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No abstract uploaded!

No abstract uploaded!

Let $$\mathcal{H}^{1,1} (T^1 )$$ denote the Hardy space of real-valued functions on the unit circle with weak derivatives in the usual real Hardy space $$\mathcal{H}^1 (T^1 )$$ . It is shown that when the weak derivative of a locally Lipschitz continuous function$f$has bounded variation on compact sets the Nemytskii operator$F$, defined by$F(u)=f·u$, maps $$\mathcal{H}^{1,1} (T^1 )$$ continuously into itself. A further condition sufficient for the continuous Fréchet differentiability of$F$is then added.

In recent years, sparse coding has been widely used in many applications ranging from image processing to pattern recognition. Most existing sparse coding based applications require solving a class of challenging non-smooth and non-convex optimization problems. Despite the fact that many numerical methods have been developed for solving these problems, it remains an open problem to find a numerical method which is not only empirically fast, but also has mathematically guaranteed strong convergence. In this paper, we propose an alternating iteration scheme for solving such problems. A rigorous convergence analysis shows that the proposed method satisfies the global convergence property: the whole sequence of iterates is convergent and converges to a critical point. Besides the theoretical soundness, the practical benefit of the proposed method is validated in applications including image restoration and recognition. Experiments show that the proposed method achieves similar results with less computation when compared to widely used methods such as K-SVD.