Lingxiao Li

Lingxiao Li, Haobo Zhang, Ruohao Fan et al.

Drug discovery is a lengthy and resource-intensive process composed of multiple stages. Among these stages, lead optimization plays a critical role in transforming early hit compounds into viable drug candidates. This stage requires improving ADMET-related properties through subtle structural refinement while preserving key molecular substructures responsible for binding affinity to disease targets. Recent advances in artificial intelligence have shown promise in accelerating various aspects of drug discovery; however, most existing approaches to lead optimization rely on one-step molecular optimization, which fail to account for the long-term consequences of sequential design decisions. To address this limitation, we propose TRACE, a trajectory-aware, LLM-reasoning agent for molecular lead optimization that formulates tool selection as a sequential decision-making problem over action trajectories. Given a lead molecule and an optimization objective, TRACE makes trajectory-aware decisions over molecular optimization tools, enabling forward-looking refinement under structural constraints. Experiments on multiple ADMET optimization tasks show that our agent achieves higher optimization success, larger property improvements, and higher validity, while preserving molecular similarity compared to baseline models.

Lingxiao Li, Samuel Hurault, Justin Solomon

We present a discretization-free scalable framework for solving a large class of mass-conserving partial differential equations (PDEs), including the time-dependent Fokker-Planck equation and the Wasserstein gradient flow. The main observation is that the time-varying velocity field of the PDE solution needs to be self-consistent: it must satisfy a fixed-point equation involving the probability flow characterized by the same velocity field. Instead of directly minimizing the residual of the fixed-point equation with neural parameterization, we use an iterative formulation with a biased gradient estimator that bypasses significant computational obstacles with strong empirical performance. Compared to existing approaches, our method does not suffer from temporal or spatial discretization, covers a wider range of PDEs, and scales to high dimensions. Experimentally, our method recovers analytical solutions accurately when they are available and achieves superior performance in high dimensions with less training time compared to alternatives.

Lingxiao Li, Qiang Liu, Anna Korba et al.

Sampling from a target measure whose density is only known up to a normalization constant is a fundamental problem in computational statistics and machine learning. In this paper, we present a new optimization-based method for sampling called mollified interaction energy descent (MIED). MIED minimizes a new class of energies on probability measures called mollified interaction energies (MIEs). These energies rely on mollifier functions -- smooth approximations of the Dirac delta originated from PDE theory. We show that as the mollifier approaches the Dirac delta, the MIE converges to the chi-square divergence with respect to the target measure and the gradient flow of the MIE agrees with that of the chi-square divergence. Optimizing this energy with proper discretization yields a practical first-order particle-based algorithm for sampling in both unconstrained and constrained domains. We show experimentally that for unconstrained sampling problems our algorithm performs on par with existing particle-based algorithms like SVGD, while for constrained sampling problems our method readily incorporates constrained optimization techniques to handle more flexible constraints with strong performance compared to alternatives.

Changhui Yao, Yunpan Ma, Lingxiao Li

In recent years, high-order finite element methods on high-order meshes have attracted considerable attention. This work investigates the isoparametric upwind discontinuous Galerkin method for the radiation transport equation on a bounded domain with a piecewise $C^{k+1}$ smooth curved boundary. We use the isoparametric mapping to approximate the curved domain and construct a curved upwind discontinuous Galerkin scheme. The first-order hyperbolic nature and the complexity introduced by non-affine transformation, lead to additional difficulties for geometric approximation, numerical stability and the optimal error estimate. To address these issues, with the help of an isoparametric auxiliary operator, we first prove that the bilinear form is continuous with respect to the DG norm when its first argument is the isoparametric projection error. Then the geometric approximation error of inflow boundary of original domain is precisely estimated. The error order between discrete normal vectors and the continuous ones are also proven. Finally, the rigorous analysis yields an optimal convergence rate in the DG norm. Two- and three-dimensional numerical tests are conducted to support the theoretical results.

Yuanyuan Zhang, Runwei Guan, Lingxiao Li et al.

