We consider a novel algorithm, for the completion of partially observed low-rank tensors, where each entry of the tensor can be chosen from a discrete finite alphabet set, such as in common image processing problems, where the entries represent the RGB values. The proposed low-rank tensor completion (TC) method builds on the conventional nuclear norm (NN) minimization-based low-rank TC paradigm, through the addition of a discrete-aware regularizer, which enforces discreteness in the objective of the problem, by an $\ell_0$-norm regularizer that is approximated by a continuous and differentiable function normalized via fractional programming (FP) under a proximal gradient (PG) framework, in order to solve the proposed problem. Simulation results demonstrate the superior performance of the new method both in terms of normalized mean square error (NMSE) and convergence, compared to the conventional state of-the-art (SotA) techniques, including NN minimization approaches, as well as a mixture of the latter with a matrix factorization approach.

翻译：我们提出一种新颖算法，用于补全部分观测的低秩张量，其中张量的每个元素可从离散有限字母集合中选取，例如在常见的图像处理问题中，元素代表RGB值。所提出的低秩张量补全方法建立在基于核范数最小化的传统低秩张量补全范式基础上，通过引入离散感知正则化项，在问题目标中强制实现离散性。该正则化项采用ℓ₀范数正则化器，通过分数规划归一化的连续可微函数进行近似，并在近端梯度框架下求解所提出的问题。仿真结果表明，与包括核范数最小化方法及其与矩阵分解方法混合的传统最先进技术相比，新方法在归一化均方误差和收敛性方面均表现出优越性能。