Semismooth Newton Coordinate Descent Algorithm for Elastic-Net Penalized Huber Loss Regression and Quantile Regression

We propose an algorithm, semismooth Newton coordinate descent (SNCD), for the elastic-net penalized Huber loss regression and quantile regression in high dimensional settings. Unlike existing coordinate descent type algorithms, the SNCD updates each regression coefficient and its corresponding subgradient simultaneously in each iteration. It combines the strengths of the coordinate descent and the semismooth Newton algorithm, and effectively solves the computational challenges posed by dimensionality and nonsmoothness. We establish the convergence properties of the algorithm. In addition, we present an adaptive version of the "strong rule" for screening predictors to gain extra efficiency. Through numerical experiments, we demonstrate that the proposed algorithm is very efficient and scalable to ultra-high dimensions. We illustrate the application via a real data example

Amplitude Relations in Non-linear Sigma Model

In this paper, we investigate tree-level scattering amplitude relations in $U(N)$ non-linear sigma model. We use Cayley parametrization. As was shown in the recent works [23,24] both on-shell amplitudes and off-shell currents with odd points have to vanish under Cayley parametrization. We prove the off-shell $U(1)$ identity and fundamental BCJ relation for even-point currents. By taking the on-shell limits of the off-shell relations, we show that the color-ordered tree amplitudes with even points satisfy $U(1)$-decoupling identity and fundamental BCJ relation, which have the same formations within Yang-Mills theory. We further state that all the on-shell general KK, BCJ relations as well as the minimal-basis expansion are also satisfied by color-ordered tree amplitudes. As a consequence of the relations among color-ordered amplitudes, the total $2m$-point tree amplitudes satisfy DDM form of color decomposition as well as KLT relation.Comment: 27 pages, 8 figures, 4 tables, JHEP style, improved versio

Gauge invariance induced relations and the equivalence between distinct approaches to NLSM amplitudes

In this paper, we derive generalized Bern-Carrasco-Johansson relations for color-ordered Yang-Mills amplitudes by imposing gauge invariance conditions and dimensional reduction appropriately on the new discovered graphic expansion of Einstein-Yang-Mills amplitudes. These relations are also satisfied by color-ordered amplitudes in other theories such as color-scalar theory, bi-scalar theory and nonlinear sigma model (NLSM). As an application of the gauge invariance induced relations, we further prove that the three types of BCJ numerators in NLSM , which are derived from Feynman rules, Abelian Z-theory and Cachazo-He- Yuan formula respectively, produce the same total amplitudes. In other words, the three distinct approaches to NLSM amplitudes are equivalent to each other.Comment: 40pages, 2 figure