Radar-based contactless cardiac monitoring has become a popular research direction recently, but the fine-grained electrocardiogram (ECG) signal is still hard to reconstruct from millimeter-wave radar signal. The key obstacle is to decouple the cardiac activities in the electrical domain (i.e., ECG) from that in the mechanical domain (i.e., heartbeat), and most existing research only uses pure data-driven methods to map such domain transformation as a black box. Therefore, this work first proposes a signal model for domain transformation, and then a novel deep learning framework called radarODE is designed to fuse the temporal and morphological features extracted from radar signals and generate ECG. In addition, ordinary differential equations are embedded in radarODE as a decoder to provide morphological prior, helping the convergence of the model training and improving the robustness under body movements. After being validated on the dataset, the proposed radarODE achieves better performance compared with the benchmark in terms of missed detection rate, root mean square error, Pearson correlation coefficient with the improvement of 9%, 16% and 19%, respectively. The validation results imply that radarODE is capable of recovering ECG signals from radar signals with high fidelity and can be potentially implemented in real-life scenarios.

Lingxiao Li, Yi Zhang, Shuhui Wang

Few-shot image generation is a challenging task since it aims to generate diverse new images for an unseen category with only a few images. Existing methods suffer from the trade-off between the quality and diversity of generated images. To tackle this problem, we propose Hyperbolic Attribute Editing~(HAE), a simple yet effective method. Unlike other methods that work in Euclidean space, HAE captures the hierarchy among images using data from seen categories in hyperbolic space. Given a well-trained HAE, images of unseen categories can be generated by moving the latent code of a given image toward any meaningful directions in the Poincaré disk with a fixing radius. Most importantly, the hyperbolic space allows us to control the semantic diversity of the generated images by setting different radii in the disk. Extensive experiments and visualizations demonstrate that HAE is capable of not only generating images with promising quality and diversity using limited data but achieving a highly controllable and interpretable editing process.

Yihao Wang, Ruiqi Song, Lingxiao Li et al.

Linguistic steganalysis (LS) tasks aim to detect whether a text contains secret information. Existing LS methods focus on the deep-learning model design and they achieve excellent results in ideal data. However, they overlook the unique user characteristics, leading to weak performance in social networks. And a few stegos here that further complicate detection. We propose the UP4LS, a framework with the User Profile for enhancing LS in realistic scenarios. Three kinds of user attributes like writing habits are explored to build the profile. For each attribute, the specific feature extraction module is designed. The extracted features are mapped to high-dimensional user features via the deep-learning model of the method to be improved. The content feature is extracted by the language model. Then user and content features are integrated. Existing methods can improve LS results by adding the UP4LS framework without changing their deep-learning models. Experiments show that UP4LS can significantly enhance the performance of LS-task baselines in realistic scenarios, with the overall Acc increased by 25%, F1 increased by 51%, and SOTA results. The improvement is especially pronounced in fewer stegos. Additionally, UP4LS also sets the stage for the related-task SOTA methods to efficient LS.

Ryan Burgert, Yuancheng Xu, Wenqi Xian et al.

Generative modeling aims to transform random noise into structured outputs. In this work, we enhance video diffusion models by allowing motion control via structured latent noise sampling. This is achieved by just a change in data: we pre-process training videos to yield structured noise. Consequently, our method is agnostic to diffusion model design, requiring no changes to model architectures or training pipelines. Specifically, we propose a novel noise warping algorithm, fast enough to run in real time, that replaces random temporal Gaussianity with correlated warped noise derived from optical flow fields, while preserving the spatial Gaussianity. The efficiency of our algorithm enables us to fine-tune modern video diffusion base models using warped noise with minimal overhead, and provide a one-stop solution for a wide range of user-friendly motion control: local object motion control, global camera movement control, and motion transfer. The harmonization between temporal coherence and spatial Gaussianity in our warped noise leads to effective motion control while maintaining per-frame pixel quality. Extensive experiments and user studies demonstrate the advantages of our method, making it a robust and scalable approach for controlling motion in video diffusion models. Video results are available on our webpage: https://eyeline-labs.github.io/Go-with-the-Flow. Source code and model checkpoints are available on GitHub: https://github.com/Eyeline-Labs/Go-with-the-Flow.

