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：本文针对结构化场景下部分观测低秩矩阵的补全问题，提出一种新颖算法，其中每个元素可取自有限离散字母表（如常见推荐系统中的情形）。所提出的低秩矩阵补全方法是现有最优离散感知矩阵补全方法的改进变体——该方法并非以$\ell_1$-范数替换，而是通过近端梯度框架下分数规划归一化的连续可微函数，逼近$\ell_0$-范数正则化项来强制离散性。仿真结果表明，与现有最优技术及基于$\ell_1$-范数的早期离散感知矩阵补全方法相比，新方法展现出更优性能。