arXiv subjects

Hu Ding

Publications and source records attributed to Hu Ding.

At least 19 recordsLinked to original sources

arXiv2026

Uniform Capacitated Clustering with Few Extra Centers

In uniform capacitated $(k,z)$-clustering, we open $k$ centers and assign every client to a center so that no center serves more than $U$ clients, minimizing the sum of the $z$-th powers of the assignment distances; $k$-median and $k$-means are the cases $z=1$ and $z=2$. We study algorithms that may open $k+\lfloor\varepsilon k\rfloor$ centers for a fixed $\varepsilon\in(0,1]$ and must respect the capacities exactly. In low dimensions, we give deterministic algorithms that are, except for $k$-median in the plane, the first polynomial-time algorithms with $(1+\varepsilon)k$ centers whose ratio is bounded independently of $\varepsilon$. In Euclidean space of any dimension, with clients of bounded doubling dimension, the ratio is $\mathcal{B}_z-(\mathcal{B}_z-1)\varepsilon+δ$ for every fixed $δ>0$, where $\mathcal{B}_z<3^z$ for $z>1$; this is $3-2\varepsilon+δ$ for $k$-median and $5-4\varepsilon+δ$ for $k$-means. In doubling metrics, the ratio is $3^z-(3^z-1)\varepsilon+δ$ when centers may share a location, and a constant otherwise, such as $\min\{7-4\varepsilon,\,3+2\ln(1/\varepsilon)\}+δ$ for $k$-median. In general metrics, deleting centers greedily gives ratio $3+2\ln(1/\varepsilon)+δ$ for $k$-median when centers may share a location and $7+2\ln(1/\varepsilon)+δ$ otherwise, improving the previous $O(\varepsilon^{-2}\log(1/\varepsilon))$. Our main tool is a rescue lemma. Splitting a near-optimal uncapacitated solution gives at most $2k-1$ blocks of at most $U$ clients, and giving a block a private center saves its share of the extra cost of following an optimal solution. Since a center of the optimal solution can be closed once all its blocks are rescued, $\lfloor\varepsilon k\rfloor$ extra centers save almost an $\varepsilon$-fraction of this extra cost instead of an $\varepsilon/2$-fraction.

cs.CG↗

arXiv2026

A Tight 3-Approximation for Fair $k$-Means

Fairness in clustering has attracted sustained research interest, motivated by the need to ensure equitable representation of protected groups in machine learning applications. We study fair $k$-means clustering in Euclidean space, where the proportion of each protected group in every cluster must lie within specified lower and upper bounds. These constraints make it challenging to determine both cluster centers and point assignments. Our algorithm first splits the input into many small fair clusters and then merges them into $k$ clusters with a weighted $k$-means algorithm. The merge is cheap only when the small clusters do not move points away from their optimal centers, so we add a displacement penalty to the split, which charges each point for how far the small clusters move it. Given a $ρ$-approximate algorithm for weighted $k$-means and any $ε>0$, our algorithm returns a fractional solution whose cost is at most $3ρ+ε$ times the optimal integral fair cost. With a PTAS as the subroutine, the approximation ratio becomes $3+O(ε)$, improving the previous factor of $5+O(ε)$. This ratio is the best possible: unless P=NP, no polynomial-time algorithm achieves the ratio $3-δ$ for any fixed $δ>0$, even in one dimension. The solution satisfies all fairness constraints exactly and can be rounded to an integral assignment with a bounded additive violation of fairness and no increase in cost. The same approximation guarantee extends to the $k$-sparse Wasserstein barycenter problem.

cs.CG↗

arXiv2026

The Stretch Factor of Planar Delaunay Triangulations Is Less Than 1.5949

Delaunay triangulations are the classical plane spanners: the shortest path between two vertices is at most a constant times as long as the segment between them. Determining the smallest such constant, the worst-case stretch factor of Delaunay triangulations, is a long-standing open problem; the best known lower and upper bounds were 1.5932, due to Xia and Zhang, and 1.998, due to Xia. We focus on the upper bound. Xia's analysis replaces the triangulation by a chain of disks and compares the length along the whole chain with the straight-line distance at once, by induction with a potential function. We propose a new framework, progress accounting, which compares the contribution of each single disk to the chain length with the progress of that disk along the straight line. This turns the global analysis into a local one, which can be made very precise. The key to the local analysis is our pencil lemma: it shows that the comparison for each disk needs only four angles, whatever the rest of the chain looks like. The resulting conditions on one function of four angles are verified by a computer-assisted proof in exact rational and interval arithmetic; corner lemmas handle the degenerate configurations where the conditions are tight. As a result, every Delaunay triangulation of a finite planar point set has stretch factor less than 1.5949. The gap to the best known lower bound shrinks from about 0.405 to less than 0.0017, by a factor of more than 200. This also shows that every finite planar point set has a plane spanner with stretch factor less than 1.5949, improving the previous bound of 1.998.

