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Machine learning tools often rely on embedding text as vectors of realnumbers. In this paper, we study how the semantic structure of language isencoded in the algebraic structure of such embeddings. Specifically, we look ata notion of ``semantic independence'' capturing the idea that, e.g.,``eggplant'' and ``tomato'' are independent given ``vegetable''. Although suchexamples are intuitive, it is difficult to formalize such a notion of semanticindependence. The key observation here is that any sensible formalizationshould obey a set of so-called independence axioms, and thus any algebraicencoding of this structure should also obey these axioms. This leads usnaturally to use partial orthogonality as the relevant algebraic structure. Wedevelop theory and methods that allow us to demonstrate that partialorthogonality does indeed capture semantic independence. Complementary to this,we also introduce the concept of independence preserving embeddings whereembeddings preserve the conditional independence structures of a distribution,and we prove the existence of such embeddings and approximations to them.

Uncovering Meanings of Embeddings via Partial Orthogonality