This paper considers regularized block multiconvex optimization, where the feasible set and objective function are generally nonconvex but convex in each block of variables. It also accepts nonconvex blocks and requires these blocks to be updated by proximal minimization. We review some interesting applications and propose a generalized block coordinate descent method. Under certain conditions, we show that any limit point satisfies the Nash equilibrium conditions. Furthermore, we establish global convergence and estimate the asymptotic convergence rate of the method by assuming a property based on the Kurdyka-Lojasiewicz inequality. The proposed algorithms are tested on nonnegative matrix and tensor factorization, as well as matrix and tensor recovery from incomplete observations. The tests include synthetic data and hyperspectral data, as well as image sets from the CBCL and ORL databases. Compared to the existing state-of-the-art algorithms, the proposed algorithms demonstrate superior performance in both speed and solution quality. The MATLAB code of nonnegative matrix/tensor decomposition and completion, along with a few demos, are accessible from the authors' homepages.

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Among all the interesting spaces in topology, the spheres are no doubt beautiful objects and of most consideration since antiquity. Any nontrivial observations of them are of course very important. From the categorical point of view, we should not just focus on the objects themselves, but the morphisms between them as well. For this purpose, in algebraic topology, we do want to classify the set of continuous morphisms between spheres under the equivalent relation named homotopy, which describes a continuous deformation between two continuous maps. Let Sn be the n-sphere and k (X) be the set of homotopic equivalent based maps from Sk to X. For the reason that Sk is a double suspension when k 2, the set is actually an abelian group. A natural question is, what are these abelian groups?

Lecture notes : Introduction to Cyclic Covers in Algebraic Geometry