cs.CG↗

arXiv2026

The Alignment Illusion in Multimodal Large Language Models

Layer-wise visual-text similarity in Multimodal Large Language Models (MLLMs) is widely interpreted as evidence that the language model progressively integrates visual content into a shared representation space. This reading rests on the assumption that scalar alignment scores reflect content-level cross-modal interaction. To test this assumption, we apply controlled interventions to the visual stream. Across 13 MLLMs from five families spanning 0.5B to 72B parameters, replacing projector-output visual tokens with Gaussian noise sharply reduces task accuracy, yet four standard scalar measures (CKA, SVCCA, MIR, and the leading principal-angle cosine) fail to consistently separate the corrupted stream from the original. We call this failure the alignment illusion and trace it to the shared language-model pathway: anisotropic MLP down-projections pull visual and text tokens toward common output directions, producing weight-induced alignment. Because this component is essentially one-dimensional, we introduce the principal-angle gap (PA gap), defined as the difference between the top two principal-angle cosines, which separates weight-induced similarity from multi-directional visual structure. Under graded visual corruption, the PA gap tracks task accuracy more consistently than the scalar scores we consider; under a structured but irrelevant image, it further exposes regimes in which internal geometry and task accuracy come apart. Internal visual-text alignment in MLLMs is therefore best read as a geometric diagnostic of the visual stream inside the language model rather than a direct proxy for content-level cross-modal interaction, and is most informative when calibrated by controlled task evidence.

cs.CV↗

arXiv2026

MIRROR: Aligning Semantic Relations from Language to Image via Gromov--Wasserstein

Multimodal Large Language Models (MLLMs) inherit rich relational priors from their language backbones, yet often fail when asked to apply these relationships in visual contexts. We trace this failure to a structural blind spot: projection-based alignment trains each visual token to carry the right semantics, but never asks whether the relationships between concepts survive the crossing from language to vision. To address this, we propose MIRROR (Mapping Inter-concept Relations from language to visual Representation via Optimal-transport-based Regularization), a geometric regularization framework that transfers relational priors from language to vision by exploiting the rich relational structure encoded in language representations. Specifically, we derive a surrogate loss from the proposed Semi-Inverse Gromov-Wasserstein (SI-GW) problem, an inverse geometric problem that aligns visual representations with language-derived relational priors. We show that this formulation admits a unique closed-form solution that prescribes the ideal visual relational structure implied by language geometry and cross-modal coupling. The structure of the formulation also enables efficient computation, making it applicable to long token sequences. Applying SI-GW inside decoder-only Transformers requires careful design. We introduce targeted strategies at the layer, head, and token levels to ensure stable extraction without additional parameters or inference cost. MIRROR improves relational consistency while preserving performance on general vision-language tasks.

cs.CV↗

arXiv2026

Sample-and-Search: An Effective Algorithm for Learning-Augmented k-Median Clustering in High dimensions

In this paper, we investigate the learning-augmented $k$-median clustering problem, which aims to improve the performance of traditional clustering algorithms by preprocessing the point set with a predictor of error rate $α\in [0,1)$. This preprocessing step assigns potential labels to the points before clustering. We introduce an algorithm for this problem based on a simple yet effective sampling method, which substantially improves upon the time complexities of existing algorithms. Moreover, we mitigate their exponential dependency on the dimensionality of the Euclidean space. Lastly, we conduct experiments to compare our method with several state-of-the-art learning-augmented $k$-median clustering methods. The experimental results suggest that our proposed approach can significantly reduce the computational complexity in practice, while achieving a lower clustering cost.

cs.DS↗

arXiv2025

Survey on Recent Progress of AI for Chemistry: Methods, Applications, and Opportunities

The development of artificial intelligence (AI) techniques has brought revolutionary changes across various realms. In particular, the use of AI-assisted methods to accelerate chemical research has become a popular and rapidly growing trend, leading to numerous groundbreaking works. In this paper, we provide a comprehensive review of current AI techniques in chemistry from a computational perspective, considering various aspects in the design of methods. We begin by discussing the characteristics of data from diverse sources, followed by an overview of various representation methods. Next, we review existing models for several topical tasks in the field, and conclude by highlighting some key challenges that warrant further attention.

