{0,1}-Brauer Configuration Algebras and Their Applications in Graph Energy Theory

The energy E(G) of a graph G is the sum of the absolute values of its adjacency matrix. In contrast, the trace norm of a digraph Q, which is the sum of the singular values of the corresponding adjacency matrix, is the oriented version of the energy of a graph. It is worth pointing out that one of the main problems in this theory consists of determining appropriated bounds of these types of energies for significant classes of graphs, digraphs and matrices, provided that, in general, finding out their exact values is a problem of great difficulty. In this paper, the trace norm of a {0,1}-Brauer configuration is introduced. It is estimated and computed by associating suitable families of graphs and posets to Brauer configuration algebras.

Common Positive Solution of Two Nonlinear Matrix Equations Using Fixed Point Results

We discuss a pair of nonlinear matrix equations (NMEs) of the form X=R1+∑i=1kAi*F(X)Ai, X=R2+∑i=1kBi*G(X)Bi, where R1,R2∈P(n), Ai,Bi∈M(n), i=1,⋯,k, and the operators F,G:P(n)→P(n) are continuous in the trace norm. We go through the necessary criteria for a common positive definite solution of the given NME to exist. We develop the concept of a joint Suzuki-implicit type pair of mappings to meet the requirement and achieve certain existence findings under weaker assumptions. Some concrete instances are provided to show the validity of our findings. An example is provided that contains a randomly generated matrix as well as convergence and error analysis. Furthermore, we offer graphical representations of average CPU time analysis for various initializations.

Partial Multi-Label Learning with Noisy Label Identification

Partial multi-label learning (PML) deals with problems where each instance is assigned with a candidate label set, which contains multiple relevant labels and some noisy labels. Recent studies usually solve PML problems with the disambiguation strategy, which recovers ground-truth labels from the candidate label set by simply assuming that the noisy labels are generated randomly. In real applications, however, noisy labels are usually caused by some ambiguous contents of the example. Based on this observation, we propose a partial multi-label learning approach to simultaneously recover the ground-truth information and identify the noisy labels. The two objectives are formalized in a unified framework with trace norm and ℓ1 norm regularizers. Under the supervision of the observed noise-corrupted label matrix, the multi-label classifier and noisy label identifier are jointly optimized by incorporating the label correlation exploitation and feature-induced noise model. Extensive experiments on synthetic as well as real-world data sets validate the effectiveness of the proposed approach.