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.

My primary concerns are listed below: A. Proof 1. a) In line 379 (supplementary material) Theorem 2.3 from 5 cannot be directly used to establish RIP for \sum_t \sqrt{wt} At(\Delta) as \sum_t wt At(\Delta) _2 2 NOT \sum_t \sqrt{wt} At(\Delta) _2 2. However, as theorem 3.1 requires bound on only the former, the proof in [5] can be extended. The proof needs some work though. B. Theory 2. The results are derived for exact global solution to (2) which is a non-convex optimization. The paper is incomplete without the suggested future work on analyzing alternating minimization. I believe the analysis will follow through with a bit of linear algebra machinery, but the exact expression of the joint error term arising from Thm 3.4 and 3.8 and the weighted power iteration is more useful towards understanding the tradeoff of sample complexity and accuracy using the dynamic estimation in practical implementations.

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.