Linear Spaces of Meanings: the Compositional Language of VLMs

We investigate compositional structures in vector data embeddings from pre-trained vision-language models (VLMs). Traditionally, compositionality has been associated with algebraic operations on embeddings of words from a pre-existing vocabulary. In contrast, we seek to approximate label representations from a text encoder as combinations of a smaller set of vectors in the embedding space. These vectors can be seen as "ideal words" which can be used to generate new concepts in an efficient way. We present a theoretical framework for understanding linear compositionality, drawing connections with mathematical representation theory and previous definitions of disentanglement. We provide theoretical and empirical evidence that ideal words provide good compositional approximations of composite concepts and can be more effective than token-based decompositions of the same concepts.