ICML 2026 Papers

International Conference on Machine Learning

Decision breakdown

6,341papers

Counted from the official paper list; may differ slightly from the number the conference announced.

  • Spotlight
    5368.5%
  • Poster
    5,80591.5%

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Off-Policy Evaluation with Strategic Agents via Local Disclosure

Kiet Q. H. Vo, Abbavaram Gowtham Reddy, Julian Rodemann, Siu Lun Chau, Krikamol Muandet

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We study off-policy evaluation (OPE) under strategic behavior where decision subjects (or agents) respond to a decision maker's policy by strategically modifying their covariates. Such behavior induces a policy-dependent covariate shift, breaking the standard assumption in existing methods that covariates are exogenous to the policy. Related work addresses this challenge by imposing strong assumptions such as repeated interactions or full knowledge of agents’ response behavior, substantially limiting its applicability to OPE. In contrast, we consider a one-shot OPE setting where the decision maker has only partial knowledge of the agents' response behavior. Our key insight is that disclosing local information through post-hoc explanations reveals agents’ pre-strategic covariates prior to adaptation, mitigating the information loss induced by strategic behavior. Leveraging this structure, we estimate a statistical model for the agents’ responses and construct a doubly robust estimator for policy value. By assuming that the agents' cost sensitivity follows a conditional log-normal distribution, we establish consistency of the proposed estimator and validate our approach empirically. More broadly, our results highlight how interaction design can mitigate information asymmetry by revealing otherwise hidden structure in agents' strategic responses.

Poster
Do Activation Verbalization Methods Convey Privileged Information?

Millicent Li, Alberto Mario Ceballos Arroyo, Giordano Rogers, Naomi Saphra, Byron C Wallace

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Recent interpretability methods have proposed to translate LLM internal representations into natural language descriptions using a second *verbalizer* LLM. This is intended to illuminate how the target model represents and operates on inputs. But do such *activation verbalization* approaches actually provide *privileged* knowledge about the internal workings of the target model, or do they merely convey information about the inputs provided to it? We critically evaluate popular verbalization methods and datasets used in prior work and find that one can perform well on such benchmarks without access to target model internals, suggesting that these datasets are not ideal for evaluating verbalization methods. We then run controlled experiments which reveal that verbalizations often reflect the parametric knowledge of the verbalizer LLM that generated them, rather than the knowledge of the target LLM whose activations are decoded. Taken together, our results indicate a need for targeted benchmarks and experimental controls to rigorously assess whether verbalization methods provide meaningful insights into the operations of LLMs.

Poster
Multimodal Fact-Level Attribution for Verifiable Reasoning

David Wan, Han Wang, Ziyang Wang, Elias Stengel-Eskin, Hyunji Lee, Mohit Bansal

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Multimodal large language models (MLLMs) are increasingly used for real-world tasks involving multi-step reasoning and long-form generation, where reliability requires grounding model outputs in heterogeneous input sources and verifying individual factual claims. However, existing multimodal grounding benchmarks and evaluation methods focus on simplified, observation-based scenarios or limited modalities and fail to assess attribution in complex multimodal reasoning. We introduce MuRGAt (Multimodal Reasoning with Grounded Attribution), a benchmark for evaluating fact-level multimodal attribution in settings that require reasoning beyond direct observation. Given inputs spanning video, audio, and other modalities, MuRGAt requires models to generate answers with explicit reasoning and precise citations, where each citation specifies both modality and temporal segments. To enable reliable assessment, we introduce an automatic evaluation framework that strongly correlates with human judgments. Benchmarking with human and automated scores reveals that even strong MLLMs frequently hallucinate citations despite correct reasoning. Moreover, we observe a key trade-off: increasing reasoning depth or enforcing structured grounding often degrades accuracy, highlighting a significant gap between internal reasoning and verifiable attribution.

