When Locally Linear Embedding Hits Boundary

Based on the Riemannian manifold model, we study the asymptotic behavior of a widely applied unsupervised learning algorithm, locally linear embedding (LLE), when the point cloud is sampled from a compact, smooth manifold with boundary. We show several peculiar behaviors of LLE near the boundary that are different from those diffusion-based algorithms. Particularly, LLE converges to a mixed-type differential operator with degeneracy. This study leads to an alternative boundary detection algorithm and two potential approaches to recover the Dirichlet Laplace-Beltrami operator.Comment: 11 Figure

Think globally, fit locally under the Manifold Setup: Asymptotic Analysis of Locally Linear Embedding

Since its introduction in 2000, the locally linear embedding (LLE) has been widely applied in data science. We provide an asymptotical analysis of the LLE under the manifold setup. We show that for the general manifold, asymptotically we may not obtain the Laplace-Beltrami operator, and the result may depend on the non-uniform sampling, unless a correct regularization is chosen. We also derive the corresponding kernel function, which indicates that the LLE is not a Markov process. A comparison with the other commonly applied nonlinear algorithms, particularly the diffusion map, is provided, and its relationship with the locally linear regression is also discussed.Comment: 78 pages, 4 figures. We add a short discussion about thr relation between espilon and the intrinsic geometry of the manifold. We add a new section about K nearest neighborhood (KNN) and a new subsection about error in variable. We provide more numerical example

A Question Type Driven and Copy Loss Enhanced Frameworkfor Answer-Agnostic Neural Question Generation

The answer-agnostic question generation is a significant and challenging task, which aims to automatically generate questions for a given sentence but without an answer. In this paper, we propose two new strategies to deal with this task: question type prediction and copy loss mechanism. The question type module is to predict the types of questions that should be asked, which allows our model to generate multiple types of questions for the same source sentence. The new copy loss enhances the original copy mechanism to make sure that every important word in the source sentence has been copied when generating questions. Our integrated model outperforms the state-of-the-art approach in answer-agnostic question generation, achieving a BLEU-4 score of 13.9 on SQuAD. Human evaluation further validates the high quality of our generated questions. We will make our code public available for further research

A Statistical Study of H I Gas in Nearby Narrow-Line AGN-Hosting Galaxies

As a quenching mechanism, AGN feedback could suppress on-going star formation in their host galaxies. On the basis of a sample of galaxies selected from ALFALFA HI survey, the dependence of their HI mass M[HI], stellar mass M[*] & HI-to-stellar mass ratio M[HI]/M[*] on various tracers of AGN activity are presented and analyzed in this paper. Almost all the AGN-hostings in this sample are gas-rich galaxies, and there is no any evidence to be shown to indicate that the AGN activity could increase/decrease either M[HI] or M[HI]/M[*]. The cold neutral gas can not be fixed positions accurately just based on available HI data due to the large beam size of ALFALFA survey. In addition, even though AGN-hostings are more easily detected by HI survey compared with absorption line galaxies, these two types of galaxies show similar star formation history. If an AGN-hosting would ultimately evolve into an old red galaxy with few cold gas, then when and how the gas has been exhausted have to be solved by future hypotheses and observations.Comment: 34 pages, 8 figures, published in A

Problems on Foundations of General Relativity

It was generally believed that, in general relativity, the fundamental laws of nature should be invariant or covariant under a general coordinate transformation. In general relativity, the equivalence principle tells us the existence of a local inertial coordinate system and the fundamental laws in the local inertial coordinate system which are the same as those in inertial reference system. Then, after a general coordinate transformation, the fundamental laws of nature in arbitrary coordinate system or in arbitrary curved space-time can be obtained. However, through a simple example, we find that, under a general coordinate transformation, basic physical equations in general relativity do not transform covariantly, especially they do not preserve their forms under the transformation from a local inertial coordinate system to a curved space-time. The origination of the violation of the general covariance is then studied, and a general theory on general coordinate transformations is developed. Because of the the existence of the non-homogeneous term, the fundamental laws of nature in arbitrary curved space-time can not be expressed by space-time metric, physical observable and their derivatives. In other words, basic physical equations obtained from the equivalence principle and the principle of general covariance are different from those in general relativity. Both the equivalence principle and the principle of general covariance can not be treated as foundations of general relativity. So, what are the foundations of General Relativity? Such kind of essential problems on General Relativity can be avoided in the physical picture of gravity. Quantum gauge theory of gravity, which is founded in the physics picture of gravity, does not have such kind of fundamental problems.Comment: 26 pages, no figur

