Analysis of Nuclear Norm Regularization for Full-rank Matrix Completion

In this paper, we provide a theoretical analysis of the nuclear-norm regularized least squares for full-rank matrix completion. Although similar formulations have been examined by previous studies, their results are unsatisfactory because only additive upper bounds are provided. Under the assumption that the top eigenspaces of the target matrix are incoherent, we derive a relative upper bound for recovering the best low-rank approximation of the unknown matrix. Our relative upper bound is tighter than previous additive bounds of other methods if the mass of the target matrix is concentrated on its top eigenspaces, and also implies perfect recovery if it is low-rank. The analysis is built upon the optimality condition of the regularized formulation and existing guarantees for low-rank matrix completion. To the best of our knowledge, this is first time such a relative bound is proved for the regularized formulation of matrix completion

Eight-potential-well order-disorder ferroelectric model and effects of random fields

An eight-potential-well order-disorder ferroelectric model was presented and the phase transition was studied under the mean-field approximation. It was shown that the two-body interactions are able to account for the first-order and the second order phase transitions. With increasing the random fields in the system, a first-order phase transition is transformed into a second-order phase transition, and furthermore, a second-order phase transition is inhibited. However, proper random fields can promote the spontaneous appearance of a first-order phase transition by increasing the overcooled temperature. The connections of the model with relaxors were discussed.Comment: 8 pages, 5 figures. Submitted to Applied Physics Letter

Dichlorido(dimethyl­formamide-κO)[1,4,7-tris­(2-cyano­ethyl)-1,4,7-triaza­cyclo­nonane-κ3 N 1,N 4,N 7]cobalt(II)

The title compound, [CoCl2(C15H24N6)(C3H7NO)], crystallizes as a monomeric complex. The coordination environment around the CoII center could be described as a distorted octa­hedron consisting of three N donors from the facially coordinating triaza macrocyclic ligand, one O donor from dimethyl­formamide and two Cl− ions. Neutral complex mol­ecules are associated via inter­molecular C—H⋯Cl hydrogen bonds to form two-dimensional layers. C—H⋯O hydrogen bonds are also present

Evolving small-world networks with geographical attachment preference

We introduce a minimal extended evolving model for small-world networks which is controlled by a parameter. In this model the network growth is determined by the attachment of new nodes to already existing nodes that are geographically close. We analyze several topological properties for our model both analytically and by numerical simulations. The resulting network shows some important characteristics of real-life networks such as the small-world effect and a high clustering.Comment: 11 pages, 4 figure