Poster
Bridging the Grounding Gap in VideoQA via Typed Memory for Language-based Belief-State Reasoning

Saman Forouzandeh, Wei Peng, Xinghuo Yu, Mahdi Jalili

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VideoQA models can be accurate yet often fail to align answers with the correct video segments (the \emph{grounding gap}). We introduce \textbf{LINGUA} (\textbf{L}anguage-based \textbf{IN}ference for \textbf{G}rounded Video \textbf{U}nderstanding \textbf{A}gent), a memory-based agent that performs grounded VideoQA by reasoning in an explicit \emph{linguistic belief state}. LINGUA uses five mechanisms: (1) event-driven perception (retains 8--12\% of frames while preserving 94\% of question-relevant events); (2) typed memory for episodic narratives, semantic affordances, and procedural scripts; (3) Belief-Action-Verification loops with postcondition and temporal checks; (4) meta reflection with contrastive refinement; and (5) Bayesian reliability tracking for continual learning without gradient updates. Built with Gemma3-4B (Ollama, 4-bit), LINGUA outperforms strong baselines on five VideoQA benchmarks, reaching 82.4\% on NExT-QA and 42.3\% Acc@GQA on NExT-GQA (answer + IoU$\geq$0.5 temporal localization), while running 2.6$\times$ faster than dense-frame methods. In continual learning over 100 videos, accuracy rises from 45.2\% (first 10) to 61.8\% (last 10) without catastrophic forgetting, indicating online adaptation via memory refinement.

Poster
Active Attacks: Red-teaming LLMs via Adaptive Environments

Taeyoung Yun, Pierre-Luc St-Charles, Jinkyoo Park, Yoshua Bengio, Minsu Kim

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We address the challenge of automatically generating diverse attack prompts for large language models (LLMs) that elicit harmful behaviors (e.g., insults, sexual content) and are used for safety fine-tuning. While several prior approaches train LLMs with reinforcement learning (RL) to generate such prompts using only a toxicity classifier as a reward, existing diversity-seeking RL methods often collapse to limited modes: once high-reward prompts are found, exploration of new regions is discouraged. Inspired by the active learning paradigm that encourages adaptive exploration, we introduce \textbf{Active Attacks}, a novel RL-based red-teaming algorithm that adapts its attacks as the victim evolves. By periodically safety fine-tuning the victim LLM with collected attack prompts, we naturally induce an \emph{easy-to-hard exploration curriculum}, where the attacker progresses beyond easy modes toward increasingly difficult ones. We observe that this simple plug-and-play module, which seamlessly integrates into existing RL objectives, unexpectedly outperformed prior RL-based methods, improving cross-attack success rates against GFlowNets, the previous state-of-the-art, from 0.07\% to 31.28\% (a relative gain of more than 400×) with only a 6\% increase in computation.

Poster
Physics-Informed Pre-training on Efficient Electron-Density Images for Organic Material Property Prediction

Zhixiang Cheng, hongxin xiang, Mingquan Liu, Tengfei Ma, Yingzhuo Tu, Wenjie Du, Bosheng Song, Yiping Liu, xiangxiang Zeng

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Precise property prediction of organic materials is pivotal for next-generation electronic and energy devices. In density functional theory (DFT), the electron density (ED) serves as the fundamental determinant of material properties. Yet, establishing it as an input modality for material property prediction has been impeded by two practical barriers: scarce large-scale ED data and the enormous computational complexity of ED representation. To bridge these gaps, we introduce VisionED, an efficient physics-informed model pre-trained on electron-density images. We curate a dataset of 2 million molecules and represent ED as multi-shot images that efficiently encode both geometric and electronic structure. VisionED is then pre-trained on 12 million multi-shot ED images via cross-scale, physics-informed pretext tasks. Empirical evaluations on photovoltaic and organic chromophore datasets show that VisionED outperforms state-of-the-art baselines by up to 27.0\%, exhibiting superior robustness under distribution shifts and data scarcity. Notably, the model generalizes to unseen device-scale applications, successfully recovering experimental trends and mixing-ratio effects in ternary blends with an average accuracy of 92.77\%. Moreover, relative to the previous ED point cloud, the ED image improves performance by 26.2\% with 2.6$\times$ fewer memory and 4.6$\times$ lower time. The code and data are available at https://github.com/ZhixiangCheng/VisionED.