Energy Storage as Public Asset

Energy storage has exhibited great potential in providing flexibility in power system to meet critical peak demand and thus reduce the overall generation cost, which in turn stabilizes the electricity prices. In this work, we exploit the opportunities for the independent system operator (ISO) to invest and manage storage as public asset, which could systematically provide benefits to the public. Assuming a quadratic generation cost structure, we apply parametric analysis to investigate the ISO's problem of economic dispatch, given variant quantities of storage investment. This investment is beneficial to users on expectation. However, it may not necessarily benefit everyone. We adopt the notion of marginal system cost impact (MCI) to measure each user's welfare and show its relationship with the conventional locational marginal price. We find interesting convergent characteristics for MCI. Furthermore, we perform $k$-means clustering to classify users for effective user profiling and conduct numerical studies on both prototype and IEEE test systems to verify our theoretical conclusions

Uncertainty Relations in the Presence of Quantum Memory for Mutually Unbiased Measurements

In [Berta 2014 Entanglement], uncertainty relations in the presence of quantum memory was formulated for mutually unbiased bases using conditional collision entropy. In this paper, we generalize their results to the mutually unbiased measurements. Our primary result is an equality between the amount of uncertainty for a set of measurements and the amount of entanglement of the measured state, both of which are quantified by the conditional collision entropy. Implications of this equality relation are discussed. We further show that similar equality relation can be obtained for generalized symmetric informationally complete measurements. We also derive an interesting equality for arbitrary orthogonal basis of the space of Hermitian, traceless operators.Comment: 8 pages, comments are welcome

Community Evolution of Social Network: Feature, Algorithm and Model

Researchers have devoted themselves to exploring static features of social networks and further discovered many representative characteristics, such as power law in the degree distribution and assortative value used to differentiate social networks from nonsocial ones. However, people are not satisfied with these achievements and more and more attention has been paid on how to uncover those dynamic characteristics of social networks, especially how to track community evolution effectively. With these interests, in the paper we firstly display some basic but dynamic features of social networks. Then on its basis, we propose a novel core-based algorithm of tracking community evolution, CommTracker, which depends on core nodes to establish the evolving relationships among communities at different snapshots. With the algorithm, we discover two unique phenomena in social networks and further propose two representative coefficients: GROWTH and METABOLISM by which we are also able to distinguish social networks from nonsocial ones from the dynamic aspect. At last, we have developed a social network model which has the capabilities of exhibiting two necessary features above.Comment: 16 pages,7 figure

Evolution of oxygen and nitrogen abundances and nitrogen production mechanism in massive star-forming galaxies

Utilizing the observational data of 55,318 star-forming galaxies (SFGs) selected from the catalog of MPA-JHU emission-line measurements for the SDSS DR8, we investigate the galaxy downsizing effect of their O and N enrichments, and the nitrogen production mechanism in them. We show the redshift evolution of O and N abundances and specific star formation rates for different galaxy mass ranges, demonstrating the galaxy downsizing effect caused by less massive progenitors of less massive galaxies. The O and N abundances do not remain constant for different galaxy mass ranges, and the enrichment (and hence star formation) decreases with increasing galaxy stellar mass. We find evidence of the O enrichment for galaxies with stellar masses $M_{*}>10^{11.0}$ (in units of $M_{\odot}$), i.e. $\Delta({\log}({\rm O/H})) \sim 0.10$ and $\Delta({\log}({\rm N/H})) \sim 0.28$ from redshift 0.023 to 0.30. Based on the evolutionary schematic model of N/O ratios in Coziol et al., who proposed the scheme that the production of nitrogen is the consequence of a sequence of bursts in SFGs, we conclude that the nitrogen production is dominated by the intermediate-mass stars, which dominate the secondary synthesis in SFGs. However, for galaxies with $M_{*}>10^{10.35}$ we find evidence of enhanced N/O abundance ratios, which are significantly above the secondary synthesis line. This suggests that outflows of massive stars, which deplete oxygen efficiently, are more important in massive galaxies. Finally we find an excellent linear relation between $M_{*}$ and log(N/O), indicating that the N/O abundance ratio is a good indicator of the stellar mass in a SFG and may be used as a standard candle for studying cosmology, if confirmed with further studies.Comment: Updated to match the published version in MNRAS. 25 pages, 12 figures, 2 table

Optimal Liquidation in a Finite Time Regime Switching Model with Permanent and Temporary Pricing Impact

In this paper we discuss the optimal liquidation over a finite time horizon until the exit time. The drift and diffusion terms of the asset price are general functions depending on all variables including control and market regime. There is also a local nonlinear transaction cost associated to the liquidation. The model deals with both the permanent impact and the temporary impact in a regime switching framework. The problem can be solved with the dynamic programming principle. The optimal value function is the unique continuous viscosity solution to the HJB equation and can be computed with the finite difference method.Comment: 17 page