Support Matrix Machine via Joint $\ell_{2,1}$ and Nuclear Norm Minimization Under Matrix Completion Framework for Classification of Corrupted Data

Traditional support vector machines (SVMs) are fragile in the presence of outliers; even a single corrupt data point can arbitrarily alter the quality of the approximation. If even a small fraction of columns is corrupted, then classification performance will inevitably deteriorate. This article considers the problem of high-dimensional data classification, where a number of the columns are arbitrarily corrupted. An efficient S upport M atrix M achine that simultaneously performs matrix Re covery (SSMRe) is proposed, i.e. feature selection and classification through joint minimization of $\ell_{2,1}$ (the nuclear norm of $L$ ). The data are assumed to consist of a low-rank clean matrix plus a sparse noisy matrix. SSMRe works under incoherence and ambiguity conditions and is able to recover an intrinsic matrix of higher rank in the presence of data densely corrupted. The objective function is a spectral extension of the conventional elastic net; it combines the property of matrix recovery along with low rank and joint sparsity to deal with complex high-dimensional noisy data. Furthermore, SSMRe leverages structural information, as well as the intrinsic structure of data, avoiding the inevitable upper bound. Experimental results on different real-time applications, supported by the theoretical analysis and statistical testing, show significant gain for BCI, face recognition, and person identification datasets, especially in the presence of outliers, while preserving a reasonable number of support vectors.