We consider the problem of recovering an unknown low-rank matrix X with (possibly) non-orthogonal, effectively sparse rank-1 decomposition from measurements y gathered in a linear measurement process A. We propose a variational formulation that lends itself to alternating minimization and whose global minimizers provably approximate X up to noise level. Working with a variant of robust injectivity, we derive reconstruction guarantees for various choices of A including sub-gaussian, Gaussian rank-1, and heavy-tailed measurements. Numerical experiments support the validity of our theoretical considerations.

翻译：我们考虑从线性测量过程A收集的测量值y中恢复未知低秩矩阵X（可能具有非正交、有效稀疏的秩1分解）的问题。我们提出了一种适用于交替最小化的变分公式，其全局极小化器在噪声水平内可靠地逼近X。基于鲁棒单射性的一种变体，我们推导了针对包括亚高斯、高斯秩1和重尾测量在内的多种A的重建保证。数值实验支持了理论分析的有效性。