Asymptotic behavior of positively curved steady Ricci Solitons

In this paper, we analyze the asymptotic behavior of $\kappa$-noncollapsed and positively curved steady Ricci solitons and prove that any $n$-dimensional $\kappa$-noncollapsed steady K\"ahler-Ricci soliton with non-negative sectional curvature must be flat.Comment: This is a final version. We added some details in the proofs of Proposition 4.3 and Corollary 4.

Loss Tomography from Tree Topologies to General Topologies

Loss tomography has received considerable attention in recent years and a number of estimators based on maximum likelihood (ML) or Bayesian principles have been proposed. Almost all of the estimators are devoted to the tree topology despite the general topology is more common in practice. There has been few likelihood function devoted to the general topology, not to mention the estimator. To overcome this, two sets of sufficient statistics for the tree and general topologies, respectively, are proposed in this paper. Using the statistics, two likelihood functions, one for a topology, are proposed here and subsequently two likelihood equations for the general topology, one is link-based and the other is path-based, are obtained. In addition, a dependence between subtrees in terms of their estimates is identified for the general topology and a divide-and-conquer strategy is proposed to deal with the dependence, which divides a general network into two types of independent trees. Further, two algorithms, one for a type of the independent trees, are proposed to estimate the loss rates of each type.Comment: will be submitted for publication. arXiv admin note: substantial text overlap with arXiv:1009.255

Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature

In this paper, we give a delay estimate of scalar curvature for a complete non-compact expanding (or steady) gradient Ricci soliton with nonnegative Ricci curvature. As an application, we prove that any complete non-compact expanding (or steady) gradient K\"{a}hler-Ricci solitons with positively pinched Ricci curvature should be Ricci flat. The result answers a question in case of K\"{a}hler-Ricci solitons proposed by Chow, Lu and Ni in a book.Comment: 18 page

Classification of gradient steady Ricci solitons with linear curvature decay

In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional $\kappa$-noncollpased steady Ricci solitons with nonnegative sectional curvature under the linear curvature decay

A Variational Approach on Level sets and Linear Convergence of Variable Bregman Proximal Gradient Method for Nonconvex Optimization Problems

We develop a new variational approach on level sets aiming towards convergence rate analysis of a variable Bregman proximal gradient (VBPG) method for a broad class of nonsmooth and nonconvex optimization problems. With this new approach, we are able to extend the concepts of Bregman proximal mapping and their corresponding Bregman proximal envelops, Bregman proximal gap function to nonconvex setting. Properties of these mappings and functions are examined. An aim of this work is to provide a solid foundation on which further design and analysis of VBPG for more general nonconvex optimization problems are possible. Another aim is to provide a unified theory on linear convergence of VBPG with a particular interest towards proximal gradient methods. Centrol to our analysis for achieving the above goals is an error bound in terms of level sets and subdifferentials (level-set subdifferential error bound) along with its links to other level-set error bounds. As a consequence, we have established a number of positive results. These newly established results not only enable us to show that any accumulation of the sequence generated by VBPG is at least a critical point of the limiting subdifferential or even a critical point of the proximal subdifferential with a fixed Bregman function in each iteration, but also provide a fresh perspective that allows us to explore inner-connections among many known sufficient conditions for linear convergence of various first-order methods. Along the way, we are able to derive a number of verifiable conditions for level-set error bounds to hold, obtain linear convergence of VBPG, and derive necessary conditions and sufficient conditions for linear convergence relative to a level set for nonsmooth and nonconvex optimization problems

Higher dimensional steady Ricci solitons with linear curvature decay

We prove that any noncompact $\kappa$-noncollapsed steady Ricci soliton with nonnegative curvature operator must be rotationally symmetric if it has a linear curvature decay.Comment: We improve the main results in the previous version of paper without the assumption of positive Ricci curvatur

Steady Ricci solitons with horizontally ϵ\epsilon-pinched Ricci curvature

In this paper, we prove that any $\kappa$-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally $\epsilon$-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any $\kappa$-noncollapsed gradient steady Ricci soliton $(M^n, g,f)$ with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature $R(x)$ satisfies $\lim_{r(x)\rightarrow\infty}R(x)f(x)=C_0\sup_{x\in M}R(x)$ with $C_0>\frac{n-2}{2}$.Comment: Corollary 3.11 is adde

A note on compact κ\kappa-solutions of K\"{a}hler-Ricci flow

In this paper, we give a complete classification of $\kappa$-solutions of K\"{a}haler-Ricci flow on compact complex manifolds. Namely, they must be quotients of products of irreducible compact Hermitian symmetric manifolds.Comment: 7 page

Using Conditional Generative Adversarial Networks to Generate Ground-Level Views From Overhead Imagery

This paper develops a deep-learning framework to synthesize a ground-level view of a location given an overhead image. We propose a novel conditional generative adversarial network (cGAN) in which the trained generator generates realistic looking and representative ground-level images using overhead imagery as auxiliary information. The generator is an encoder-decoder network which allows us to compare low- and high-level features as well as their concatenation for encoding the overhead imagery. We also demonstrate how our framework can be used to perform land cover classification by modifying the trained cGAN to extract features from overhead imagery. This is interesting because, although we are using this modified cGAN as a feature extractor for overhead imagery, it incorporates knowledge of how locations look from the ground.Comment: 5 pages. arXiv admin note: text overlap with arXiv:1806.0512

Fine-Grained Land Use Classification at the City Scale Using Ground-Level Images

We perform fine-grained land use mapping at the city scale using ground-level images. Mapping land use is considerably more difficult than mapping land cover and is generally not possible using overhead imagery as it requires close-up views and seeing inside buildings. We postulate that the growing collections of georeferenced, ground-level images suggest an alternate approach to this geographic knowledge discovery problem. We develop a general framework that uses Flickr images to map 45 different land-use classes for the City of San Francisco. Individual images are classified using a novel convolutional neural network containing two streams, one for recognizing objects and another for recognizing scenes. This network is trained in an end-to-end manner directly on the labeled training images. We propose several strategies to overcome the noisiness of our user-generated data including search-based training set augmentation and online adaptive training. We derive a ground truth map of San Francisco in order to evaluate our method. We demonstrate the effectiveness of our approach through geo-visualization and quantitative analysis. Our framework achieves over 29% recall at the individual land parcel level which represents a strong baseline for the challenging 45-way land use classification problem especially given the noisiness of the image data