In this article, we propose a new method for low-rank completion of a large sparse matrix, subject to non-negativity constraint. As a challenging prototype of this problem, we have in mind the well-known Netflix problem. Our method is based on the derivation of a constrained gradient system and its numerical integration. The methods we propose are based on the constrained minimization of a functional associated to the low-rank completion matrix having minimal distance to the given matrix. In the main 2-level method, the low-rank matrix is expressed in the form of the non-negative factorization X = εUV^T, where the factors U and V are assumed to be normalized with unit Frobenius norm. In the inner level—for a given ε—we minimize the functional; in the outer level, we tune the parameter ε until we reach a solution. Numerical experiments on well-known large test matrices show the effectiveness of the method when compared with other algorithms available in the literature.

中文翻译：

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一种非负的低秩完成的有效方法

在本文中，我们提出了一种在非负约束条件下低秩完成大型稀疏矩阵的新方法。作为此问题的具有挑战性的原型，我们考虑了众所周知的Netflix问题。我们的方法基于约束梯度系统的推导及其数值积分。我们提出的方法基于与低阶完成矩阵相关联的函数的约束最小化，该函数与给定矩阵的距离最小。在主2层的方法，所述低秩矩阵在非负因子分解的形式表示X = ε ü V ^Ť，其中所述因素ù和V假设已使用单位Frobenius范数标准化。在内部级别（对于给定的ε），我们将功能最小化；在最外层，我们调整参数ε直到找到一个解。与文献中提供的其他算法相比，在著名的大型测试矩阵上进行的数值实验证明了该方法的有效性。