Hongwei Lin

Hongwei Lin, Yunyang Xiong, Xiao Wang et al.

In this paper, we present the isogeometric least-squares collocation (IGA-L) method, which determines the numerical solution by making the approximate differential operator fit the real differential operator in a least-squares sense. The number of collocation points employed in IGA-L can be larger than that of the unknowns. Theoretical analysis and numerical examples presented in this paper show the superiority of IGA-L over state-of-the-art collocation methods. First, a small increase in the number of collocation points in IGA-L leads to a large improvement in the accuracy of its numerical solution. Second, IGA-L method is more flexible and more stable, because the number of collocation points in IGA-L is variable. Third, IGA-L is convergent in some cases of singular parameterization. Moreover, the consistency and convergence analysis are also developed in this paper.

Ziang Cui, Mengran Yu, Tianjiao Li et al.

Large Language Models (LLMs) have achieved remarkable strides in multilingual translation but are hindered by a systemic cross-lingual verbosity bias, rendering them unsuitable for strict time-constrained tasks like subtitling and dubbing. Current prompt-engineering approaches struggle to resolve this conflict between semantic fidelity and rigid temporal feasibility. To bridge this gap, we first introduce Sand-Glass, a benchmark specifically designed to evaluate translation under syllable-level duration constraints. Furthermore, we propose HOMURA, a reinforcement learning framework that explicitly optimizes the trade-off between semantic preservation and temporal compliance. By employing a KL-regularized objective with a novel dynamic syllable-ratio reward, HOMURA effectively "tames" the output length. Experimental results demonstrate that our method significantly outperforms strong LLM baselines, achieving precise length control that respects linguistic density hierarchies without compromising semantic adequacy.

Chenkai Xu, Hongwei Lin

With the development of the Internet and other digital technologies, the speed of data generation has become considerably faster than the speed of data processing. Because big data typically contain massive redundant information, it is possible to significantly simplify a big data set while maintaining the key information it contains. In this paper, we develop a big data simplification method based on the eigenvalues and eigenfunctions of the Laplace-Beltrami operator (LBO). Specifically, given a data set that can be considered as an unorganized data point set in high-dimensional space, a discrete LBO defined on the big data set is constructed and its eigenvalues and eigenvectors are calculated. Then, the local extremum and the saddle points of the eigenfunctions are proposed to be the feature points of a data set in high-dimensional space, constituting a simplified data set. Moreover, we develop feature point detection methods for the functions defined on an unorganized data point set in high-dimensional space, and devise metrics for measuring the fidelity of the simplified data set to the original set. Finally, examples and applications are demonstrated to validate the efficiency and effectiveness of the proposed methods, demonstrating that data set simplification is a method for processing a maximum-sized data set using a limited data processing capability.

Zhetong Dong, Hongwei Lin, Chi Zhou

Many attempts have been made in recent decades to integrate machine learning (ML) and topological data analysis. A prominent problem in applying persistent homology to ML tasks is finding a vector representation of a persistence diagram (PD), which is a summary diagram for representing topological features. From the perspective of data fitting, a stable vector representation, namely, persistence B-spline grid (PBSG), is proposed based on the efficient technique of progressive-iterative approximation for least-squares B-spline function fitting. We theoretically prove that the PBSG method is stable with respect to the metric of 1-Wasserstein distance defined on the PD space. The proposed method was tested on a synthetic data set, data sets of randomly generated PDs, data of a dynamical system, and 3D CAD models, showing its effectiveness and efficiency

Hongwei Lin, Qi Cao, Xiaoting Zhang

Developed in [Deng and Lin, 2014], Least-Squares Progressive Iterative Approximation (LSPIA) is an efficient iterative method for solving B-spline curve and surface least-squares fitting systems. In [Deng and Lin 2014], it was shown that LSPIA is convergent when the iterative matrix is nonsingular. In this paper, we will show that LSPIA is still convergent even the iterative matrix is singular.