Sensitivity of low-rank matrix recovery

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. This is one customary approach to build recommender systems. We give an algorithm for computing the associated condition number and demonstrate experimentally how the number of measurements affects it. In addition, we study the condition number of the best 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 its condition number, which shows that it depends on the relative singular value gap between the r th and ( r + 1 ) th singular values of the input matrix.