arXiv subjects

Xiaolong Li

Publications and source records attributed to Xiaolong Li.

At least 109 records · Page 6Linked to original sources

arXiv2023

Kähler manifolds and the curvature operator of the second kind

This article aims to investigate the curvature operator of the second kind on Kähler manifolds. The first result states that an $m$-dimensional Kähler manifold with $\frac{3}{2}(m^2-1)$-nonnegative (respectively, $\frac{3}{2}(m^2-1)$-nonpositive) curvature operator of the second kind must have constant nonnegative (respectively, nonpositive) holomorphic sectional curvature. The second result asserts that a closed $m$-dimensional Kähler manifold with $\left(\frac{3m^3-m+2}{2m}\right)$-positive curvature operator of the second kind has positive orthogonal bisectional curvature, thus being biholomorphic to $\mathbb{CP}^m$. We also prove that $\left(\frac{3m^3+2m^2-3m-2}{2m}\right)$-positive curvature operator of the second kind implies positive orthogonal Ricci curvature. Our approach is pointwise and algebraic.

math.DG↗

arXiv2023

Manifolds with nonnegative curvature operator of the second kind

We investigate the curvature operator of the second kind on Riemannian manifolds and prove several classification results. The first one asserts that a closed Riemannian manifold with three-positive curvature operator of the second kind is diffeomorphic to a spherical space form, improving a recent result of Cao-Gursky-Tran assuming two-positivity. The second one states that a closed Riemannian manifold with three-nonnegative curvature operator of the second kind is either diffeomorphic to a spherical space form, or flat, or isometric to a quotient of a compact irreducible symmetric space. This settles the nonnegativity part of Nishikawa's conjecture under a weaker assumption.

math.DG↗

arXiv2022

Number-Adaptive Prototype Learning for 3D Point Cloud Semantic Segmentation

3D point cloud semantic segmentation is one of the fundamental tasks for 3D scene understanding and has been widely used in the metaverse applications. Many recent 3D semantic segmentation methods learn a single prototype (classifier weights) for each semantic class, and classify 3D points according to their nearest prototype. However, learning only one prototype for each class limits the model's ability to describe the high variance patterns within a class. Instead of learning a single prototype for each class, in this paper, we propose to use an adaptive number of prototypes to dynamically describe the different point patterns within a semantic class. With the powerful capability of vision transformer, we design a Number-Adaptive Prototype Learning (NAPL) model for point cloud semantic segmentation. To train our NAPL model, we propose a simple yet effective prototype dropout training strategy, which enables our model to adaptively produce prototypes for each class. The experimental results on SemanticKITTI dataset demonstrate that our method achieves 2.3% mIoU improvement over the baseline model based on the point-wise classification paradigm.

cs.CV↗

arXiv2022

Tracking and Reconstructing Hand Object Interactions from Point Cloud Sequences in the Wild

In this work, we tackle the challenging task of jointly tracking hand object pose and reconstructing their shapes from depth point cloud sequences in the wild, given the initial poses at frame 0. We for the first time propose a point cloud based hand joint tracking network, HandTrackNet, to estimate the inter-frame hand joint motion. Our HandTrackNet proposes a novel hand pose canonicalization module to ease the tracking task, yielding accurate and robust hand joint tracking. Our pipeline then reconstructs the full hand via converting the predicted hand joints into a template-based parametric hand model MANO. For object tracking, we devise a simple yet effective module that estimates the object SDF from the first frame and performs optimization-based tracking. Finally, a joint optimization step is adopted to perform joint hand and object reasoning, which alleviates the occlusion-induced ambiguity and further refines the hand pose. During training, the whole pipeline only sees purely synthetic data, which are synthesized with sufficient variations and by depth simulation for the ease of generalization. The whole pipeline is pertinent to the generalization gaps and thus directly transferable to real in-the-wild data. We evaluate our method on two real hand object interaction datasets, e.g. HO3D and DexYCB, without any finetuning. Our experiments demonstrate that the proposed method significantly outperforms the previous state-of-the-art depth-based hand and object pose estimation and tracking methods, running at a frame rate of 9 FPS.

