arXiv subjects
Publications and source records attributed to Nan Li.
We show that if $X$ is a limit of $n$-dimensional Riemannian manifolds with Ricci curvature bounded below and $γ$ is a limit geodesic in $X$ then along the interior of $γ$ same scale measure metric tangent cones $T_{γ(t)}X$ are Hölder continuous with respect to measured Gromov-Hausdorff topology and have the same dimension in the sense of Colding-Naber.
Consider a binary classification problem in which the learner is given a labeled training set, an unlabeled test set, and is restricted to choosing exactly $k$ test points to output as positive predictions. Problems of this kind---{\it transductive precision@$k$}---arise in information retrieval, digital advertising, and reserve design for endangered species. Previous methods separate the training of the model from its use in scoring the test points. This paper introduces a new approach, Transductive Top K (TTK), that seeks to minimize the hinge loss over all training instances under the constraint that exactly $k$ test instances are predicted as positive. The paper presents two optimization methods for this challenging problem. Experiments and analysis confirm the importance of incorporating the knowledge of $k$ into the learning process. Experimental evaluations of the TTK approach show that the performance of TTK matches or exceeds existing state-of-the-art methods on 7 UCI datasets and 3 reserve design problem instances.
Inspired by the multiverse scenario, we study a heterotic dark energy model in which there are two parts, the first being the cosmological constant and the second being the holographic dark energy, thus this model is named the $Λ$HDE model. By studying the $Λ$HDE model theoretically, we find that the parameters $d$ and $Ω_{hde}$ are divided into a few domains in which the fate of the universe is quite different. We investigate dynamical behaviors of this model, and especially the future evolution of the universe. We perform fitting analysis on the cosmological parameters in the $Λ$HDE model by using the recent observational data. We find the model yields $χ^2_{\rm min}=426.27$ when constrained by $\rm Planck+SNLS3+BAO+HST$, comparable to the results of the HDE model (428.20) and the concordant $Λ$CDM model (431.35). At 68.3\% CL, we obtain $-0.07<Ω_{\Lambda0}<0.68$ and correspondingly $0.04<Ω_{hde0}<0.79$, implying at present there is considerable degeneracy between the holographic dark energy and cosmological constant components in the $Λ$HDE model.
We prove a Lipschitz-Volume rigidity theorem in Alexandrov geometry, that is, if a 1-Lipschitz map $f\colon X=\amalg X_\ell\to Y$ between Alexandrov spaces preserves volume, then it is a path isometry and an isometry when restricted to the interior of $X$. We furthermore characterize the metric structure on $Y$ with respect to $X$ when $f$ is also onto. This implies the converse of Petrunin's Gluing Theorem: if a gluing of two Alexandrov spaces via a bijection between their boundaries produces an Alexandrov space, then the bijection must be an isometry.
We introduce a notion of probabilistic convexity and generalize some classical globalization theorems in Alexandrov geometry. A weighted Alexandrov's lemma is developed as a basic tool.
We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.
The United Nations released official probabilistic population projections (PPP) for all countries for the first time in July 2014. These were obtained by projecting the period total fertility rate (TFR) and life expectancy at birth ($e_0$) using Bayesian hierarchical models, yielding a large set of future trajectories of TFR and $e_0$ for all countries and future time periods to 2100, sampled from their joint predictive distribution. Each trajectory was then converted to age-specific mortality and fertility rates, and population was projected using the cohort-component method. This yielded a large set of trajectories of future age- and sex-specific population counts and vital rates for all countries. In this paper we describe the methodology used for deriving the age-specific mortality and fertility rates in the 2014 PPP, we identify limitations of these methods, and we propose several methodological improvements to overcome them. The methods presented in this paper are implemented in the publicly available bayesPop R package.
Our universe hosts various large-scale structures from voids to galaxy clusters, so it would be interesting to find some simple and reasonable measure to describe the inhomogeneities in the universe. We explore two different methods for this purpose: the Kullback-Leibler entropy and the Weyl curvature tensor. These two quantities characterize the deviation of the actual distribution of matter from the unperturbed background. We calculate these two measures in the spherically symmetric Lemaitre-Tolman-Bondi model in the dust universe. Both exact and perturbative calculations are presented, and we observe that these two measures are in proportion up to second order.
Many believe that in-field hardware faults are too rare in practice to justify the need for Logic Built-In Self-Test (LBIST) in a design. Until now, LBIST was primarily used in safety-critical applications. However, this may change soon. First, even if costly methods like burn-in are applied, it is no longer possible to get rid of all latent defects in devices at leading-edge technology. Second, demands for high reliability spread to consumer electronics as smartphones replace our wallets and IDs. However, today many ASIC vendors are reluctant to use LBIST. In this paper, we describe the needs for successful deployment of LBIST in the industrial practice and discuss how these needs can be addressed. Our work is hoped to attract a wider attention to this important research topic.