physics.chem-ph↗

arXiv2025

Towards Metric DBSCAN: Exact, Approximate, and Streaming Algorithms

DBSCAN is a popular density-based clustering algorithm that has many different applications in practice. However, the running time of DBSCAN in high-dimensional space or general metric space ({\em e.g.,} clustering a set of texts by using edit distance) can be as large as quadratic in the input size. Moreover, most of existing accelerating techniques for DBSCAN are only available for low-dimensional Euclidean space. In this paper, we study the DBSCAN problem under the assumption that the inliers (the core points and border points) have a low intrinsic dimension (which is a realistic assumption for many high-dimensional applications), where the outliers can locate anywhere in the space without any assumption. First, we propose a $k$-center clustering based algorithm that can reduce the time-consuming labeling and merging tasks of DBSCAN to be linear. Further, we propose a linear time approximate DBSCAN algorithm, where the key idea is building a novel small-size summary for the core points. Also, our algorithm can be efficiently implemented for streaming data and the required memory is independent of the input size. Finally, we conduct our experiments and compare our algorithms with several popular DBSCAN algorithms. The experimental results suggest that our proposed approach can significantly reduce the computational complexity in practice.

cs.DS↗

arXiv2025

Uncertainty-Aware Label Refinement on Hypergraphs for Personalized Federated Facial Expression Recognition

Most facial expression recognition (FER) models are trained on large-scale expression data with centralized learning. Unfortunately, collecting a large amount of centralized expression data is difficult in practice due to privacy concerns of facial images. In this paper, we investigate FER under the framework of personalized federated learning, which is a valuable and practical decentralized setting for real-world applications. To this end, we develop a novel uncertainty-Aware label refineMent on hYpergraphs (AMY) method. For local training, each local model consists of a backbone, an uncertainty estimation (UE) block, and an expression classification (EC) block. In the UE block, we leverage a hypergraph to model complex high-order relationships between expression samples and incorporate these relationships into uncertainty features. A personalized uncertainty estimator is then introduced to estimate reliable uncertainty weights of samples in the local client. In the EC block, we perform label propagation on the hypergraph, obtaining high-quality refined labels for retraining an expression classifier. Based on the above, we effectively alleviate heterogeneous sample uncertainty across clients and learn a robust personalized FER model in each client. Experimental results on two challenging real-world facial expression databases show that our proposed method consistently outperforms several state-of-the-art methods. This indicates the superiority of hypergraph modeling for uncertainty estimation and label refinement on the personalized federated FER task. The source code will be released at https://github.com/mobei1006/AMY.

cs.CV↗

arXiv2024

Relax and Merge: A Simple Yet Effective Framework for Solving Fair $k$-Means and $k$-sparse Wasserstein Barycenter Problems

The fairness of clustering algorithms has gained widespread attention across various areas, including machine learning, In this paper, we study fair $k$-means clustering in Euclidean space. Given a dataset comprising several groups, the fairness constraint requires that each cluster should contain a proportion of points from each group within specified lower and upper bounds. Due to these fairness constraints, determining the optimal locations of $k$ centers is a quite challenging task. We propose a novel ``Relax and Merge'' framework that returns a $(1+4ρ+ O(ε))$-approximate solution, where $ρ$ is the approximate ratio of an off-the-shelf vanilla $k$-means algorithm and $O(ε)$ can be an arbitrarily small positive number. If equipped with a PTAS of $k$-means, our solution can achieve an approximation ratio of $(5+O(ε))$ with only a slight violation of the fairness constraints, which improves the current state-of-the-art approximation guarantee. Furthermore, using our framework, we can also obtain a $(1+4ρ+O(ε))$-approximate solution for the $k$-sparse Wasserstein Barycenter problem, which is a fundamental optimization problem in the field of optimal transport, and a $(2+6ρ)$-approximate solution for the strictly fair $k$-means clustering with no violation, both of which are better than the current state-of-the-art methods. In addition, the empirical results demonstrate that our proposed algorithm can significantly outperform baseline approaches in terms of clustering cost.

