document: Learning incoherent sparse and low-rank patterns from multiple tasks We consider the problem of learning incoherent sparse and low-rank patterns from multiple tasks . Our approach is based on a linear multi-task learning formulation , in which the sparse and low-rank patterns are induced by a cardinality regularization term and a low-rank constraint , respectively . This formulation is non-convex ; we convert it into its convex surrogate , which can be routinely solved via semidefinite programming for small-size problems . We propose to employ the general projected gradient scheme to efficiently solve such a convex surrogate ; however , in the optimization formulation , the objective function is non-differentiable and the feasible domain is non-trivial . We present the procedures for computing the projected gradient and ensuring the global convergence of the projected gradient scheme . The computation of projected gradient involves a constrained optimization problem ; we show that the optimal solution to such a problem can be obtained via solving an unconstrained optimization subproblem and an Euclidean projection subproblem . In addition , we present two projected gradient algorithms and discuss their rates of convergence . Experimental results on benchmark data sets demonstrate the effectiveness of the proposed multi-task learning formulation and the efficiency of the proposed projected gradient algorithms .
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