We consider a novel algorithm, for the completion of partially observed low-rank matrices in a structured setting where each entry can be chosen from a finite discrete alphabet set, such as in common recommender systems. The proposed low-rank matrix completion (MC) method is an improved variation of state-of-the-art (SotA) discrete aware matrix completion method which we previously proposed, in which discreteness is enforced by an $\ell_0$-norm regularizer, not by replaced with the $\ell_1$-norm, but instead approximated by a continuous and differentiable function normalized via fractional programming (FP) under a proximal gradient (PG) framework. Simulation results demonstrate the superior performance of the new method compared to the SotA techniques as well as the earlier $\ell_1$-norm-based discrete-aware matrix completion approach.

翻译：我们提出一种新颖算法，用于在结构化场景中补全部分观测的低秩矩阵，其中每个条目可从有限离散字母表集合中选择，例如常见推荐系统中的情形。所提出的低秩矩阵补全（MC）方法是对我们先前提出的先进离散感知矩阵补全方法的改进变体。该方法通过ℓ₀范数正则化项而非ℓ₁范数替代来强制离散性，并利用分数规划归一化的连续可微函数进行近似，在近端梯度框架下实现优化。仿真结果表明，新方法相较于先进技术以及早期基于ℓ₁范数的离散感知矩阵补全方法具有更优越的性能。