Approximate Set Union Via Approximate Randomization

We develop an randomized approximation algorithm for the size of set union problem \arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert, which given a list of sets $A_1,...,A_m$ with approximate set size $m_i$ for $A_i$ with $m_i\in \left((1-\beta_L)|A_i|, (1+\beta_R)|A_i|\right)$, and biased random generators with Prob(x=\randomElm(A_i))\in \left[{1-\alpha_L\over |A_i|},{1+\alpha_R\over |A_i|}\right] for each input set $A_i$ and element $x\in A_i,$ where $i=1, 2, ..., m$. The approximation ratio for \arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert is in the range $[(1-\epsilon)(1-\alpha_L)(1-\beta_L), (1+\epsilon)(1+\alpha_R)(1+\beta_R)]$ for any $\epsilon\in (0,1)$, where $\alpha_L, \alpha_R, \beta_L,\beta_R\in (0,1)$. The complexity of the algorithm is measured by both time complexity, and round complexity. The algorithm is allowed to make multiple membership queries and get random elements from the input sets in one round. Our algorithm makes adaptive accesses to input sets with multiple rounds. Our algorithm gives an approximation scheme with O(\setCount\cdot(\log \setCount)^{O(1)}) running time and $O(\log m)$ rounds, where $m$ is the number of sets. Our algorithm can handle input sets that can generate random elements with bias, and its approximation ratio depends on the bias. Our algorithm gives a flexible tradeoff with time complexity O\left(\setCount^{1+\xi}\right) and round complexity $O\left({1\over \xi}\right)$ for any $\xi\in(0,1)$

Penalized Estimation of Directed Acyclic Graphs From Discrete Data

Bayesian networks, with structure given by a directed acyclic graph (DAG), are a popular class of graphical models. However, learning Bayesian networks from discrete or categorical data is particularly challenging, due to the large parameter space and the difficulty in searching for a sparse structure. In this article, we develop a maximum penalized likelihood method to tackle this problem. Instead of the commonly used multinomial distribution, we model the conditional distribution of a node given its parents by multi-logit regression, in which an edge is parameterized by a set of coefficient vectors with dummy variables encoding the levels of a node. To obtain a sparse DAG, a group norm penalty is employed, and a blockwise coordinate descent algorithm is developed to maximize the penalized likelihood subject to the acyclicity constraint of a DAG. When interventional data are available, our method constructs a causal network, in which a directed edge represents a causal relation. We apply our method to various simulated and real data sets. The results show that our method is very competitive, compared to many existing methods, in DAG estimation from both interventional and high-dimensional observational data.Comment: To appear in Statistics and Computin

The disappearance of a narrow Mg II absorption system in quasar SDSS J165501.31+260517.4

In this letter, we present for the first time, the discovery of the disappearance of a narrow Mg II $\lambda\lambda2796,2803$ absorption system from the spectra of quasar SDSS J165501.31+260517.4 ($z_{\rm e}=1.8671$). This absorber is located at $z_{\rm abs} =1.7877$, and has a velocity offset of $8,423\rm ~km~s^{-1}$ with respect to the quasar. According to the velocity offset and the line variability, this narrow Mg II $\lambda\lambda2796,2803$ absorption system is likely intrinsic to the quasar. Since the corresponding UV continuum emission and the absorption lines of another narrow Mg II $\lambda\lambda2796,2803$ absorption system at $z_{\rm abs}=1.8656$ are very stable, we think that the disappearance of the absorption system is unlikely to be caused by the change in ionization of absorption gas. Instead, it likely arises from the motion of the absorption gas across the line of sight

Land Use Rights, Government Land Supply, and the Pattern of Redevelopment in Shanghai

This paper reviews the urban redevelopment activities in Shanghai as the land market reforms were introduced. We focus on the impact of land use institutions on the spatial pattern of these activities. Sites for private real estate redevelopment were supplied by individual districts in the city. But the need for districts to pay for the resettlement of displaced residents contributed to a spatial mismatch between the supply of redevelopment sites and the market demand for commercial real estate space. Resettlement costs are highest at the high demand locations. State owned enterprises and institutions occupying land allocated by the state also engaged in real estate development. Whereas the density of private redevelopment was sensitive to the volume of commercial activities in a district, this does not appear to have been important in determining the location of the significant increase in the stock of commercial space resulting from development by local enterprises and institutions. This growth shows considerable decentralization between 1993 and 1996,indicative of spatially inefficient redevelopment activities by land-rich state enterprises.Institutions, Land Market, Economic Transition, Urban Redevelopment, Shanghai