  Recent work has explored methods for learning continuous vector space word
representations reflecting the underlying semantics of words. Simple vector
space arithmetic using cosine distances has been shown to capture certain types
of analogies, such as reasoning about plurals from singulars, past tense from
present tense, etc. In this paper, we introduce a new approach to capture
analogies in continuous word representations, based on modeling not just
individual word vectors, but rather the subspaces spanned by groups of words.
We exploit the property that the set of subspaces in n-dimensional Euclidean
space form a curved manifold space called the Grassmannian, a quotient subgroup
of the Lie group of rotations in n- dimensions. Based on this mathematical
model, we develop a modified cosine distance model based on geodesic kernels
that captures relation-specific distances across word categories. Our
experiments on analogy tasks show that our approach performs significantly
better than the previous approaches for the given task.