Poster
Unifying Dataset Pruning and Distillation for Efficient Large-scale Compression

Lingao Xiao, Songhua Liu, Yang He, Xinchao Wang

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Dataset pruning (DP) and dataset distillation (DD) fundamentally differ in their outputs: DP selects original image subsets, while DD generates synthetic images. Recently, DD's increasing reliance on original images suggests a convergence of the two directions. To investigate this convergence trend, we propose a unified dataset compression (DC) benchmark. This benchmark reveals an interesting trade-off for soft-label-DD: while soft labels provide valuable information, they can make the distillation process less essential, as distilled images may not always outperform random subsets. In addition, the benchmark reveals that in current stages, dataset pruning outperforms dataset distillation at small dataset sizes. Given these observations, we explore hard-label-DC as a complementary approach that emphasizes image quality while offering substantial storage efficiency. Our PCA (Prune, Combine, and Augment) is the first framework that does not rely on soft labels but instead focuses on image quality. (1) "P'' means selecting easy samples based on dataset pruning metrics, (2) "C'' indicates combining these samples effectively, and (3) "A'' is to apply constrained image augmentation during training.

Poster
Beyond Continuity: Simulation-free Reconstruction of Discrete Branching Dynamics from Single-cell Snapshots

Junda Ying, Yuxuan Wang, Bowen Yang, Peijie Zhou, Lei Zhang

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Inferring cellular trajectories from destructive snapshots is complicated by the challenges of stochasticity and non-conservative mass dynamics such as cell proliferation and apoptosis. Existing unbalanced Optimal Transport (OT) methods treat mass as a continuous fluid, performing inference at the population level. However, this macroscopic view often fails to capture the discrete, jump-like nature of birth-death events at single-cell resolution, which is essential for understanding lineage branching and fate decisions. We present **Unbalanced Schrödinger Bridge (USB)**, a simulation-free framework for learning underlying dynamics that effectively integrates both stochastic and unbalanced effects which also models the discrete, jump-like birth–death dynamics at single-cell resolution. Theoretically, USB provides a tractable solution to the Branching Schrödinger Bridge (BSB) problem, offering a rigorous microscopic interpretation where individual cells undergo both Brownian motion and discrete birth-death jumps. Technically, the method implements an efficient solver by introducing a simulation-free training objective that effectively scales to high-dimensional omics data. Empirically, we demonstrate on both simulated and real-world datasets that USB not only achieves trajectory reconstruction performance better than or comparable to deterministic baselines but also uniquely enables realistic discrete simulation of birth-death dynamics at single-cell resolution.

Poster
GradPower: Powering Gradients for Faster Language Model Pre-Training

Jinbo Wang, Mingze Wang, Jiaqi Zhang, Wei Wang, Peng Pei, Xunliang Cai, Weinan E, Lei Wu

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We propose **GradPower**, a lightweight gradient-transformation technique for accelerating language model pre-training. Given a gradient vector $\boldsymbol{g}=(g_ {i})_ {i}$, GradPower first applies the elementwise `sign-power` transformation: $ \varphi_ p(\boldsymbol{g}) = \left({\rm sign}(g_ i)|g_ i|^p\right)_ {i} $ for a fixed $p>0$, and then feeds the transformed gradient into a base optimizer. Notably, GradPower requires only a **single-line code change** and no modifications to the base optimizer’s internal logic, including the hyperparameters. When applied to AdamW (termed **AdamWPower**), GradPower consistently achieves lower terminal loss across diverse architectures (LLaMA, Qwen2MoE), parameter scales (66M to 2B), datasets (C4, OpenWebText), and learning-rate schedules (cosine, warmup-stable-decay). The most pronounced gains are observed when training modern mixture-of-experts models with warmup-stable-decay schedules. GradPower also integrates seamlessly with other state-of-the-art optimizers, such as Muon, yielding further improvements. Finally, we provide theoretical analyses that reveal the underlying mechanism of GradPower and highlight the influence of gradient noise.

Poster
Adaptive Bandit Algorithms for Contextual Matching Markets

Shiyun Lin, Simon Mauras, Vianney Perchet, Nadav Merlis

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We study bandit learning in matching markets, where players and arms constitute the two market sides, and the players' utilities are linear in the arm contexts. In each round, new arms arrive with observable contexts. Then, the algorithm matches them to players, aiming to minimize each player's regret against a *stable matching benchmark*. This contextual structure creates significant complexity: subtle context shifts can slightly alter one player's utility while completely reconfiguring the underlying benchmark, causing large regret spikes for others. We address this in two settings: *stochastic* contexts, drawn from a latent distribution, and *adversarial* contexts, which may be arbitrary. For the stochastic case, we introduce a novel minimum preference gap to capture learning difficulty and provide a fully adaptive algorithm with an instance-dependent poly-logarithmic regret upper bound. We also establish matching instance-independent regret upper and lower bounds under a mild distributional assumption. For the adversarial setting, we propose a tractable regret notion that remains valid under arbitrary contexts and achieves an instance-independent sublinear regret bound via an adaptive algorithm.