cs.CV↗

arXiv2022

Product manifolds and the curvature operator of the second kind

We investigate the curvature operator of the second kind on product Riemannian manifolds and obtain some optimal rigidity results. For instance, we prove that the universal cover of an $n$-dimensional non-flat complete locally reducible Riemannian manifold with $(n+\frac{n-2}{n})$-nonnegative (respectively, $(n+\frac{n-2}{n})$-nonpositive) curvature operator of the second kind must be isometric to $\mathbb{S}^{n-1}\times \mathbb{R}$ (respectively, $\mathbb{H}^{n-1}\times \mathbb{R}$) up to scaling. We also prove analogous optimal rigidity results for $\mathbb{S}^{n_1}\times \mathbb{S}^{n_2}$ and $\mathbb{H}^{n_1}\times \mathbb{H}^{n_2}$, $n_1,n_2 \geq 2$, among product Riemannian manifolds, as well as for $\mathbb{CP}^{m_1}\times \mathbb{CP}^{m_2}$ and $\mathbb{CH}^{m_1}\times \mathbb{CH}^{m_2}$, $m_1,m_2\geq 1$, among product Kähler manifolds. Our approach is pointwise and algebraic.

math.DG↗

arXiv2022

Robin heat kernel comparison on manifolds

We investigate the heat kernel with Robin boundary condition and prove comparison theorems for heat kernel on geodesic balls and on minimal submanifolds. We also prove an eigenvalue comparison theorem for the first Robin eigenvalues on minimal submanifolds. This generalizes corresponding results for the Dirichlet and Neumann heat kernels.

math.DG↗

arXiv2022

The second Robin eigenvalue in non-compact rank-1 symmetric spaces

In this paper, we prove a quantitative spectral inequality for the second Robin eigenvalue in non-compact rank-1 symmetric spaces. In particular, this shows that for bounded domains in non-compact rank-1 symmetric spaces, the geodesic ball maximises the second Robin eigenvalue among domains of the same volume, with negative Robin parameter in the regime connecting the first nontrivial Neumann and Steklov eigenvalues. This result generalises the work of Freitas and Laugesen in the Euclidean setting [FL21] as well as our previous work in the hyperbolic space [LWW20].

math.DG↗

arXiv2022

Manifolds with $4\frac{1}{2}$-positive curvature operator of the second kind

We show that a closed four-manifold with $4\frac{1}{2}$-positive curvature operator of the second kind is diffeomorphic to a spherical space form. The curvature assumption is sharp as both $\mathbb{CP}^2$ and $\mathbb{S}^3 \times \mathbb{S}^1$ have $4\frac{1}{2}$-nonnegative curvature operator of the second kind. In higher dimensions $n\geq 5$, we show that closed Riemannian manifolds with $4\frac{1}{2}$-positive curvature operator of the second kind are homeomorphic to spherical space forms. These results are proved by showing that $4\frac{1}{2}$-positive curvature operator of the second kind implies both positive isotropic curvature and positive Ricci curvature. Rigidity results for $4\frac{1}{2}$-nonnegative curvature operator of the second kind are also obtained.

math.DG↗

arXiv2022

Rubin Observatory LSST Transients and Variable Stars Roadmap

The Vera C. Rubin Legacy Survey of Space and Time holds the potential to revolutionize time domain astrophysics, reaching completely unexplored areas of the Universe and mapping variability time scales from minutes to a decade. To prepare to maximize the potential of the Rubin LSST data for the exploration of the transient and variable Universe, one of the four pillars of Rubin LSST science, the Transient and Variable Stars Science Collaboration, one of the eight Rubin LSST Science Collaborations, has identified research areas of interest and requirements, and paths to enable them. While our roadmap is ever-evolving, this document represents a snapshot of our plans and preparatory work in the final years and months leading up to the survey's first light.

astro-ph.IM↗

arXiv2022

Toward automated detection of light echoes in synoptic surveys: considerations on the application of the Deep Convolutional Neural Networks

Light Echoes (LEs) are the reflections of astrophysical transients off of interstellar dust. They are fascinating astronomical phenomena that enable studies of the scattering dust as well as of the original transients. LEs, however, are rare and extremely difficult to detect as they appear as faint, diffuse, time-evolving features. The detection of LEs still largely relies on human inspection of images, a method unfeasible in the era of large synoptic surveys. The Vera C. Rubin Observatory Legacy Survey of Space and Time, LSST, will generate an unprecedented amount of astronomical imaging data at high spatial resolution, exquisite image quality, and over tens of thousands of square degrees of sky: an ideal survey for LEs. However, the Rubin data processing pipelines are optimized for the detection of point-sources and will entirely miss LEs. Over the past several years, Artificial Intelligence (AI) object detection frameworks have achieved and surpassed real-time, human-level performance. In this work, we prepare a dataset from the ATLAS telescope and test a popular AI object detection framework, You Only Look Once, or YOLO, developed in the computer vision community, to demonstrate the potential of AI in the detection of LEs in astronomical images. We find that an AI framework can reach human-level performance even with a size- and quality-limited dataset. We explore and highlight challenges, including class imbalance and label incompleteness, and roadmap the work required to build an end-to-end pipeline for the automated detection and study of LEs in high-throughput astronomical surveys.

