  Black-box probing models can reliably extract linguistic features like tense,
number, and syntactic role from pretrained word representations. However, the
manner in which these features are encoded in representations remains poorly
understood. We present a systematic study of the linear geometry of
contextualized word representations in ELMO and BERT. We show that a variety of
linguistic features (including structured dependency relationships) are encoded
in low-dimensional subspaces. We then refine this geometric picture, showing
that there are hierarchical relations between the subspaces encoding general
linguistic categories and more specific ones, and that low-dimensional feature
encodings are distributed rather than aligned to individual neurons. Finally,
we demonstrate that these linear subspaces are causally related to model
behavior, and can be used to perform fine-grained manipulation of BERT's output
distribution.
