Two types of interacting dark energy models are investigated using the type Ia supernova (SNIa), observational $H(z)$ data (OHD), cosmic microwave background (CMB) shift parameter and the secular Sandage-Loeb (SL) test. We find that the inclusion of
SL test can obviously provide more stringent constraint on the parameters in both models. For the constant coupling model, the interaction term including the SL test is estimated at $delta=-0.01 pm 0.01 (1sigma) pm 0.02 (2sigma)$, which has been improved to be only a half of original scale on corresponding errors. Comparing with the combination of SNIa and OHD, we find that the inclusion of SL test directly reduces the best-fit of interaction from 0.39 to 0.10, which indicates that the higher-redshift observation including the SL test is necessary to track the evolution of interaction. For the varying coupling model, we reconstruct the interaction $delta (z)$, and find that the interaction is also negative similar as the constant coupling model. However, for high redshift, the interaction generally vanishes at infinity. The constraint result also shows that the $Lambda$CDM model still behaves a good fit to the observational data, and the coincidence problem is still quite severe. However, the phantom-like dark energy with $w_X<-1$ is slightly favored over the $Lambda$CDM model.
In this paper, we propose a novel image interpolation algorithm, which is formulated via combining both the local autoregressive (AR) model and the nonlocal adaptive 3-D sparse model as regularized constraints under the regularization framework. Esti
mating the high-resolution image by the local AR regularization is different from these conventional AR models, which weighted calculates the interpolation coefficients without considering the rough structural similarity between the low-resolution (LR) and high-resolution (HR) images. Then the nonlocal adaptive 3-D sparse model is formulated to regularize the interpolated HR image, which provides a way to modify these pixels with the problem of numerical stability caused by AR model. In addition, a new Split-Bregman based iterative algorithm is developed to solve the above optimization problem iteratively. Experiment results demonstrate that the proposed algorithm achieves significant performance improvements over the traditional algorithms in terms of both objective quality and visual perception
We study the problem of high-dimensional variable selection via some two-step procedures. First we show that given some good initial estimator which is $ell_{infty}$-consistent but not necessarily variable selection consistent, we can apply the nonne
gative Garrote, adaptive Lasso or hard-thresholding procedure to obtain a final estimator that is both estimation and variable selection consistent. Unlike the Lasso, our results do not require the irrepresentable condition which could fail easily even for moderate $p_n$ (Zhao and Yu, 2007) and it also allows $p_n$ to grow almost as fast as $exp(n)$ (for hard-thresholding there is no restriction on $p_n$). We also study the conditions under which the Ridge regression can be used as an initial estimator. We show that under a relaxed identifiable condition, the Ridge estimator is $ell_{infty}$-consistent. Such a condition is usually satisfied when $p_nle n$ and does not require the partial orthogonality between relevant and irrelevant covariates which is needed for the univariate regression in (Huang et al., 2008). Our numerical studies show that when using the Lasso or Ridge as initial estimator, the two-step procedures have a higher sparsity recovery rate than the Lasso or adaptive Lasso with univariate regression used in (Huang et al., 2008).
