  In real-world applications, it is important for machine learning algorithms
to be robust against data outliers or corruptions. In this paper, we focus on
improving the robustness of a large class of learning algorithms that are
formulated as low-rank semi-definite programming (SDP) problems. Traditional
formulations use square loss, which is notorious for being sensitive to
outliers. We propose to replace this with more robust noise models, including
the $\ell_1$-loss and other nonconvex losses. However, the resultant
optimization problem becomes difficult as the objective is no longer convex or
smooth. To alleviate this problem, we design an efficient algorithm based on
majorization-minimization. The crux is on constructing a good optimization
surrogate, and we show that this surrogate can be efficiently obtained by the
alternating direction method of multipliers (ADMM). By properly monitoring
ADMM's convergence, the proposed algorithm is empirically efficient and also
theoretically guaranteed to converge to a critical point. Extensive experiments
are performed on four machine learning applications using both synthetic and
real-world data sets. Results show that the proposed algorithm is not only fast
but also has better performance than the state-of-the-art.
