arXiv subjects

Kuan Xu

Publications and source records attributed to Kuan Xu.

35 records · Page 2Linked to original sources

arXiv2025

A new banded Petrov--Galerkin spectral method

We propose a Petrov--Galerkin spectral method for ODEs with variable coefficients. When the variable coefficients are smooth, the new method yields a strictly banded linear system, which can be efficiently constructed and solved in linear complexity. The performance advantage of our method is demonstrated through benchmarking against Mortensen's Galerkin method and the ultraspherical spectral method. Furthermore, we introduce a systematic approach for designing the recombined basis and establish that our new method serves as a unifying framework that encompasses all existing banded Galerkin spectral methods. This significantly addresses the ongoing challenge of developing recombined bases and sparse Galerkin spectral method. Additionally, the accelerating techniques presented in this paper can also enhance the performance of the ultraspherical spectral method.

math.NA↗

arXiv2025

ChorusCVR: Chorus Supervision for Entire Space Post-Click Conversion Rate Modeling

Post-click conversion rate (CVR) estimation is a vital task in many recommender systems of revenue businesses, e.g., e-commerce and advertising. In a perspective of sample, a typical CVR positive sample usually goes through a funnel of exposure to click to conversion. For lack of post-event labels for un-clicked samples, CVR learning task commonly only utilizes clicked samples, rather than all exposed samples as for click-through rate (CTR) learning task. However, during online inference, CVR and CTR are estimated on the same assumed exposure space, which leads to a inconsistency of sample space between training and inference, i.e., sample selection bias (SSB). To alleviate SSB, previous wisdom proposes to design novel auxiliary tasks to enable the CVR learning on un-click training samples, such as CTCVR and counterfactual CVR, etc. Although alleviating SSB to some extent, none of them pay attention to the discrimination between ambiguous negative samples (un-clicked) and factual negative samples (clicked but un-converted) during modelling, which makes CVR model lacks robustness. To full this gap, we propose a novel ChorusCVR model to realize debiased CVR learning in entire-space.

cs.IR↗

arXiv2024

Understanding the ultraspherical spectral method

The ultraspherical spectral method features high accuracy and fast solution. In this article, we determine the sources of error arising from the ultraspherical spectral method and derive its effective condition number, which explains why its backward error is consistent with a numerical method with bounded condition number. In addition, we show the cause for the Cauchy error to go below the machine epsilon and decay eventually to exact zero, revealing the fact that the Cauchy error can be misleading when used as an indicator of convergence and accuracy. The analysis in this work can be readily extended to other spectral methods, when applicable, and to the solution of PDEs.

math.NA↗

arXiv2024

Spectral approximation of convolution operators of Fredholm type

We have developed a method for constructing spectral approximations for convolution operators of Fredholm type. The algorithm we propose is numerically stable and takes advantage of the recurrence relations satisfied by the entries of such a matrix approximation. When used for computing the Fredholm convolution of two given functions, such approximations produce the convolution more rapidly than the state-of-the-art methods. The proposed approximation also leads to a spectral method for solving the Fredholm convolution integral equations and enables the computation of eigenvalues and pseudospectra of Fredholm convolution operators, which is otherwise intractable with existing techniques.

math.NA↗

arXiv2023

Enhancing Column Generation by Reinforcement Learning-Based Hyper-Heuristic for Vehicle Routing and Scheduling Problems

Column generation (CG) is a vital method to solve large-scale problems by dynamically generating variables. It has extensive applications in common combinatorial optimization, such as vehicle routing and scheduling problems, where each iteration step requires solving an NP-hard constrained shortest path problem. Although some heuristic methods for acceleration already exist, they are not versatile enough to solve different problems. In this work, we propose a reinforcement learning-based hyper-heuristic framework, dubbed RLHH, to enhance the performance of CG. RLHH is a selection module embedded in CG to accelerate convergence and get better integer solutions. In each CG iteration, the RL agent selects a low-level heuristic to construct a reduced network only containing the edges with a greater chance of being part of the optimal solution. In addition, we specify RLHH to solve two typical combinatorial optimization problems: Vehicle Routing Problem with Time Windows (VRPTW) and Bus Driver Scheduling Problem (BDSP). The total cost can be reduced by up to 27.9\% in VRPTW and 15.4\% in BDSP compared to the best lower-level heuristic in our tested scenarios, within equivalent or even less computational time. The proposed RLHH is the first RL-based CG method that outperforms traditional approaches in terms of solution quality, which can promote the application of CG in combinatorial optimization.