Lingxiao Li

We propose a new mixed finite element method for the three-dimensional steady magnetohydrodynamic (MHD) kinematics equations for which the velocity of the fluid is given. Although prescribing the velocity field leads to a simpler model than full MHD equations, its conservative and efficient numerical methods are still active research topic. The distinctive feature of our discrete scheme is that the divergence-free conditions for current density and magnetic induction are both satisfied. To reach this goal, we use magnetic vector potential to represent magnetic induction and resort to H(div)-conforming element to discretize the current density. We develop an preconditioned iterative solver based on a block preconditioner for the algebraic systems arising from the discretization. Several numerical experiments are implemented to verify the divergence-free properties, the convergence rate of the finite element scheme and the robustness of the preconditioner.

Zeqi Xiao, Yiwei Zhao, Lingxiao Li et al.

We investigate whether video generative models can exhibit visuospatial intelligence, a capability central to human cognition, using only visual data. To this end, we present Video4Spatial, a framework showing that video diffusion models conditioned solely on video-based scene context can perform complex spatial tasks. We validate on two tasks: scene navigation - following camera-pose instructions while remaining consistent with 3D geometry of the scene, and object grounding - which requires semantic localization, instruction following, and planning. Both tasks use video-only inputs, without auxiliary modalities such as depth or poses. With simple yet effective design choices in the framework and data curation, Video4Spatial demonstrates strong spatial understanding from video context: it plans navigation and grounds target objects end-to-end, follows camera-pose instructions while maintaining spatial consistency, and generalizes to long contexts and out-of-domain environments. Taken together, these results advance video generative models toward general visuospatial reasoning.

Yuan Qing, Kunyu Zheng, Lingxiao Li et al.

Recent advances in video generation have made AI-synthesized content increasingly difficult to distinguish from real footage. We propose a physics-based authentication signature that real cameras produce naturally, but that generative models cannot faithfully reproduce. Our approach exploits the Moiré effect: the interference fringes formed when a camera views a compact two-layer grating structure. We derive the Moiré motion invariant, showing that fringe phase and grating image displacement are linearly coupled by optical geometry, independent of viewing distance and grating structure. A verifier extracts both signals from video and tests their correlation. We validate the invariant on both real-captured and AI-generated videos from multiple state-of-the-art generators, and find that real and AI-generated videos produce significantly different correlation signatures, suggesting a robust means of differentiating them. Our work demonstrates that deterministic optical phenomena can serve as physically grounded, verifiable signatures against AI-generated video.

Lingxiao Li, Haobo Zhang, Bin Chen et al.

Text-conditioned molecular generation aims to translate natural-language descriptions into chemical structures, enabling scientists to specify functional groups, scaffolds, and physicochemical constraints without handcrafted rules. Diffusion-based models, particularly latent diffusion models (LDMs), have recently shown promise by performing stochastic search in a continuous latent space that compactly captures molecular semantics. Yet existing methods rely on one-shot conditioning, where the entire prompt is encoded once and applied throughout diffusion, making it hard to satisfy all the requirements in the prompt. We discuss three outstanding challenges of one-shot conditioning generation, including the poor interpretability of the generated components, the failure to generate all substructures, and the overambition in considering all requirements simultaneously. We then propose three principles to address those challenges, motivated by which we propose Chain-of-Generation (CoG), a training-free multi-stage latent diffusion framework. CoG decomposes each prompt into curriculum-ordered semantic segments and progressively incorporates them as intermediate goals, guiding the denoising trajectory toward molecules that satisfy increasingly rich linguistic constraints. To reinforce semantic guidance, we further introduce a post-alignment learning phase that strengthens the correspondence between textual and molecular latent spaces. Extensive experiments on benchmark and real-world tasks demonstrate that CoG yields higher semantic alignment, diversity, and controllability than one-shot baselines, producing molecules that more faithfully reflect complex, compositional prompts while offering transparent insight into the generation process.

