We characterize the first-order sensitivity of approximately recovering a low-rank matrix from linear measurements, a standard problem in compressed sensing. A special case covered by our analysis is approximating an incomplete matrix by a low-rank matrix. We give an algorithm for computing the associated condition number and demonstrate experimentally how the number of linear measurements affects it. In addition, we study the condition number of the rank-r matrix approximation problem. It measures in the Frobenius norm by how much an infinitesimal perturbation to an arbitrary input matrix is amplified in the movement of its best rank-r approximation. We give an explicit formula for the condition number, which shows that it does depend on the relative singular value gap between the rth and (r+1)th singular values of the input matrix.

翻译：我们从线性测量中大致恢复低级矩阵的一阶敏感度,这是压缩感测的一个标准问题。我们分析涉及的一个特殊情况是,一个低级矩阵接近一个不完整的矩阵。我们给出计算相关条件数字的算法,并实验性地展示线性测量数量如何影响它。此外,我们研究了级-r矩阵近似问题的条件号。它测量了Frobenius规范中任意输入矩阵的极小扰动量,其最佳排序-r近似量增加了多少。我们给出了条件号的明确公式,表明它确实取决于输入矩阵的rth和(r+1)单值之间的相对单值差。