cs.LG↗

arXiv2023

PyPose v0.6: The Imperative Programming Interface for Robotics

PyPose is an open-source library for robot learning. It combines a learning-based approach with physics-based optimization, which enables seamless end-to-end robot learning. It has been used in many tasks due to its meticulously designed application programming interface (API) and efficient implementation. From its initial launch in early 2022, PyPose has experienced significant enhancements, incorporating a wide variety of new features into its platform. To satisfy the growing demand for understanding and utilizing the library and reduce the learning curve of new users, we present the fundamental design principle of the imperative programming interface, and showcase the flexible usage of diverse functionalities and modules using an extremely simple Dubins car example. We also demonstrate that the PyPose can be easily used to navigate a real quadruped robot with a few lines of code.

cs.RO↗

arXiv2023

AirVO: An Illumination-Robust Point-Line Visual Odometry

This paper proposes an illumination-robust visual odometry (VO) system that incorporates both accelerated learning-based corner point algorithms and an extended line feature algorithm. To be robust to dynamic illumination, the proposed system employs the convolutional neural network (CNN) and graph neural network (GNN) to detect and match reliable and informative corner points. Then point feature matching results and the distribution of point and line features are utilized to match and triangulate lines. By accelerating CNN and GNN parts and optimizing the pipeline, the proposed system is able to run in real-time on low-power embedded platforms. The proposed VO was evaluated on several datasets with varying illumination conditions, and the results show that it outperforms other state-of-the-art VO systems in terms of accuracy and robustness. The open-source nature of the proposed system allows for easy implementation and customization by the research community, enabling further development and improvement of VO for various applications.

cs.RO↗

arXiv2023

Solving nonlinear ODEs with the ultraspherical spectral method

We extend the ultraspherical spectral method to solving nonlinear ODE boundary value problems. We propose to use the inexact Newton-GMRES framework for which an effective preconditioner can be constructed and a fast Jacobian-vector multiplication can be effected, thanks to the structured operators of the ultraspherical spectral method. With a mixed-precision implementation, the inexact Newton-GMRES-ultraspherical framework exhibits extraordinary speed and accuracy, as we show by extensive numerical experiments.

math.NA↗

arXiv2023

Solving time-dependent PDEs with the ultraspherical spectral method

We apply the ultraspherical spectral method to solving time-dependent PDEs by proposing two approaches to discretization based on the method of lines and show that these approaches produce approximately same results. We analyze the stability, the error, and the computational cost of the proposed method. In addition, we show how adaptivity can be incorporated to offer adequate spatial resolution efficiently. Both linear and nonlinear problems are considered. We also explore time integration using exponential integrators with the ultraspherical spatial discretization. Comparisons with the Chebyshev pseudospectral method are given along the discussion and they show that the ultraspherical spectral method is a competitive candidate for the spatial discretization of time-dependent PDEs.

math.NA↗

arXiv2023

PyPose: A Library for Robot Learning with Physics-based Optimization

Deep learning has had remarkable success in robotic perception, but its data-centric nature suffers when it comes to generalizing to ever-changing environments. By contrast, physics-based optimization generalizes better, but it does not perform as well in complicated tasks due to the lack of high-level semantic information and reliance on manual parametric tuning. To take advantage of these two complementary worlds, we present PyPose: a robotics-oriented, PyTorch-based library that combines deep perceptual models with physics-based optimization. PyPose's architecture is tidy and well-organized, it has an imperative style interface and is efficient and user-friendly, making it easy to integrate into real-world robotic applications. Besides, it supports parallel computing of any order gradients of Lie groups and Lie algebras and $2^{\text{nd}}$-order optimizers, such as trust region methods. Experiments show that PyPose achieves more than $10\times$ speedup in computation compared to the state-of-the-art libraries. To boost future research, we provide concrete examples for several fields of robot learning, including SLAM, planning, control, and inertial navigation.

cs.RO↗

arXiv2023

A Pre-training Framework for Knowledge Graph Completion

Knowledge graph completion (KGC) is one of the effective methods to identify new facts in knowledge graph. Except for a few methods based on graph network, most of KGC methods trend to be trained based on independent triples, while are difficult to take a full account of the information of global network connection contained in knowledge network. To address these issues, in this study, we propose a simple and effective Network-based Pre-training framework for knowledge graph completion (termed NetPeace), which takes into account the information of global network connection and local triple relationships in knowledge graph. Experiments show that in NetPeace framework, multiple KGC models yields consistent and significant improvements on benchmarks (e.g., 36.45% Hits@1 and 27.40% MRR improvements for TuckER on FB15k-237), especially dense knowledge graph. On the challenging low-resource task, NetPeace that benefits from the global features of KG achieves higher performance (104.03% MRR and 143.89% Hit@1 improvements at most) than original models.

cs.AI↗

arXiv2023

Knowledge Graph Completion based on Tensor Decomposition for Disease Gene Prediction