Yunlong Li, Fei Wang, Lingxiao Li

The incompressible magnetohydrodynamic (MHD) equations are fundamental in many scientific and engineering applications. However, their strong nonlinearity and dual divergence-free constraints make them highly challenging for conventional numerical solvers. To overcome these difficulties, we propose a Structure-Preserving Randomized Neural Network (SP-RaNN) that automatically and exactly satisfies the divergence-free conditions. Unlike deep neural network (DNN) approaches that rely on expensive nonlinear and nonconvex optimization, SP-RaNN reformulates the training process into a linear least-squares system, thereby eliminating nonconvex optimization. The method linearizes the governing equations through Picard or Newton iterations, discretizes them at collocation points within the domain and on the boundaries using finite-difference schemes, and solves the resulting linear system via a linear least-squares procedure. By design, SP-RaNN preserves the intrinsic mathematical structure of the equations within a unified space-time framework, ensuring both stability and accuracy. Numerical experiments on the Navier-Stokes, Maxwell, and MHD equations demonstrate that SP-RaNN achieves higher accuracy, faster convergence, and exact enforcement of divergence-free constraints compared with both traditional numerical methods and DNN-based approaches. This structure-preserving framework provides an efficient and reliable tool for solving complex PDE systems while rigorously maintaining their underlying physical laws.

Yitong Deng, Winnie Lin, Lingxiao Li et al.

Adapting pretrained image-based diffusion models to generate temporally consistent videos has become an impactful generative modeling research direction. Training-free noise-space manipulation has proven to be an effective technique, where the challenge is to preserve the Gaussian white noise distribution while adding in temporal consistency. Recently, Chang et al. (2024) formulated this problem using an integral noise representation with distribution-preserving guarantees, and proposed an upsampling-based algorithm to compute it. However, while their mathematical formulation is advantageous, the algorithm incurs a high computational cost. Through analyzing the limiting-case behavior of their algorithm as the upsampling resolution goes to infinity, we develop an alternative algorithm that, by gathering increments of multiple Brownian bridges, achieves their infinite-resolution accuracy while simultaneously reducing the computational cost by orders of magnitude. We prove and experimentally validate our theoretical claims, and demonstrate our method's effectiveness in real-world applications. We further show that our method readily extends to the 3-dimensional space.

Yihao Wang, Lingxiao Li, Yifan Tang et al.

Intelligent transportation systems (ITS) use advanced technologies such as artificial intelligence to significantly improve traffic flow management efficiency, and promote the intelligent development of the transportation industry. However, if the data in ITS is attacked, such as tampering or forgery, it will endanger public safety and cause social losses. Therefore, this paper proposes a watermarking that can verify the integrity of copyright in response to the needs of ITS, termed ITSmark. ITSmark focuses on functions such as extracting watermarks, verifying permission, and tracing tampered locations. The scheme uses the copyright information to build the multi-bit space and divides this space into multiple segments. These segments will be assigned to tokens. Thus, the next token is determined by its segment which contains the copyright. In this way, the obtained data contains the custom watermark. To ensure the authorization, key parameters are encrypted during copyright embedding to obtain cipher data. Only by possessing the correct cipher data and private key, can the user entirely extract the watermark. Experiments show that ITSmark surpasses baseline performances in data quality, extraction accuracy, and unforgeability. It also shows unique capabilities of permission verification and tampered location tracing, which ensures the security of extraction and the reliability of copyright verification. Furthermore, ITSmark can also customize the watermark embedding position and proportion according to user needs, making embedding more flexible.

Lingxiao Li, Kaixiong Gong, Weihong Li et al.

This paper introduces Bifröst, a novel 3D-aware framework that is built upon diffusion models to perform instruction-based image composition. Previous methods concentrate on image compositing at the 2D level, which fall short in handling complex spatial relationships ($\textit{e.g.}$, occlusion). Bifröst addresses these issues by training MLLM as a 2.5D location predictor and integrating depth maps as an extra condition during the generation process to bridge the gap between 2D and 3D, which enhances spatial comprehension and supports sophisticated spatial interactions. Our method begins by fine-tuning MLLM with a custom counterfactual dataset to predict 2.5D object locations in complex backgrounds from language instructions. Then, the image-compositing model is uniquely designed to process multiple types of input features, enabling it to perform high-fidelity image compositions that consider occlusion, depth blur, and image harmonization. Extensive qualitative and quantitative evaluations demonstrate that Bifröst significantly outperforms existing methods, providing a robust solution for generating realistically composited images in scenarios demanding intricate spatial understanding. This work not only pushes the boundaries of generative image compositing but also reduces reliance on expensive annotated datasets by effectively utilizing existing resources in innovative ways.

