We consider the low rank matrix completion problem over finite fields. This problem has been extensively studied in the domain of real/complex numbers, however, to the best of authors' knowledge, there exists merely one efficient algorithm to tackle the problem in the binary field, due to Saunderson et al. [1]. In this paper, we improve upon the theoretical guarantees for the algorithm provided in [1]. Furthermore, we formulate a new graphical model for the matrix completion problem over the finite field of size $q$, $\Bbb{F}_q$, and present a message passing (MP) based approach to solve this problem. The proposed algorithm is the first one for the considered matrix completion problem over finite fields of arbitrary size. Our proposed method has a significantly lower computational complexity, reducing it from $O(n^{2r+3})$ in [1] down to $O(n^2)$ (where, the underlying matrix has dimension $n \times n$ and $r$ denotes its rank), while also improving the performance.

翻译：我们研究有限域上的低秩矩阵完备化问题。该问题在实/复数域中已被广泛研究，然而据作者所知，在二元域中仅存在一种有效算法，由Saunderson等人[1]提出。本文改进了文献[1]中算法的理论保证。此外，我们为大小为$q$的有限域$\Bbb{F}_q$上的矩阵完备化问题构建了一个新的图模型，并提出基于消息传递(MP)的方法来解决该问题。所提算法是首个针对任意大小有限域上矩阵完备化问题的解决方案。我们的方法显著降低了计算复杂度，从[1]中的$O(n^{2r+3})$降至$O(n^2)$（其中底层矩阵维度为$n \times n$，$r$表示其秩），同时性能也得到提升。