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The paper introduces a multiway tensor generalization of the bigraphical lasso which uses a two-way sparse Kronecker sum multivariate normal model for the precision matrix to model parsimoniously conditional dependence relationships of matrix variate data based on the Cartesian product of graphs. We call this tensor graphical lasso generalization TeraLasso. We demonstrate by using theory and examples that the TeraLasso model can be accurately and scalably estimated from very limited data samples of high dimensional variables with multiway co-ordinates such as space, time and replicates. Statistical consistency and statistical rates of convergence are established for both the bigraphical lasso and TeraLasso estimators of the precision matrix and estimators of its support (non-sparsity) set respectively. We propose a scalable composite gradient descent algorithm and analyse the computational convergence rate, showing that the composite gradient descent algorithm is guaranteed to converge at a geometric rate to the global minimizer of the TeraLasso objective function. Finally, we illustrate TeraLasso by using both simulation and experimental data from a meteorological data set, showing that we can accurately estimate precision matrices and recover meaningful conditional dependence graphs from high dimensional complex data sets.

The energy resource is an essential input of economic growth, which has an important impact on the ecological environment and social welfare. From the perspective of social welfare, considering the radial and non-radial characteristics of different input and output indicators, and the inseparability of the energy input and undesirable output, this study employs the non-separable hybrid DEA (Data Envelopment Analysis) model to evaluate the total energy efficiency of Chinese provinces between 2012 and 2016. Furthermore, this study calculates the energy saving and emission reduction potentials of China. The results reveal that the average total factor energy efficiency in China from 2012 to 2016 is 0.694, which means that there are still 30.6% energy efficiency losses. There is great potential for China to save energy, reduce pollutant emissions, and increase the output of social welfare. There are great differences in the total factor energy efficiency among provinces. The average energy saving potential of the whole country is 60.5%. If the energy efficiency of all provinces can reach the frontier, the whole country can save more than half of the energy consumption. The highest national average emission reduction potential is SO2, followed by dust, CO2, and NOX. The implication of the conclusion is that in the development of regional economy, we cannot sacrifice the social welfare and sustainable development and take the growth rate of GDP as the only objective. Different energy saving and emission reduction policies should be put forward according to the characteristics of different provinces.