cs.LG↗

arXiv2024

An Effective Dynamic Gradient Calibration Method for Continual Learning

Continual learning (CL) is a fundamental topic in machine learning, where the goal is to train a model with continuously incoming data and tasks. Due to the memory limit, we cannot store all the historical data, and therefore confront the ``catastrophic forgetting'' problem, i.e., the performance on the previous tasks can substantially decrease because of the missing information in the latter period. Though a number of elegant methods have been proposed, the catastrophic forgetting phenomenon still cannot be well avoided in practice. In this paper, we study the problem from the gradient perspective, where our aim is to develop an effective algorithm to calibrate the gradient in each updating step of the model; namely, our goal is to guide the model to be updated in the right direction under the situation that a large amount of historical data are unavailable. Our idea is partly inspired by the seminal stochastic variance reduction methods (e.g., SVRG and SAGA) for reducing the variance of gradient estimation in stochastic gradient descent algorithms. Another benefit is that our approach can be used as a general tool, which is able to be incorporated with several existing popular CL methods to achieve better performance. We also conduct a set of experiments on several benchmark datasets to evaluate the performance in practice.

cs.LG↗

arXiv2024

Approximate Algorithms For $k$-Sparse Wasserstein Barycenter With Outliers

Wasserstein Barycenter (WB) is one of the most fundamental optimization problems in optimal transportation. Given a set of distributions, the goal of WB is to find a new distribution that minimizes the average Wasserstein distance to them. The problem becomes even harder if we restrict the solution to be ``$k$-sparse''. In this paper, we study the $k$-sparse WB problem in the presence of outliers, which is a more practical setting since real-world data often contains noise. Existing WB algorithms cannot be directly extended to handle the case with outliers, and thus it is urgently needed to develop some novel ideas. First, we investigate the relation between $k$-sparse WB with outliers and the clustering (with outliers) problems. In particular, we propose a clustering based LP method that yields constant approximation factor for the $k$-sparse WB with outliers problem. Further, we utilize the coreset technique to achieve the $(1+ε)$-approximation factor for any $ε>0$, if the dimensionality is not high. Finally, we conduct the experiments for our proposed algorithms and illustrate their efficiencies in practice.

cs.LG↗

arXiv2024

A Novel Skip Orthogonal List for Dynamic Optimal Transport Problem

Optimal transport is a fundamental topic that has attracted a great amount of attention from the optimization community in the past decades. In this paper, we consider an interesting discrete dynamic optimal transport problem: can we efficiently update the optimal transport plan when the weights or the locations of the data points change? This problem is naturally motivated by several applications in machine learning. For example, we often need to compute the optimal transport cost between two different data sets; if some changes happen to a few data points, should we re-compute the high complexity cost function or update the cost by some efficient dynamic data structure? We are aware that several dynamic maximum flow algorithms have been proposed before, however, the research on dynamic minimum cost flow problem is still quite limited, to the best of our knowledge. We propose a novel 2D Skip Orthogonal List together with some dynamic tree techniques. Although our algorithm is based on the conventional simplex method, it can efficiently find the variable to pivot within expected $O(1)$ time, and complete each pivoting operation within expected $O(|V|)$ time where $V$ is the set of all supply and demand nodes. Since dynamic modifications typically do not introduce significant changes, our algorithm requires only a few simplex iterations in practice. So our algorithm is more efficient than re-computing the optimal transport cost that needs at least one traversal over all $|E| = O(|V|^2)$ variables, where $|E|$ denotes the number of edges in the network. Our experiments demonstrate that our algorithm significantly outperforms existing algorithms in the dynamic scenarios.

cs.DS↗

arXiv2023

Is Simple Uniform Sampling Effective for Center-Based Clustering with Outliers: When and Why?

Real-world datasets often contain outliers, and the presence of outliers can make the clustering problems to be much more challenging. In this paper, we propose a simple uniform sampling framework for solving three representative center-based clustering with outliers problems: $k$-center/median/means clustering with outliers. Our analysis is fundamentally different from the previous (uniform and non-uniform) sampling based ideas. To explain the effectiveness of uniform sampling in theory, we introduce a measure of "significance" and prove that the performance of our framework depends on the significance degree of the given instance. In particular, the sample size can be independent of the input data size $n$ and the dimensionality $d$, if we assume the given instance is "significant", which is in fact a fairly reasonable assumption in practice. Due to its simplicity, the uniform sampling approach also enjoys several significant advantages over the non-uniform sampling approaches in practice. To the best of our knowledge, this is the first work that systematically studies the effectiveness of uniform sampling from both theoretical and experimental aspects.

