RESUMO
A novel learning framework of nonparallel hyperplanes support vector machines (NPSVMs) is proposed for binary classification and multiclass classification. This framework not only includes twin SVM (TWSVM) and its many deformation versions but also extends them into multiclass classification problem when different parameters or loss functions are chosen. Concretely, we discuss the linear and nonlinear cases of the framework, in which we select the hinge loss function as example. Moreover, we also give the primal problems of several extension versions of TWSVM's deformation versions. It is worth mentioning that, in the decision function, the Euclidean distance is replaced by the absolute value |w (T) x + b|, which keeps the consistency between the decision function and the optimization problem and reduces the computational cost particularly when the kernel function is introduced. The numerical experiments on several artificial and benchmark datasets indicate that our framework is not only fast but also shows good generalization.
RESUMO
We investigate the stationary spectral density of fractional bistable Klein-Kramers systems. First, we deduce a dissipation-fluctuation relation between the stationary spectral density at thermal equilibrium and the linear response of the system to an applied perturbation. Second, we describe how to obtain the linear dynamic susceptibility from the method of moments, and thus we derive the fluctuating spectral density from the dissipation-fluctuation relation. Finally, we exhibit the structure of this fluctuating spectral distribution and explore the effect of the subdiffusion on it. Compared with the standard bistable Klein-Kramers systems, our observation on the spectral distribution in fractional systems reveals that the subdiffusion weakens the oscillatory components of the intrawell oscillation and the above-barrier motion. This phenomenon should reflect a fact that the particles tend to stand still in separate wells in subdiffusive processes.