Cheng Meng

Mengyu Li, Jun Yu, Hongteng Xu et al.

As a valid metric of metric-measure spaces, Gromov-Wasserstein (GW) distance has shown the potential for matching problems of structured data like point clouds and graphs. However, its application in practice is limited due to the high computational complexity. To overcome this challenge, we propose a novel importance sparsification method, called \textsc{Spar-GW}, to approximate GW distance efficiently. In particular, instead of considering a dense coupling matrix, our method leverages a simple but effective sampling strategy to construct a sparse coupling matrix and update it with few computations. The proposed \textsc{Spar-GW} method is applicable to the GW distance with arbitrary ground cost, and it reduces the complexity from $O(n^4)$ to $O(n^{2+δ})$ for an arbitrary small $δ>0$. Theoretically, the convergence and consistency of the proposed estimation for GW distance are established under mild regularity conditions. In addition, this method can be extended to approximate the variants of GW distance, including the entropic GW distance, the fused GW distance, and the unbalanced GW distance. Experiments show the superiority of our \textsc{Spar-GW} to state-of-the-art methods in both synthetic and real-world tasks.

Tao Li, Cheng Meng, Hongteng Xu et al.

Distribution comparison plays a central role in many machine learning tasks like data classification and generative modeling. In this study, we propose a novel metric, called Hilbert curve projection (HCP) distance, to measure the distance between two probability distributions with low complexity. In particular, we first project two high-dimensional probability distributions using Hilbert curve to obtain a coupling between them, and then calculate the transport distance between these two distributions in the original space, according to the coupling. We show that HCP distance is a proper metric and is well-defined for probability measures with bounded supports. Furthermore, we demonstrate that the modified empirical HCP distance with the $L_p$ cost in the $d$-dimensional space converges to its population counterpart at a rate of no more than $O(n^{-1/2\max\{d,p\}})$. To suppress the curse-of-dimensionality, we also develop two variants of the HCP distance using (learnable) subspace projections. Experiments on both synthetic and real-world data show that our HCP distance works as an effective surrogate of the Wasserstein distance with low complexity and overcomes the drawbacks of the sliced Wasserstein distance.

Mengyu Li, Jun Yu, Tao Li et al.

Sinkhorn algorithm has been used pervasively to approximate the solution to optimal transport (OT) and unbalanced optimal transport (UOT) problems. However, its practical application is limited due to the high computational complexity. To alleviate the computational burden, we propose a novel importance sparsification method, called Spar-Sink, to efficiently approximate entropy-regularized OT and UOT solutions. Specifically, our method employs natural upper bounds for unknown optimal transport plans to establish effective sampling probabilities, and constructs a sparse kernel matrix to accelerate Sinkhorn iterations, reducing the computational cost of each iteration from $O(n^2)$ to $\widetilde{O}(n)$ for a sample of size $n$. Theoretically, we show the proposed estimators for the regularized OT and UOT problems are consistent under mild regularity conditions. Experiments on various synthetic data demonstrate Spar-Sink outperforms mainstream competitors in terms of both estimation error and speed. A real-world echocardiogram data analysis shows Spar-Sink can effectively estimate and visualize cardiac cycles, from which one can identify heart failure and arrhythmia. To evaluate the numerical accuracy of cardiac cycle prediction, we consider the task of predicting the end-systole time point using the end-diastole one. Results show Spar-Sink performs as well as the classical Sinkhorn algorithm, requiring significantly less computational time.

Cheng Meng, Wenxin Le, Xinyi Li et al.

Existing methods for detection rule generation are tightly coupled to specific input-output combinations, requiring dedicated pipelines for each. We formalize this problem as a unified mapping f:C*L->R and characterize optimal rules through semantic distance. We propose UniRule, an agentic RAG framework built on dual semantic projection spaces: detection intent and detection logic. This design enables retrieval and generation across arbitrary contexts and target languages within a single system. Experiments across 12 scenarios (3 languages, 4 context types, 12,000 pairwise comparisons) show that UniRule significantly outperforms pure LLM generation with a Bradley-Terry coefficient of 0.52, validating semantic projection as an effective abstraction for unified rule generation. Together, the formalization, method, and evaluation provide an initial framework for studying detection rule generation as a unified task.

Xuanhua He, Tianyu Yang, Ke Cao et al.

