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Word embeddings are commonly obtained as optimizers of a criterion function$f$ of a text corpus, but assessed on word-task performance using a differentevaluation function $g$ of the test data. We contend that a possible source ofdisparity in performance on tasks is the incompatibility between classes oftransformations that leave $f$ and $g$ invariant. In particular, wordembeddings defined by $f$ are not unique; they are defined only up to a classof transformations to which $f$ is invariant, and this class is larger than theclass to which $g$ is invariant. One implication of this is that the apparentsuperiority of one word embedding over another, as measured by word taskperformance, may largely be a consequence of the arbitrary elements selectedfrom the respective solution sets. We provide a formal treatment of the aboveidentifiability issue, present some numerical examples, and discuss possibleresolutions.

Invariance and identifiability issues for word embeddings