  Sequential word order is important when processing text. Currently, neural
networks (NNs) address this by modeling word position using position
embeddings. The problem is that position embeddings capture the position of
individual words, but not the ordered relationship (e.g., adjacency or
precedence) between individual word positions. We present a novel and
principled solution for modeling both the global absolute positions of words
and their order relationships. Our solution generalizes word embeddings,
previously defined as independent vectors, to continuous word functions over a
variable (position). The benefit of continuous functions over variable
positions is that word representations shift smoothly with increasing
positions. Hence, word representations in different positions can correlate
with each other in a continuous function. The general solution of these
functions is extended to complex-valued domain due to richer representations.
We extend CNN, RNN and Transformer NNs to complex-valued versions to
incorporate our complex embedding (we make all code available). Experiments on
text classification, machine translation and language modeling show gains over
both classical word embeddings and position-enriched word embeddings. To our
knowledge, this is the first work in NLP to link imaginary numbers in
complex-valued representations to concrete meanings (i.e., word order).