Lingxiao Li, Raaz Dwivedi, Lester Mackey · harvard, mit

Modern compression methods can summarize a target distribution $\mathbb{P}$ more succinctly than i.i.d. sampling but require access to a low-bias input sequence like a Markov chain converging quickly to $\mathbb{P}$. We introduce a new suite of compression methods suitable for compression with biased input sequences. Given $n$ points targeting the wrong distribution and quadratic time, Stein kernel thinning (SKT) returns $\sqrt{n}$ equal-weighted points with $\widetilde{O}(n^{-1/2})$ maximum mean discrepancy (MMD) to $\mathbb{P}$. For larger-scale compression tasks, low-rank SKT achieves the same feat in sub-quadratic time using an adaptive low-rank debiasing procedure that may be of independent interest. For downstream tasks that support simplex or constant-preserving weights, Stein recombination and Stein Cholesky achieve even greater parsimony, matching the guarantees of SKT with as few as $\text{poly-log}(n)$ weighted points. Underlying these advances are new guarantees for the quality of simplex-weighted coresets, the spectral decay of kernel matrices, and the covering numbers of Stein kernel Hilbert spaces. In our experiments, our techniques provide succinct and accurate posterior summaries while overcoming biases due to burn-in, approximate Markov chain Monte Carlo, and tempering.

Yihao Wang, Ruiqi Song, Lingxiao Li et al.

Existing linguistic steganography schemes often overlook the conditional probability (CP) of tokens in the candidate pool, allocating the one coding to all tokens, which results in identical selection likelihoods. This approach leads to the selection of low-CP tokens, degrading the quality of stegos and making them more detectable. This paper proposes a scheme based on the interval allocated, called DAIRstega. DAIRstega first uses a portion of the read secret to build the roulette area. Then, this scheme uses the idea of the roulette wheel and takes the CPs of tokens as the main basis for allocating the roulette area (i.e., the interval length). Thus, tokens with larger CPs are allocated more area. The secret will have an increased likelihood of selecting a token with a higher CP. During allocation, we designed some allocation functions and three constraints to optimize the process. Additionally, DAIRstega supports prompt-based controllable generation of stegos. Rich experiments show that the proposed embedding way and DAIRstega perform better than the existing ways and baselines, which shows strong perceptual, statistical, and semantic concealment, as well as anti-steganalysis ability. It can also generate high-quality longer stegos, addressing the deficiencies in this task. DAIRstega is confirmed to have potential as a secure watermarking, offering insights for its development.

Lingxiao Li, Salar Rahili, Yiwei Zhao

Language Models (LMs) are increasingly being used for code generation, but ensuring the correctness of generated programs remains a significant challenge. Although imperfect code may be acceptable during software development with human oversight, domains such as video games and robotics require one-shot correctness for runtime-critical components. We present a constrained decoding algorithm for generating semantically correct programs that incorporates a context-sensitive parser, which, at each step, outputs a regular expression that satisfies a critical non-extensible property to guide the generation of the next token sequence that can continue to a correct program. To build such a context-sensitive parser, we propose a framework of a dynamic tree of parsers (ToP) during parsing, where each parser corresponds to a modular context-free grammar enriched with contextual information such as variable scopes and type constraints, with tree branches representing ambiguity in the future code segment. We demonstrate our approach through sLua, a strongly typed variant of Lua, showing that our method can generate semantically correct programs conforming to any prescribed scripting API. We further show that, with careful design, our semantic guarantees extend to runtime correctness, as validated in the application of generating game mechanics for a roguelike video game.

Yin Hua, Zhiqiang Liu, Mingyang Chen et al.

In natural language processing (NLP) and computer vision (CV), the successful application of foundation models across diverse tasks has demonstrated their remarkable potential. However, despite the rich structural and textual information embedded in knowledge graphs (KGs), existing research of foundation model for KG has primarily focused on their structural aspects, with most efforts restricted to in-KG tasks (e.g., knowledge graph completion, KGC). This limitation has hindered progress in addressing more challenging out-of-KG tasks. In this paper, we introduce MERRY, a foundation model for general knowledge graph reasoning, and investigate its performance across two task categories: in-KG reasoning tasks (e.g., KGC) and out-of-KG tasks (e.g., KG question answering, KGQA). We not only utilize the structural information, but also the textual information in KGs. Specifically, we propose a multi-perspective Conditional Message Passing (CMP) encoding architecture to bridge the gap between textual and structural modalities, enabling their seamless integration. Additionally, we introduce a dynamic residual fusion module to selectively retain relevant textual information and a flexible edge scoring mechanism to adapt to diverse downstream tasks. Comprehensive evaluations on 28 datasets demonstrate that MERRY outperforms existing baselines in most scenarios, showcasing strong reasoning capabilities within KGs and excellent generalization to out-of-KG tasks such as KGQA.

