  Analyzing the urban trajectory in cities has become an important topic in
data mining. How can we model the human mobility consisting of stay and travel
from the raw trajectory data? How can we infer such a mobility model from the
single trajectory information? How can we further generalize the mobility
inference to accommodate the real-world trajectory data that is sparsely
sampled over time?
  In this paper, based on formal and rigid definitions of the stay/travel
mobility, we propose a single trajectory inference algorithm that utilizes a
generic long-tailed sparsity pattern in the large-scale trajectory data. The
algorithm guarantees a 100\% precision in the stay/travel inference with a
provable lower-bound in the recall. Furthermore, we introduce an
encoder-decoder learning architecture that admits multiple trajectories as
inputs. The architecture is optimized for the mobility inference problem
through customized embedding and learning mechanism. Evaluations with three
trajectory data sets of 40 million urban users validate the performance
guarantees of the proposed inference algorithm and demonstrate the superiority
of our deep learning model, in comparison to well-known sequence learning
methods. On extremely sparse trajectories, the deep learning model achieves a
2$\times$ overall accuracy improvement from the single trajectory inference
algorithm, through proven scalability and generalizability to large-scale
versatile training data.
