Robust estimation of highly corrupted low-rank matrix via alternating direction method of multiplier

Low-rank matrices play a central role in modelling and computational methods for signal processing and large-scale data analysis. Real-world observed data are often sampled from low-dimensional subspaces, but with sample-specific corruptions (i.e. outliers) or random noises. In many applications where low-rank matrices arise, these matrices cannot be fully sampled or directly observed, and one encounters the problem of recovering the matrix given only incomplete and indirect observations. The authors aim to recover a low-rank component from incomplete and indirect observations and correct the possible errors. A new low-rank matrix recovery formula based on generalised Tikhonov regularisation and its solution algorithm are proposed. The proposed method determines the low-rank component for performing matrix recovery from highly corrupted observations. The authors' recommended algorithm reduces not only the outliers but also random corruptions in the recovering task. The experimental results obtained using both synthetic and real application data demonstrate the efficacy of the proposed method.