  Low-rank matrix factorizations arise in a wide variety of applications --
including recommendation systems, topic models, and source separation, to name
just a few. In these and many other applications, it has been widely noted that
by incorporating temporal information and allowing for the possibility of
time-varying models, significant improvements are possible in practice.
However, despite the reported superior empirical performance of these dynamic
models over their static counterparts, there is limited theoretical
justification for introducing these more complex models. In this paper we aim
to address this gap by studying the problem of recovering a dynamically
evolving low-rank matrix from incomplete observations. First, we propose the
locally weighted matrix smoothing (LOWEMS) framework as one possible approach
to dynamic matrix recovery. We then establish error bounds for LOWEMS in both
the {\em matrix sensing} and {\em matrix completion} observation models. Our
results quantify the potential benefits of exploiting dynamic constraints both
in terms of recovery accuracy and sample complexity. To illustrate these
benefits we provide both synthetic and real-world experimental results.