Lingxiao Li, Kaixuan Fan, Boqing Gong et al.

Few-shot image generation aims to generate diverse and high-quality images for an unseen class given only a few examples in that class. A key challenge in this task is balancing category consistency and image diversity, which often compete with each other. Moreover, existing methods offer limited control over the attributes of newly generated images. In this work, we propose Hyperbolic Diffusion Autoencoders (HypDAE), a novel approach that operates in hyperbolic space to capture hierarchical relationships among images from seen categories. By leveraging pre-trained foundation models, HypDAE generates diverse new images for unseen categories with exceptional quality by varying stochastic subcodes or semantic codes. Most importantly, the hyperbolic representation introduces an additional degree of control over semantic diversity through the adjustment of radii within the hyperbolic disk. Extensive experiments and visualizations demonstrate that HypDAE significantly outperforms prior methods by achieving a better balance between preserving category-relevant features and promoting image diversity with limited data. Furthermore, HypDAE offers a highly controllable and interpretable generation process.

Lingxiao Li, Yifan Wang, Xinyan Gao et al.

Chain-of-Thought (CoT) prompting has proven remarkably effective for eliciting complex reasoning in large language models (LLMs). Yet, its potential in multimodal large language models (MLLMs) remains largely untapped, hindered by the absence of large-scale datasets that capture the rich, spatially grounded reasoning intrinsic to visual understanding. Existing visual-CoT resources are typically small, domain-specific, or lack the human-like stepwise structure necessary for compositional visual reasoning. In this paper, we introduce VisReason, a large-scale dataset designed to advance visual Chain-of-Thought reasoning. VisReason comprises 489K annotated examples spanning four diverse domains, each featuring multi-round, human-like rationales that guide MLLMs through interpretable visual reasoning steps. Building upon this, we curate VisReason-Pro, a 165K subset produced with a stronger expert-level GPT annotator, enriched with detailed reasoning traces and 3D spatial grounding via depth-informed annotations. Fine-tuning the state-of-the-art Qwen2.5-VL model on VisReason and VisReason-Pro yields substantial improvements in step-by-step visual reasoning accuracy, interpretability, and cross-benchmark generalization. These results demonstrate that VisReason equips MLLMs with more systematic and generalizable reasoning capabilities. We envision VisReason as a cornerstone for cultivating human-like visual reasoning, paving the way toward the next generation of multimodal intelligence.

Wei Hao, Zixi Wang, Lauren Hong et al.

ML models are increasingly being pushed to mobile devices, for low-latency inference and offline operation. However, once the models are deployed, it is hard for ML operators to track their accuracy, which can degrade unpredictably (e.g., due to data drift). We design the first end-to-end system for continuously monitoring and adapting models on mobile devices without requiring feedback from users. Our key observation is that often model degradation is due to a specific root cause, which may affect a large group of devices. Therefore, once the system detects a consistent degradation across a large number of devices, it employs a root cause analysis to determine the origin of the problem and applies a cause-specific adaptation. We evaluate the system on two computer vision datasets, and show it consistently boosts accuracy compared to existing approaches. On a dataset containing photos collected from driving cars, our system improves the accuracy on average by 15%.

Alexander Korotin, Vage Egiazarian, Lingxiao Li et al.

Wasserstein barycenters have become popular due to their ability to represent the average of probability measures in a geometrically meaningful way. In this paper, we present an algorithm to approximate the Wasserstein-2 barycenters of continuous measures via a generative model. Previous approaches rely on regularization (entropic/quadratic) which introduces bias or on input convex neural networks which are not expressive enough for large-scale tasks. In contrast, our algorithm does not introduce bias and allows using arbitrary neural networks. In addition, based on the celebrity faces dataset, we construct Ave, celeba! dataset which can be used for quantitative evaluation of barycenter algorithms by using standard metrics of generative models such as FID.

