A refined convergence analysis of pDCAe with applications to simultaneous sparse recovery and outlier detection

We consider the problem of minimizing a difference-of-convex (DC) function, which can be written as the sum of a smooth convex function with Lipschitz gradient, a proper closed convex function and a continuous possibly nonsmooth concave function. We refine the convergence analysis in [38] for the proximal DC algorithm with extrapolation (pDCAe) and show that the whole sequence generated by the algorithm is convergent when the objective is level-bounded, em without imposing differentiability assumptions in the concave part. Our analysis is based on a new potential function and we assume such a function is a Kurdyka-Lojasiewicz (KL) function. We also establish a relationship between our KL assumption and the one used in [38]. Finally, we demonstrate how the pDCAe can be applied to a class of simultaneous sparse recovery and outlier detection problems arising from robust compressed sensing in signal processing and least trimmed squares regression in statistics. Specifically, we show that the objectives of these problems can be written as level-bounded DC functions whose concave parts are em typically nonsmooth. Moreover, for a large class of loss functions and regularizers, the KL exponent of the corresponding potential function are shown to be 1/2, which implies that the pDCAe is locally linearly convergent when applied to these problems. Our numerical experiments show that the pDCAe usually outperforms the proximal DC algorithm with nonmonotone linesearch [24, Appendix A] in both CPU time and solution quality for this particular application.