Completing Low-Rank Matrices with Corrupted Samples from Few Coefficients in General Basis
This work addresses subspace recovery in signal processing and information theory, offering a more general and robust approach compared to existing methods, though it appears incremental in extending prior robust matrix completion techniques.
The paper tackles the problem of robust matrix completion from corrupted and missing data in general basis, proving exact recovery of the range space for matrices with rank and corruption levels up to O(min{m,n}/log^3(m+n)). It proposes a universal regularization parameter and an efficient algorithm, with experiments validating the theory.
Subspace recovery from corrupted and missing data is crucial for various applications in signal processing and information theory. To complete missing values and detect column corruptions, existing robust Matrix Completion (MC) methods mostly concentrate on recovering a low-rank matrix from few corrupted coefficients w.r.t. standard basis, which, however, does not apply to more general basis, e.g., Fourier basis. In this paper, we prove that the range space of an $m\times n$ matrix with rank $r$ can be exactly recovered from few coefficients w.r.t. general basis, though $r$ and the number of corrupted samples are both as high as $O(\min\{m,n\}/\log^3 (m+n))$. Our model covers previous ones as special cases, and robust MC can recover the intrinsic matrix with a higher rank. Moreover, we suggest a universal choice of the regularization parameter, which is $λ=1/\sqrt{\log n}$. By our $\ell_{2,1}$ filtering algorithm, which has theoretical guarantees, we can further reduce the computational cost of our model. As an application, we also find that the solutions to extended robust Low-Rank Representation and to our extended robust MC are mutually expressible, so both our theory and algorithm can be applied to the subspace clustering problem with missing values under certain conditions. Experiments verify our theories.