Accurate identification of disease genes has consistently been one of the keys to decoding a disease's molecular mechanism. Most current approaches focus on constructing biological networks and utilizing machine learning, especially, deep learning to identify disease genes, but ignore the complex relations between entities in the biological knowledge graph. In this paper, we construct a biological knowledge graph centered on diseases and genes, and develop an end-to-end Knowledge graph completion model for Disease Gene Prediction using interactional tensor decomposition (called KDGene). KDGene introduces an interaction module between the embeddings of entities and relations to tensor decomposition, which can effectively enhance the information interaction in biological knowledge. Experimental results show that KDGene significantly outperforms state-of-the-art algorithms. Furthermore, the comprehensive biological analysis of the case of diabetes mellitus confirms KDGene's ability for identifying new and accurate candidate genes. This work proposes a scalable knowledge graph completion framework to identify disease candidate genes, from which the results are promising to provide valuable references for further wet experiments.

cs.AI↗

arXiv2023

SeaD: End-to-end Text-to-SQL Generation with Schema-aware Denoising

In text-to-SQL task, seq-to-seq models often lead to sub-optimal performance due to limitations in their architecture. In this paper, we present a simple yet effective approach that adapts transformer-based seq-to-seq model to robust text-to-SQL generation. Instead of inducing constraint to decoder or reformat the task as slot-filling, we propose to train seq-to-seq model with Schema aware Denoising (SeaD), which consists of two denoising objectives that train model to either recover input or predict output from two novel erosion and shuffle noises. These denoising objectives acts as the auxiliary tasks for better modeling the structural data in S2S generation. In addition, we improve and propose a clause-sensitive execution guided (EG) decoding strategy to overcome the limitation of EG decoding for generative model. The experiments show that the proposed method improves the performance of seq-to-seq model in both schema linking and grammar correctness and establishes new state-of-the-art on WikiSQL benchmark. The results indicate that the capacity of vanilla seq-to-seq architecture for text-to-SQL may have been under-estimated.

cs.CL↗

arXiv2022

AirObject: A Temporally Evolving Graph Embedding for Object Identification

Object encoding and identification are vital for robotic tasks such as autonomous exploration, semantic scene understanding, and re-localization. Previous approaches have attempted to either track objects or generate descriptors for object identification. However, such systems are limited to a "fixed" partial object representation from a single viewpoint. In a robot exploration setup, there is a requirement for a temporally "evolving" global object representation built as the robot observes the object from multiple viewpoints. Furthermore, given the vast distribution of unknown novel objects in the real world, the object identification process must be class-agnostic. In this context, we propose a novel temporal 3D object encoding approach, dubbed AirObject, to obtain global keypoint graph-based embeddings of objects. Specifically, the global 3D object embeddings are generated using a temporal convolutional network across structural information of multiple frames obtained from a graph attention-based encoding method. We demonstrate that AirObject achieves the state-of-the-art performance for video object identification and is robust to severe occlusion, perceptual aliasing, viewpoint shift, deformation, and scale transform, outperforming the state-of-the-art single-frame and sequential descriptors. To the best of our knowledge, AirObject is one of the first temporal object encoding methods. Source code is available at https://github.com/Nik-V9/AirObject.

cs.CV↗

arXiv2022

AirCode: A Robust Object Encoding Method

Object encoding and identification are crucial for many robotic tasks such as autonomous exploration and semantic relocalization. Existing works heavily rely on the tracking of detected objects but have difficulty recalling revisited objects precisely. In this paper, we propose a novel object encoding method, which is named as AirCode, based on a graph of key-points. To be robust to the number of key-points detected, we propose a feature sparse encoding and object dense encoding method to ensure that each key-point can only affect a small part of the object descriptors, leading it to be robust to viewpoint changes, scaling, occlusion, and even object deformation. In the experiments, we show that it achieves superior performance for object identification than the state-of-the-art algorithms and is able to provide reliable semantic relocalization. It is a plug-and-play module and we expect that it will play an important role in various applications.

cs.RO↗

arXiv2018

Volterra-type convolution of classical polynomials

We present a general framework for calculating the Volterra-type convolution of polynomials from an arbitrary polynomial sequence $\{P_k(x)\}_{k \geqslant 0}$ with $°P_k(x) = k$. Based on this framework, series representations for the convolutions of classical orthogonal polynomials, including Jacobi and Laguerre families, are derived, along with some relevant results pertaining to these new formulas.

math.CA↗

arXiv2018

Spectral approximation of convolution operator

We develop a unified framework for constructing matrix approximations to the convolution operator of Volterra type defined by functions that are approximated using classical orthogonal polynomials on $[-1, 1]$. The numerically stable algorithms we propose exploit recurrence relations and symmetric properties satisfied by the entries of these convolution matrices. Laguerre-based convolution matrices that approximate Volterra convolution operator defined by functions on $[0, \infty]$ are also discussed for the sake of completeness.

math.NA↗