astro-ph.IM↗

arXiv2022

From Data to Software to Science with the Rubin Observatory LSST

The Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST) dataset will dramatically alter our understanding of the Universe, from the origins of the Solar System to the nature of dark matter and dark energy. Much of this research will depend on the existence of robust, tested, and scalable algorithms, software, and services. Identifying and developing such tools ahead of time has the potential to significantly accelerate the delivery of early science from LSST. Developing these collaboratively, and making them broadly available, can enable more inclusive and equitable collaboration on LSST science. To facilitate such opportunities, a community workshop entitled "From Data to Software to Science with the Rubin Observatory LSST" was organized by the LSST Interdisciplinary Network for Collaboration and Computing (LINCC) and partners, and held at the Flatiron Institute in New York, March 28-30th 2022. The workshop included over 50 in-person attendees invited from over 300 applications. It identified seven key software areas of need: (i) scalable cross-matching and distributed joining of catalogs, (ii) robust photometric redshift determination, (iii) software for determination of selection functions, (iv) frameworks for scalable time-series analyses, (v) services for image access and reprocessing at scale, (vi) object image access (cutouts) and analysis at scale, and (vii) scalable job execution systems. This white paper summarizes the discussions of this workshop. It considers the motivating science use cases, identified cross-cutting algorithms, software, and services, their high-level technical specifications, and the principles of inclusive collaborations needed to develop them. We provide it as a useful roadmap of needs, as well as to spur action and collaboration between groups and individuals looking to develop reusable software for early LSST science.

astro-ph.IM↗

arXiv2022

Kählerity of Einstein four-manifolds

We prove that a closed oriented Einstein four-manifold is either anti-self-dual or (after passing to a double Riemann cover if necessary) Kähler-Einstein, provided that $λ_2 \geq -\frac{S}{12}$, where $λ_2$ is the middle eigenvalue of the self-dual Weyl tensor $W^+$ and $S$ is the scalar curvature. The same conclusion holds for closed oriented four-manifolds with $δW^+=0$.

math.DG↗

arXiv2022

Abundance of observable Lyapunov irregular sets

Lyapunov exponent is widely used in natural science to find chaotic signal, but its existence is seldom discussed. In the present paper, we consider the problem of whether the set of points at which Lyapunov exponent fails to exist, called the Lyapunov irregular set, has positive Lebesgue measure. The only known example with the Lyapunov irregular set of positive Lebesgue measure is a figure-8 attractor by the work of Ott and Yorke [OY2008], whose key mechanism (homoclinic loop) is easy to be broken by small perturbations. In this paper, we show that surface diffeomorphisms with a robust homoclinic tangency given by Colli and Vargas [CV2001], as well as other several known nonhyperbolic dynamics, has the Lyapunov irregular set of positive Lebesgue measure. We can construct such positive Lebesgue measure sets both as the time averages exist and do not exist on it.

math.DS↗

arXiv2021

Leveraging SE(3) Equivariance for Self-Supervised Category-Level Object Pose Estimation

Category-level object pose estimation aims to find 6D object poses of previously unseen object instances from known categories without access to object CAD models. To reduce the huge amount of pose annotations needed for category-level learning, we propose for the first time a self-supervised learning framework to estimate category-level 6D object pose from single 3D point clouds.During training, our method assumes no ground-truth pose annotations, no CAD models, and no multi-view supervision. The key to our method is to disentangle shape and pose through an invariant shape reconstruction module and an equivariant pose estimation module, empowered by SE(3) equivariant point cloud networks.The invariant shape reconstruction module learns to perform aligned reconstructions, yielding a category-level reference frame without using any annotations. In addition, the equivariant pose estimation module achieves category-level pose estimation accuracy that is comparable to some fully supervised methods. Extensive experiments demonstrate the effectiveness of our approach on both complete and partial depth point clouds from the ModelNet40 benchmark, and on real depth point clouds from the NOCS-REAL 275 dataset. The project page with code and visualizations can be found at: https://dragonlong.github.io/equi-pose.

