Dynamic matrix recovery from incomplete observations under an exact low-rank constraint
This provides theoretical justification for dynamic models in applications like recommendation systems, addressing a gap in the literature, though it is incremental as it builds on existing static low-rank recovery methods.
The paper tackles the problem of recovering dynamically evolving low-rank matrices from incomplete observations, establishing error bounds for the proposed LOWEMS framework and showing benefits in recovery accuracy and sample complexity through synthetic and real-world experiments.
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.