cs.LG↗

arXiv2023

A Data-dependent Approach for High Dimensional (Robust) Wasserstein Alignment

Many real-world problems can be formulated as the alignment between two geometric patterns. Previously, a great amount of research focus on the alignment of 2D or 3D patterns in the field of computer vision. Recently, the alignment problem in high dimensions finds several novel applications in practice. However, the research is still rather limited in the algorithmic aspect. To the best of our knowledge, most existing approaches are just simple extensions of their counterparts for 2D and 3D cases, and often suffer from the issues such as high computational complexities. In this paper, we propose an effective framework to compress the high dimensional geometric patterns. Any existing alignment method can be applied to the compressed geometric patterns and the time complexity can be significantly reduced. Our idea is inspired by the observation that high dimensional data often has a low intrinsic dimension. Our framework is a ``data-dependent'' approach that has the complexity depending on the intrinsic dimension of the input data. Our experimental results reveal that running the alignment algorithm on compressed patterns can achieve similar qualities, comparing with the results on the original patterns, but the runtimes (including the times cost for compression) are substantially lower.

cs.CV↗

arXiv2023

Coresets for Wasserstein Distributionally Robust Optimization Problems

Wasserstein distributionally robust optimization (\textsf{WDRO}) is a popular model to enhance the robustness of machine learning with ambiguous data. However, the complexity of \textsf{WDRO} can be prohibitive in practice since solving its ``minimax'' formulation requires a great amount of computation. Recently, several fast \textsf{WDRO} training algorithms for some specific machine learning tasks (e.g., logistic regression) have been developed. However, the research on designing efficient algorithms for general large-scale \textsf{WDRO}s is still quite limited, to the best of our knowledge. \textit{Coreset} is an important tool for compressing large dataset, and thus it has been widely applied to reduce the computational complexities for many optimization problems. In this paper, we introduce a unified framework to construct the $ε$-coreset for the general \textsf{WDRO} problems. Though it is challenging to obtain a conventional coreset for \textsf{WDRO} due to the uncertainty issue of ambiguous data, we show that we can compute a ``dual coreset'' by using the strong duality property of \textsf{WDRO}. Also, the error introduced by the dual coreset can be theoretically guaranteed for the original \textsf{WDRO} objective. To construct the dual coreset, we propose a novel grid sampling approach that is particularly suitable for the dual formulation of \textsf{WDRO}. Finally, we implement our coreset approach and illustrate its effectiveness for several \textsf{WDRO} problems in the experiments.

cs.LG↗

arXiv2023

Randomized Greedy Algorithms and Composable Coreset for k-Center Clustering with Outliers

In this paper, we study the problem of {\em $k$-center clustering with outliers}. The problem has many important applications in real world, but the presence of outliers can significantly increase the computational complexity. Though a number of methods have been developed in the past decades, it is still quite challenging to design quality guaranteed algorithm with low complexity for this problem. Our idea is inspired by the greedy method, Gonzalez's algorithm, that was developed for solving the ordinary $k$-center clustering problem. Based on some novel observations, we show that a simple randomized version of this greedy strategy actually can handle outliers efficiently. We further show that this randomized greedy approach also yields small coreset for the problem in doubling metrics (even if the doubling dimension is not given), which can greatly reduce the computational complexity. Moreover, together with the partial clustering framework proposed in arXiv:1703.01539 , we prove that our coreset method can be applied to distributed data with a low communication complexity. The experimental results suggest that our algorithms can achieve near optimal solutions and yield lower complexities comparing with the existing methods.

cs.LG↗

arXiv2023

Sublinear Time Algorithms for Several Geometric Optimization (With Outliers) Problems In Machine Learning

In this paper, we study several important geometric optimization problems arising in machine learning. First, we revisit the Minimum Enclosing Ball (MEB) problem in Euclidean space $\mathbb{R}^d$. The problem has been extensively studied before, but real-world machine learning tasks often need to handle large-scale datasets so that we cannot even afford linear time algorithms. Motivated by the recent studies on {\em beyond worst-case analysis}, we introduce the notion of stability for MEB, which is natural and easy to understand. Roughly speaking, an instance of MEB is stable, if the radius of the resulting ball cannot be significantly reduced by removing a small fraction of the input points. Under the stability assumption, we present two sampling algorithms for computing radius-approximate MEB with sample complexities independent of the number of input points $n$. In particular, the second algorithm has the sample complexity even independent of the dimensionality $d$. We also consider the general case without the stability assumption. We present a hybrid algorithm that can output either a radius-approximate MEB or a covering-approximate MEB. Our algorithm improves the running time and the number of passes for the previous sublinear MEB algorithms. Our method relies on two novel techniques, the Uniform-Adaptive Sampling method and Sandwich Lemma. Furthermore, we observe that these two techniques can be generalized to design sublinear time algorithms for a broader range of geometric optimization problems with outliers in high dimensions, including MEB with outliers, one-class and two-class linear SVMs with outliers, $k$-center clustering with outliers, and flat fitting with outliers. Our proposed algorithms also work fine for kernels.

cs.DS↗