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:我们考虑了以(可能)非垂直、实际上稀疏的一等分解从测量和收集到线性测量过程A中恢复未知的低位矩阵X的问题。我们建议了一种变式配方,这种配方可以进行交替最小化,其全球最小化手段可以接近X至噪音水平。我们与一个强效注射的变种一起,为A的各种选择,包括低高斯级-1和重尾测量,提供重建保障。数字实验支持我们理论考虑的有效性。