Spatial Analysis of Air Particulate Pollution Distributions and Its Relation to Real Property Value in Beijing, China

Air particulate pollution contributes the major air pollution in Beijing, China. In this research, concentrations of air particulate pollutants were measured at a total of twenty-three field locations in the urban districts of Beijing applying a laser particle counter in June and December 2015. Geographic Information System (GIS) was utilized to study the two and three-dimensional spatial distributions of air particulate pollution (PM0.5, PM1.0, PM2.5, PM5.0, PM10). Geostatistical or spatial statistical models were applied to interpolate the spatial distributions of air particulate pollution and real property values in the study area. Geographically Weighted Regression (GWR) was applied to analyze the spatial relationships of air particulate pollution and distribution of real property values. The three-dimensional analysis was conducted to illustrate vertical spatial distributions of air particulate pollution for each of the twenty-three field survey profiles in ArcGIS. Temporal distributions of air particulate pollution within 10 hours daytime at two field survey locations were analyzed. The results show that the concentrations of different sizes of air particulate pollutants in urban areas of Beijing distribute differently with different spatial patterns. The spatial distributions of real property values indicate that the highest value occurred in the northwestern and the central parts of Beijing both in the June and December 2015. There is no significant relationship of real property values and the intensity of air particulate pollution. Therefore, we suggest that the spatial distribution factors of air particulate pollution in Beijing is not a major factor for people to purchase real properties as homes

Recovery Guarantees for Quadratic Tensors with Limited Observations

We consider the tensor completion problem of predicting the missing entries of a tensor. The commonly used CP model has a triple product form, but an alternate family of quadratic models which are the sum of pairwise products instead of a triple product have emerged from applications such as recommendation systems. Non-convex methods are the method of choice for learning quadratic models, and this work examines their sample complexity and error guarantee. Our main result is that with the number of samples being only linear in the dimension, all local minima of the mean squared error objective are global minima and recover the original tensor accurately. The techniques lead to simple proofs showing that convex relaxation can recover quadratic tensors provided with linear number of samples. We substantiate our theoretical results with experiments on synthetic and real-world data, showing that quadratic models have better performance than CP models in scenarios where there are limited amount of observations available

Generalization and Equilibrium in Generative Adversarial Nets (GANs)

We show that training of generative adversarial network (GAN) may not have good generalization properties; e.g., training may appear successful but the trained distribution may be far from target distribution in standard metrics. However, generalization does occur for a weaker metric called neural net distance. It is also shown that an approximate pure equilibrium exists in the discriminator/generator game for a special class of generators with natural training objectives when generator capacity and training set sizes are moderate. This existence of equilibrium inspires MIX+GAN protocol, which can be combined with any existing GAN training, and empirically shown to improve some of them.Comment: This is an updated version of an ICML'17 paper with the same title. The main difference is that in the ICML'17 version the pure equilibrium result was only proved for Wasserstein GAN. In the current version the result applies to most reasonable training objectives. In particular, Theorem 4.3 now applies to both original GAN and Wasserstein GA

Matrix Completion and Related Problems via Strong Duality

This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and its dual have the same optimum. This has been well understood for convex optimization, but little was known for non-convex problems. We propose a novel analytical framework and show that under certain dual conditions, the optimal solution of the matrix factorization program is the same as its bi-dual and thus the global optimality of the non-convex program can be achieved by solving its bi-dual which is convex. These dual conditions are satisfied by a wide class of matrix factorization problems, although matrix factorization problems are hard to solve in full generality. This analytical framework may be of independent interest to non-convex optimization more broadly. We apply our framework to two prototypical matrix factorization problems: matrix completion and robust Principal Component Analysis (PCA). These are examples of efficiently recovering a hidden matrix given limited reliable observations of it. Our framework shows that exact recoverability and strong duality hold with nearly-optimal sample complexity guarantees for matrix completion and robust PCA