Lingxiao Li, Noam Aigerman, Vladimir G. Kim et al.

Finding multiple solutions of non-convex optimization problems is a ubiquitous yet challenging task. Most past algorithms either apply single-solution optimization methods from multiple random initial guesses or search in the vicinity of found solutions using ad hoc heuristics. We present an end-to-end method to learn the proximal operator of a family of training problems so that multiple local minima can be quickly obtained from initial guesses by iterating the learned operator, emulating the proximal-point algorithm that has fast convergence. The learned proximal operator can be further generalized to recover multiple optima for unseen problems at test time, enabling applications such as object detection. The key ingredient in our formulation is a proximal regularization term, which elevates the convexity of our training loss: by applying recent theoretical results, we show that for weakly-convex objectives with Lipschitz gradients, training of the proximal operator converges globally with a practical degree of over-parameterization. We further present an exhaustive benchmark for multi-solution optimization to demonstrate the effectiveness of our method.

Alexander Korotin, Lingxiao Li, Aude Genevay et al.

Despite the recent popularity of neural network-based solvers for optimal transport (OT), there is no standard quantitative way to evaluate their performance. In this paper, we address this issue for quadratic-cost transport -- specifically, computation of the Wasserstein-2 distance, a commonly-used formulation of optimal transport in machine learning. To overcome the challenge of computing ground truth transport maps between continuous measures needed to assess these solvers, we use input-convex neural networks (ICNN) to construct pairs of measures whose ground truth OT maps can be obtained analytically. This strategy yields pairs of continuous benchmark measures in high-dimensional spaces such as spaces of images. We thoroughly evaluate existing optimal transport solvers using these benchmark measures. Even though these solvers perform well in downstream tasks, many do not faithfully recover optimal transport maps. To investigate the cause of this discrepancy, we further test the solvers in a setting of image generation. Our study reveals crucial limitations of existing solvers and shows that increased OT accuracy does not necessarily correlate to better results downstream.

Petr Mokrov, Alexander Korotin, Lingxiao Li et al.

Wasserstein gradient flows provide a powerful means of understanding and solving many diffusion equations. Specifically, Fokker-Planck equations, which model the diffusion of probability measures, can be understood as gradient descent over entropy functionals in Wasserstein space. This equivalence, introduced by Jordan, Kinderlehrer and Otto, inspired the so-called JKO scheme to approximate these diffusion processes via an implicit discretization of the gradient flow in Wasserstein space. Solving the optimization problem associated to each JKO step, however, presents serious computational challenges. We introduce a scalable method to approximate Wasserstein gradient flows, targeted to machine learning applications. Our approach relies on input-convex neural networks (ICNNs) to discretize the JKO steps, which can be optimized by stochastic gradient descent. Unlike previous work, our method does not require domain discretization or particle simulation. As a result, we can sample from the measure at each time step of the diffusion and compute its probability density. We demonstrate our algorithm's performance by computing diffusions following the Fokker-Planck equation and apply it to unnormalized density sampling as well as nonlinear filtering.

Alexander Korotin, Lingxiao Li, Justin Solomon et al.

Wasserstein barycenters provide a geometric notion of the weighted average of probability measures based on optimal transport. In this paper, we present a scalable algorithm to compute Wasserstein-2 barycenters given sample access to the input measures, which are not restricted to being discrete. While past approaches rely on entropic or quadratic regularization, we employ input convex neural networks and cycle-consistency regularization to avoid introducing bias. As a result, our approach does not resort to minimax optimization. We provide theoretical analysis on error bounds as well as empirical evidence of the effectiveness of the proposed approach in low-dimensional qualitative scenarios and high-dimensional quantitative experiments.

Lingxiao Li, Aude Genevay, Mikhail Yurochkin et al.

