DYNAMIC LOW-RANK MATRIX RECOVERY: THEORY AND APPLICATIONS

The purpose of this work is to provide both theoretical understanding of and practical algorithms for dynamic low-rank matrix recovery. Although the benefits of exploiting dynamics in low-rank matrix recovery have been observed in many applications, the theoretical understanding of and justification for these methods is limited. This dissertation concerns two widely-used dynamics models in the context of low-rank matrix recovery: random walk dynamics and measurement induced dynamics. For random walk dynamics, we propose a locally weighted matrix smoothing (LOWEMS) framework, establish its recovery guarantee and algorithmic convergence, and discuss two practical extensions for it. For measurement induced dynamics, we propose a general DynEmb framework and demonstrate its effectiveness for the knowledge tracing application. In the end, we conduct some initial theoretical analysis on a simplified measurement induced dynamic model.Ph.D

Dynamic matrix recovery from incomplete observations under an exact low-rank constraint

A 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 matrix sensing and 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