Abstract. The synchronization problem over the special orthogonal group SO(d) consists of estimating a set of unknown rotations R 1 , R 2 , . . . , Rn from noisy measurements of a subset of their pairwise ratios R −1 i R j . The problem has found applications in computer vision, computer graphics, and sensor network localization, among others. Its least squares solution can be approximated by either spectral relaxation or semidefinite programming followed by a rounding procedure, analogous to the approximation algorithms of Max-Cut. The contribution of this paper is three-fold: First, we introduce a robust penalty function involving the sum of unsquared deviations and derive a relaxation that leads to a convex optimization problem; Second, we apply the alternating direction method to minimize the penalty function; Finally, under a specific model of the measurement noise and the measurement graph, we prove that the rotations are exactly and stably recovered, exhibiting a phase transition behavior in terms of the proportion of noisy measurements. Numerical simulations confirm the phase transition behavior for our method as well as its improved accuracy compared to existing methods.

A major challenge in single particle reconstruction from cryo-electron microscopy is to establish a reliable ab initio three-dimensional model using two-dimensional projection images with unknown orientations. Common-lines–based methods estimate the orientations without additional geometric information. However, such methods fail when the detection rate of common-lines is too low due to the high level of noise in the images. An approximation to the least squares global self-consistency error was obtained in [A. Singer and Y. Shkolnisky, SIAM J. Imaging Sci., 4 (2011), pp. 543–572] using convex relaxation by semidefinite programming. In this paper we introduce a more robust global self-consistency error and show that the corresponding optimization problem can be solved via semidefinite relaxation. In order to prevent artificial clustering of the estimated viewing directions, we further introduce a spectral norm term that is added as a constraint or as a regularization term to the relaxed minimization problem. The resulting problems are solved using either the alternating direction method of multipliers or an iteratively reweighted least squares procedure. Numerical experiments with both simulated and real images demonstrate that the proposed methods significantly reduce the orientation estimation error when the detection rate of common-lines is low.

Global warming and human land management have greatly influenced vegetation growth through both changes in spring phenology and photosynthetic primary production. This will presumably impact the velocity of vegetation greenup (Vgreenup, the daily rate of changes in vegetation productivity during greenup period), yet little is currently known about the spatio-temporal patterns of Vgreenup of global vegetation. Here, we define Vgreenup as the ratio of the amplitude of greenup (Agreenup) to the duration of greenup (Dgreenup) and derive global Vgreenup from 34-year satellite leaf area index (LAI) observations to study spatio-temporal dynamics of Vgreenup at the global, hemispheric, and ecosystem scales. We find that 19.9% of the pixels analyzed (n = 1,175,453) experienced significant trends toward higher greenup rates by an average of 0.018 m m day for 1982-2015 as compared to 8.6% of pixels with significant negative trends (p < 0.05). Global distribution and dynamics of Vgreenup show high spatial heterogeneity and ecosystem-specific patterns, which is primarily determined by the high spatial variation in Agreenup, while the temporal dynamics of Vgreenup are directly controlled by both changes in Dgreenup and Agreenup. Areas with the largest Vgreenup and largest positive trends are both observed in deciduous and mixed forests as compared to nonforest ecosystems showing both lower Vgreenup and trends. For nonforest ecosystems, human-managed ecosystems (e.g., rangelands and rainfed croplands) exhibited higher Vgreenup and positive trends than those of natural counterparts, suggesting strong imprints of human land management on terrestrial ecosystem functioning. Globally, warming has accelerated Vgreenup in temperature-constrained high latitude forest ecosystems and arctic regions, but decelerated Vgreenup in temperate and arid/semiarid areas. These results suggest that the combined effects of climate change and human land management have greatly accelerated global vegetation greenup, with important implications for changes in terrestrial ecosystem functioning and global carbon cycling.

Forest certification plays an important role in the global trade of legal, sustainably harvested timber. There is no accurate definition of how international forest certification systems impact international trade from a global perspective. This paper is intended to evaluate the influence of forest certification on international trade, so that it can provide a scientific basis for the improvement of the international forest certification systems and for the development of relevant forestry industries in different countries. First, the influence of forest certification on international trade of forest products is explained in the economic model; hence, four hypotheses are put forward. Second, to test these hypotheses, we verify the panel data of bilateral trade and forest certification of all forest products among 67 economies from 2009 to 2018 by incorporating forest certifications into the gravity model. Finally, tests by country groups and product groups were further analyzed, respectively. The results show that: (1) The extended Poisson pseudo maximum likelihood (PPML) estimation solves the problem of the heteroscedasticity and zero trade value problems of the gravity model well in the forest industry. (2) Forest certification has an export competitive effect, a trade barrier effect, as well as common language effect. (3) Forest certification has asymmetric trade effects. The export competitive effect of forest certification in developing countries is greater than that in developed countries. Forest certification has become a trade barrier for developing countries, especially in the process of trade with developed countries. The common language effect is higher during the trade between developed and developing countries. The export competitive effect of wood products is higher than that of furniture products. Forest certification has trade barrier effect on wood products in developing countries, while it has trade barrier effect on furniture products in developed countries.

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