cs.CV↗

arXiv2021

R4: A Framework for Route Representation and Route Recommendation

Route recommendation is significant in navigation service. Two major challenges for route recommendation are route representation and user representation. Different from items that can be identified by unique IDs in traditional recommendation, routes are combinations of links (i.e., a road segment and its following action like turning left) and the number of combinations could be close to infinite. Besides, the representation of a route changes under different scenarios. These facts result in severe sparsity of routes, which increases the difficulty of route representation. Moreover, link attribute deficiencies and errors affect preciseness of route representation. Because of the sparsity of routes, the interaction data between users and routes are also sparse. This makes it not easy to acquire user representation from historical user-item interactions as traditional recommendations do. To address these issues, we propose a novel learning framework R4. In R4, we design a sparse & dense network to obtain representations of routes. The sparse unit learns link ID embeddings and aggregates them to represent a route, which captures implicit route characteristics and subsequently alleviates problems caused by link attribute deficiencies and errors. The dense unit extracts implicit local features of routes from link attributes. For user representation, we utilize a series of historical navigation to extract user preference. R4 achieves remarkable performance in both offline and online experiments.

cs.AI↗

arXiv2021

Preparing to discover the unknown with Rubin LSST -- I: Time domain

Perhaps the most exciting promise of the Rubin Observatory Legacy Survey of Space and Time (LSST) is its capability to discover phenomena never before seen or predicted from theory: true astrophysical novelties, but the ability of LSST to make these discoveries will depend on the survey strategy. Evaluating candidate strategies for true novelties is a challenge both practically and conceptually: unlike traditional astrophysical tracers like supernovae or exoplanets, for anomalous objects the template signal is by definition unknown. We present our approach to solve this problem, by assessing survey completeness in a phase space defined by object color, flux (and their evolution), and considering the volume explored by integrating metrics within this space with the observation depth, survey footprint, and stellar density. With these metrics, we explore recent simulations of the Rubin LSST observing strategy across the entire observed footprint and in specific regions in the Local Volume: the Galactic Plane and Magellanic Clouds. Under our metrics, observing strategies with greater diversity of exposures and time gaps tend to be more sensitive to genuinely new phenomena, particularly over time-gap ranges left relatively unexplored by previous surveys. To assist the community, we have made all the tools developed publicly available. Extension of the scheme to include proper motions and the detection of associations or populations of interest, will be communicated in paper II of this series. This paper was written with the support of the Vera C. Rubin LSST Transients and Variable Stars and Stars, Milky Way, Local Volume Science Collaborations.

astro-ph.IM↗

arXiv2021

Denoising User-aware Memory Network for Recommendation

For better user satisfaction and business effectiveness, more and more attention has been paid to the sequence-based recommendation system, which is used to infer the evolution of users' dynamic preferences, and recent studies have noticed that the evolution of users' preferences can be better understood from the implicit and explicit feedback sequences. However, most of the existing recommendation techniques do not consider the noise contained in implicit feedback, which will lead to the biased representation of user interest and a suboptimal recommendation performance. Meanwhile, the existing methods utilize item sequence for capturing the evolution of user interest. The performance of these methods is limited by the length of the sequence, and can not effectively model the long-term interest in a long period of time. Based on this observation, we propose a novel CTR model named denoising user-aware memory network (DUMN). Specifically, the framework: (i) proposes a feature purification module based on orthogonal mapping, which use the representation of explicit feedback to purify the representation of implicit feedback, and effectively denoise the implicit feedback; (ii) designs a user memory network to model the long-term interests in a fine-grained way by improving the memory network, which is ignored by the existing methods; and (iii) develops a preference-aware interactive representation component to fuse the long-term and short-term interests of users based on gating to understand the evolution of unbiased preferences of users. Extensive experiments on two real e-commerce user behavior datasets show that DUMN has a significant improvement over the state-of-the-art baselines. The code of DUMN model has been uploaded as an additional material.

cs.IR↗

arXiv2021

Super exponential divergence of periodic points for C^1-generic partially hyperbolic homoclinic classes

A diffeomorphism f is called super exponential divergent if for every r>1, the lower limit of #Per_n(f)/r^n diverges to infinity as n tends to infinity, where Per_n(f) is the set of all periodic points of f with period n. This property is stronger than the usual super exponential growth of the number of periodic points. We show that for a three dimensional manifold M, there exists an open subset O of Diff^1(M) such that diffeomorphisms with super exponential divergent property form a dense subset of O in the C^1-topology. A relevant result of non super exponential divergence for diffeomorphisms in a locally generic subset of Diff^r(M) (r=1,2,...\infty) is also shown.

math.DS↗