Edge polytopes is a class of interesting polytope with rich algebraic and combinatorial properties, which was introduced by Ohsugi and Hibi. In this papar, we follow a previous study on cutting edge polytopes by Hibi, Li and Zhang. Instead of focusing on the algeraic properties of the subpolytopes as the previous study, in this paper, we take a closer look on the graphs whose edge polytopes are decomposable. In particular, we answer two important questions raised in the previous study about 1) the relationship between type I and type II decomposable graphs and 2) description of decomposable graphs.
Bipartite ranking aims to learn a real-valued ranking function that orders positive instances before negative instances. Recent efforts of bipartite ranking are focused on optimizing ranking accuracy at the top of the ranked list. Most existing approaches are either to optimize task specific metrics or to extend the ranking loss by emphasizing more on the error associated with the top ranked instances, leading to a high computational cost that is super-linear in the number of training instances. We propose a highly efficient approach, titled TopPush, for optimizing accuracy at the top that has computational complexity linear in the number of training instances. We present a novel analysis that bounds the generalization error for the top ranked instances for the proposed approach. Empirical study shows that the proposed approach is highly competitive to the state-of-the-art approaches and is 10-100 times faster.
The half-open hypersimplex $Δ'_{n,k}$ consists of those $x = (x_{1}, \ldots, x_{n}) \in[0,1]^n$ with $k-1<x_1+\cdots+x_n\le k$, where $0 < k \leq n$. The $f$-vector of a half-open hypersimplex and related generating functions are explicitly studied.
This work is an attempt to discover hidden structural configurations in learning activity sequences of students in Massive Open Online Courses (MOOCs). Leveraging combined representations of video clickstream interactions and forum activities, we seek to fundamentally understand traits that are predictive of decreasing engagement over time. Grounded in the interdisciplinary field of network science, we follow a graph based approach to successfully extract indicators of active and passive MOOC participation that reflect persistence and regularity in the overall interaction footprint. Using these rich educational semantics, we focus on the problem of predicting student attrition, one of the major highlights of MOOC literature in the recent years. Our results indicate an improvement over a baseline ngram based approach in capturing "attrition intensifying" features from the learning activities that MOOC learners engage in. Implications for some compelling future research are discussed.
In this work, we explore video lecture interaction in Massive Open Online Courses (MOOCs), which is central to student learning experience on these educational platforms. As a research contribution, we operationalize video lecture clickstreams of students into cognitively plausible higher level behaviors, and construct a quantitative information processing index, which can aid instructors to better understand MOOC hurdles and reason about unsatisfactory learning outcomes. Our results illustrate how such a metric inspired by cognitive psychology can help answer critical questions regarding students' engagement, their future click interactions and participation trajectories that lead to in-video & course dropouts. Implications for research and practice are discussed
A variation of Affleck-Dine mechanism was proposed to generate the observed baryon asymmetry in [1], in which the inflaton was assumed to be a complex scalar field with a weakly broken $U(1)$ symmetry, and the baryon asymmetry generation was easily unified with the stage of inflation and reheating. We adapt this mechanism to natural inflation scenarios and compare the results with those in chaotic inflation models. We compute the net particle number obtained at the end of inflation and transform it into net baryon number after reheatings. We observed that in natural inflation models, the desired baryon-to-photon ratio can be achieved equally well as in chaotic models.
Cutting a polytope is a very natural way to produce new classes of interesting polytopes. Moreover, it has been very enlightening to explore which algebraic and combinatorial properties of the orignial polytope are hereditary to its subpolytopes obtained by a cut. In this work, we put our attention to all the seperating hyperplanes for some given polytope (integral and convex) and study the existence and classification of such hyperplanes. We prove the exitence of seperating hyperplanes for the order and chain polytopes for any finite posets that are not a single chain; prove there are no such hyperplanes for any Birkhoff polytopes. Moreover, we give a complete seperating hyperplane classification for the unit cube and its subpolytopes obtained by one cut, together with some partial classification results for order and chain polytopes.
An inflationary model in the framework of noncommutative space-time may generate a nontrivial running of the scalar spectral index, but usually induces a large tensor-to-scalar ratio simultaneously. With the latest observational data from the Planck mission, we reexamine the inflationary scenarios in a noncommutative space-time. We find that either the running of the spectral index is tiny compared with the recent observational result, or the tensor-to-scalar ratio is too large to allow a sufficient number of $e$-folds. As examples, we show that the chaotic and power-law inflation models with the noncommutative effects are not favored by the current Planck data.
We study a certain family of discrete dynamical processes introduced by Novikoff, Kleinberg and Strogatz that we call flashcard games. We prove a number of results on the evolution of these games, an in particular we settle a conjecture of NKS on the frequency with which a given card appears. We introduce a number of generalizations and variations that we believe are of interest, and provide a large number of open questions and problems.