Delayed Onset and Fast Rise of Prompt Optical-UV Emission from Gamma-Ray Bursts in Molecular Clouds

Observations imply that long \gamma-ray bursts (GRBs) are originated from explosions of massive stars, therefore they may occur in the molecular clouds where their progenitors were born. We show here that the prompt optical-UV emission from GRBs may be delayed due to the dust extinction, which can well explain the observed optical delayed onset and fast rise in GRB 080319B. The density and the size of the molecular cloud around GRB 080319B are roughly constrained to be \sim10^3cm^{-3} and \sim 8pc, respectively. We also investigate the other GRBs with prompt optical-UV data, and find similar values of the densities and sizes of the local molecular clouds. The future observations of prompt optical-UV emission from GRBs in subsecond timescale, e.g., by UFFO-Pathfinder and SVOM-GWAC, will provide more evidence and probes of the local GRB environments.Comment: 15 pages, 5 figures, RAA 13 (2013) 57-70, typo correctio

Gradient Hard Thresholding Pursuit for Sparsity-Constrained Optimization

Hard Thresholding Pursuit (HTP) is an iterative greedy selection procedure for finding sparse solutions of underdetermined linear systems. This method has been shown to have strong theoretical guarantee and impressive numerical performance. In this paper, we generalize HTP from compressive sensing to a generic problem setup of sparsity-constrained convex optimization. The proposed algorithm iterates between a standard gradient descent step and a hard thresholding step with or without debiasing. We prove that our method enjoys the strong guarantees analogous to HTP in terms of rate of convergence and parameter estimation accuracy. Numerical evidences show that our method is superior to the state-of-the-art greedy selection methods in sparse logistic regression and sparse precision matrix estimation tasks