it is misleading to interpret the cosine similarity between unembeddings without reference to the classifier that produced them.