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Projected Landweber iteration for matrix completion. (English) Zbl 1225.65049

Let \(M\in\mathbb{R}^{n\times m}\) be an unknown low-rank matrix and \(\{M_{ij}:\;(i,j)\in\Omega\}\) be a sampling set of its entries, and let \(\|\cdot \|_F\), \(\|\cdot\|_*\) be the Frobenius norm and the nuclear norm on \(\mathbb{R}^{n\times m}\), respectively. The authors derive a nonlinear constrained quadratic programming problem: \(\min\{\|\mathcal{P}_\Omega(X)-\mathcal{P}_\Omega(M)\|_F^2:\;X\in \bar{B}_R\}\), where \(\mathcal{P}_\Omega\) is the orthogonal projector such that \(\mathcal{P}_\Omega(X)_{ij}=M_{ij}\) when \((i,j)\in\Omega\) and otherwise equals to zero, and the level set \(\bar{B}_R=\{X\in\mathbb{R}^{n\times m}: \|X\|_*\leq R\}\). They design a new algorithm for addressing such a matrix completion problem, namely projected Landweber iteration, based on the projected method and Landweber iteration. The convergence of the proposed algorithm is proved strictly, and numerical results show that the proposed algorithm can be fast and efficient under suitable prior conditions of \(M\).

MSC:

65F30 Other matrix algorithms (MSC2010)
15A83 Matrix completion problems
65K05 Numerical mathematical programming methods
90C20 Quadratic programming
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