Wasserstein barycenters provide a geometrically meaningful way to aggregate probability distributions, built on the theory of optimal transport. They are difficult to compute in practice, however, leading previous work to restrict their supports to finite sets of points. Leveraging a new dual formulation for the regularized Wasserstein barycenter problem, we introduce a stochastic algorithm that constructs a continuous approximation of the barycenter. We establish strong duality and use the corresponding primal-dual relationship to parametrize the barycenter implicitly using the dual potentials of regularized transport problems. The resulting problem can be solved with stochastic gradient descent, which yields an efficient online algorithm to approximate the barycenter of continuous distributions given sample access. We demonstrate the effectiveness of our approach and compare against previous work on synthetic examples and real-world applications.

Lingxiao Li, Muhammad Aamir Cheema, Hua Lu et al.

Many modern navigation systems and map-based services do not only provide the fastest route from a source location s to a target location t but also provide a few alternative routes to the users as more options to choose from. Consequently, computing alternative paths has received significant research attention. However, it is unclear which of the existing approaches generates alternative routes of better quality because the quality of these alternatives is mostly subjective. Motivated by this, in this paper, we present a user study conducted on the road networks of Melbourne, Dhaka and Copenhagen that compares the quality (as perceived by the users) of the alternative routes generated by four of the most popular existing approaches including the routes provided by Google Maps. We also present a web-based demo system that can be accessed using any internet-enabled device and allows users to see the alternative routes generated by the four approaches for any pair of selected source and target. We report the average ratings received by the four approaches and our statistical analysis shows that there is no credible evidence that the four approaches receive different ratings on average. We also discuss the limitations of this user study and recommend the readers to interpret these results with caution because certain factors may have affected the participants' ratings.

Li Zhang, Han Wang, Lingxiao Li

Pair-based metric learning has been widely adopted to learn sentence embedding in many NLP tasks such as semantic text similarity due to its efficiency in computation. Most existing works employed a sequence encoder model and utilized limited sentence pairs with a pair-based loss to learn discriminating sentence representation. However, it is known that the sentence representation can be biased when the sampled sentence pairs deviate from the true distribution of all sentence pairs. In this paper, our theoretical analysis shows that existing works severely suffered from a good pair sampling and instance weighting strategy. Instead of one time pair selection and learning on equal weighted pairs, we propose a unified locality weighting and learning framework to learn task-specific sentence embedding. Our model, SentPWNet, exploits the neighboring spatial distribution of each sentence as locality weight to indicate the informative level of sentence pair. Such weight is updated along with pair-loss optimization in each round, ensuring the model keep learning the most informative sentence pairs. Extensive experiments on four public available datasets and a self-collected place search benchmark with 1.4 million places clearly demonstrate that our model consistently outperforms existing sentence embedding methods with comparable efficiency.

Lingxiao Li, Minhyuk Sung, Anastasia Dubrovina et al.

Fitting geometric primitives to 3D point cloud data bridges a gap between low-level digitized 3D data and high-level structural information on the underlying 3D shapes. As such, it enables many downstream applications in 3D data processing. For a long time, RANSAC-based methods have been the gold standard for such primitive fitting problems, but they require careful per-input parameter tuning and thus do not scale well for large datasets with diverse shapes. In this work, we introduce Supervised Primitive Fitting Network (SPFN), an end-to-end neural network that can robustly detect a varying number of primitives at different scales without any user control. The network is supervised using ground truth primitive surfaces and primitive membership for the input points. Instead of directly predicting the primitives, our architecture first predicts per-point properties and then uses a differential model estimation module to compute the primitive type and parameters. We evaluate our approach on a novel benchmark of ANSI 3D mechanical component models and demonstrate a significant improvement over both the state-of-the-art RANSAC-based methods and the direct neural prediction.

Lingxiao Li, Weiying Zheng

In this paper, we propose a robust solver for the finite element discrete problem of the stationary incompressible magnetohydrodynamic (MHD) equations in three dimensions. By the mixed finite element method, both the velocity and the pressure are approximated by H1-conforming finite elements, while the magnetic field is approximated by H(curl)-conforming edge elements. An efficient preconditioner is proposed to accelerate the convergence of the GMRES method for solving the linearized MHD problem. We use three numerical experiments to demonstrate the effectiveness of the finite element method and the robustness of the discrete solver. The preconditioner contains the least undetermined parameters and is optimal with respect to the number of degrees of freedom. We also show the scalability of the solver for moderate physical parameters.