Current video avatar generation methods excel at identity preservation and motion alignment but lack genuine agency, they cannot autonomously pursue long-term goals through adaptive environmental interaction. We address this by introducing L-IVA (Long-horizon Interactive Visual Avatar), a task and benchmark for evaluating goal-directed planning in stochastic generative environments, and ORCA (Online Reasoning and Cognitive Architecture), the first framework enabling active intelligence in video avatars. ORCA embodies Internal World Model (IWM) capabilities through two key innovations: (1) a closed-loop OTAR cycle (Observe-Think-Act-Reflect) that maintains robust state tracking under generative uncertainty by continuously verifying predicted outcomes against actual generations, and (2) a hierarchical dual-system architecture where System 2 performs strategic reasoning with state prediction while System 1 translates abstract plans into precise, model-specific action captions. By formulating avatar control as a POMDP and implementing continuous belief updating with outcome verification, ORCA enables autonomous multi-step task completion in open-domain scenarios. Extensive experiments demonstrate that ORCA significantly outperforms open-loop and non-reflective baselines in task success rate and behavioral coherence, validating our IWM-inspired design for advancing video avatar intelligence from passive animation to active, goal-oriented behavior.

Tao Wang, Mengyu Li, Geduo Zeng et al.

3D Gaussian Splatting (3DGS) has emerged as a powerful technique for radiance field rendering, but it typically requires millions of redundant Gaussian primitives, overwhelming memory and rendering budgets. Existing compaction approaches address this by pruning Gaussians based on heuristic importance scores, without global fidelity guarantee. To bridge this gap, we propose a novel optimal transport perspective that casts 3DGS compaction as global Gaussian mixture reduction. Specifically, we first minimize the composite transport divergence over a KD-tree partition to produce a compact geometric representation, and then decouple appearance from geometry by fine-tuning color and opacity attributes with far fewer Gaussian primitives. Experiments on benchmark datasets show that our method (i) yields negligible loss in rendering quality (PSNR, SSIM, LPIPS) compared to vanilla 3DGS with only 10% Gaussians; and (ii) consistently outperforms state-of-the-art 3DGS compaction techniques. Notably, our method is applicable to any stage of vanilla or accelerated 3DGS pipelines, providing an efficient and agnostic pathway to lightweight neural rendering. The code is publicly available at https://github.com/DrunkenPoet/GHAP

Dunyao Xue, Mengyu Li, Cheng Meng et al.

In this paper, we propose a novel element-wise subset selection method for the alternating least squares (ALS) algorithm, focusing on low-rank matrix factorization involving matrices with missing values, as commonly encountered in recommender systems. While ALS is widely used for providing personalized recommendations based on user-item interaction data, its high computational cost, stemming from repeated regression operations, poses significant challenges for large-scale datasets. To enhance the efficiency of ALS, we propose a core-elements subsampling method that selects a representative subset of data and leverages sparse matrix operations to approximate ALS estimations efficiently. We establish theoretical guarantees for the approximation and convergence of the proposed approach, showing that it achieves similar accuracy with significantly reduced computational time compared to full-data ALS. Extensive simulations and real-world applications demonstrate the effectiveness of our method in various scenarios, emphasizing its potential in large-scale recommendation systems.

Cheng Meng, Yuan Ke, Jingyi Zhang et al.

This paper studies the estimation of large-scale optimal transport maps (OTM), which is a well-known challenging problem owing to the curse of dimensionality. Existing literature approximates the large-scale OTM by a series of one-dimensional OTM problems through iterative random projection. Such methods, however, suffer from slow or none convergence in practice due to the nature of randomly selected projection directions. Instead, we propose an estimation method of large-scale OTM by combining the idea of projection pursuit regression and sufficient dimension reduction. The proposed method, named projection pursuit Monge map (PPMM), adaptively selects the most ``informative'' projection direction in each iteration. We theoretically show the proposed dimension reduction method can consistently estimate the most ``informative'' projection direction in each iteration. Furthermore, the PPMM algorithm weakly convergences to the target large-scale OTM in a reasonable number of steps. Empirically, PPMM is computationally easy and converges fast. We assess its finite sample performance through the applications of Wasserstein distance estimation and generative models.

Cheng Meng, Jun Yu, Jingyi Zhang et al.

Sufficient dimension reduction is used pervasively as a supervised dimension reduction approach. Most existing sufficient dimension reduction methods are developed for data with a continuous response and may have an unsatisfactory performance for the categorical response, especially for the binary-response. To address this issue, we propose a novel estimation method of sufficient dimension reduction subspace (SDR subspace) using optimal transport. The proposed method, named principal optimal transport direction (POTD), estimates the basis of the SDR subspace using the principal directions of the optimal transport coupling between the data respecting different response categories. The proposed method also reveals the relationship among three seemingly irrelevant topics, i.e., sufficient dimension reduction, support vector machine, and optimal transport. We study the asymptotic properties of POTD and show that in the cases when the class labels contain no error, POTD estimates the SDR subspace exclusively. Empirical studies show POTD outperforms most of the state-of-the-art linear dimension reduction methods.