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Max-norm optimization for robust matrix recovery. (English) Zbl 1414.90265
In order to recover an unknown matrix \(M^0 \in \mathbb{R}^{d_1 \times d_2}\) based on some of its entries observed with noise, \(\{Y_{i_t,j_t}\}_{t=1}^n\), the authors propose a hybrid estimator defined as the solution of the optimization problem \[ \begin{cases} \text{Minimize }\frac{1}{n} \sum_{t=1}^n (Y_{i_t,j_t} - M_{i_t,j_t})^2 +\lambda \|M\|_{\max} + \mu \|M\|_*, \\ \text{subject to } M \in \mathbb{R}^{d_1 \times d_2},\; \|M\|_\infty \leq \alpha, \end{cases} \] where \(\|M\|_{\max}\), \(\|M\|_*\) and \(\|M\|_\infty\) denote the max-norm, the nuclear norm and the elementwise \(\infty\)-norm of \(M\), respectively, while \(\lambda >0\) and \(\mu \geq 0\) are tuning parameters. Efficient algorithms are provided along with numerical experiments on different datasets.

90C25 Convex programming
90C29 Multi-objective and goal programming
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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