Testing the assumptions about the geometry of sentence embedding spaces: the cosine measure need not apply

Transformer models learn to encode and decode an input text, and produce contextual token embeddings as a side-effect. The mapping from language into the embedding space maps words expressing similar concepts onto points that are close in the space. In practice, the reverse implication is also assumed: words corresponding to close points in this space are similar or related, those that are further are not. Does closeness in the embedding space extend to shared properties for sentence embeddings? We present an investigation of sentence embeddings and show that the geometry of their embedding space is not predictive of their relative performances on a variety of tasks. We compute sentence embeddings in three ways: as averaged token embeddings, as the embedding of the special [CLS] token, and as the embedding of a random token from the sentence. We explore whether there is a correlation between the distance between sentence embedding variations and their performance on linguistic tasks, and whether despite their distances, they do encode the same information in the same manner. The results show that the cosine similarity -- which treats dimensions shallowly -- captures (shallow) commonalities or differences between sentence embeddings, which are not predictive of their performance on specific tasks. Linguistic information is rather encoded in weighted combinations of different dimensions, which are not reflected in the geometry of the sentence embedding space.