Poster
MeshTok: Efficient Multi-Scale Tokenization for Scalable PDE Transformers

Yanshun Zhao, Xiaoyu Peng, Jiamin Jiang, Congcong Zhu, Jingrun Chen

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Conventional patchified Transformers operate on uniform spatial partitions, distributing computational effort evenly across the domain irrespective of local features. This inflexible tokenization scheme is inherently limited in its ability to efficiently represent and process solutions to complex PDEs. To address this, we propose MeshTok, an adaptive mesh refinement (AMR)-inspired tokenization and sequence modeling framework. This method selectively refines spatial regions exhibiting sharp gradients, transient features, or multiscale structures, generating a heterogeneous set of multiscale tokens defined on a fixed simulation grid. These tokens are processed within a unified Transformer sequence, enabling the model to simultaneously capture coarse-grained global context and fine-grained local details without requiring specialized architectural components. Although adaptive refinement moderately increases token count, it promotes a more targeted allocation of computational resources to physically informative regions, which we view as a practical inductive bias rather than a formal optimality guarantee. Experimental evaluations across multiple PDE families and benchmark datasets demonstrate that MeshTok consistently improves the efficiency-accuracy trade-off compared to uniform-grid baselines. This suggests adaptive multiscale tokenization as a scalable and generalizable design principle for neural PDE modeling. Code is available at https://github.com/SCAILab-USTC/MeshTok.

Poster
Conditional Quantile Adjusted Conformal Prediction for Time Series

Cheng Yu, Zhoufan Zhu, Ke Zhu

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Conformal prediction is challenging for time series with time-varying conditional distributions. Existing sequential conformal methods can yield volatile, non-nested prediction intervals due to noisy tail conditional quantile estimation and quantile crossing issue. To overcome these challenges, we construct prediction intervals for time series via a novel method called Conditional Quantile Adjusted Conformal Prediction (CQACP), which stabilizes sequential conformal calibration by modeling the conditional quantile curve of nonconformity score. At each time step, CQACP evaluates a base conditional quantile learner on a grid of quantile levels and fits a Cornish--Fisher approximation parameterized by conditional moments of nonconformity score with monotonicity constraints. Asymptotically, we prove the conditional validity of the prediction interval under serial dependence and show improved conditional quantile estimation accuracy. Experiments on multiple real-world datasets demonstrate that CQACP maintains accurate coverage and produces smooth, narrow, and nested prediction intervals across different significance levels and prediction models.

Poster
SSDCN: Spatial-Spectral Dual-Clustering-based Network for Hyperspectral Image Super-resolution

Yong Yang, Xuran Zhang, Shuying Huang, Xiaozheng Wang, Weiguo Wan, Hangyuan Lu

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Hyperspectral Image Single Image Super-Resolution (HSI-SISR) faces a conflict between computational efficiency and global non-local modeling. Existing Transformers suffer from quadratic complexity, while window-based methods compromise global capture. To address this, we propose the Spatial-Spectral Dual-Clustering-based Network (SSDCN). Our method introduces three innovations. First, we design a Spatial-Spectral Dual-Cluster Block (SSDCB). Replacing expensive point-to-point attention, it uses content-driven clustering to learn low-rank structural bases, achieving global modeling with linear complexity $\mathcal{O}(KN)$. Second, we propose a pyramid progressive hierarchical architecture with a Feature Reuse Reconstruction Block (FRRB). It reuses the core tensor and spectral factors from coarse levels, updating only spatial factors to minimize redundancy. Third, we propose a Pyramid Hierarchical Reconstruction Joint Loss to supervise intermediate levels, ensuring structural accuracy and preventing error accumulation. Experiments demonstrate that SSDCN surpasses SOTA methods in metrics and visual quality with significantly fewer parameters and FLOPs, achieving an optimal efficiency-performance balance.

Poster
Domain Transfer Becomes Identifiable via a Single Alignment

Sagar Shrestha, Subash Timilsina, Hoang-Son Nguyen, Xiao Fu

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Domain transfer (DT) maps source to target distributions and supports tasks such as unsupervised image-to-image translation, single-cell analysis, and cross-platform medical imaging. However, DT is fundamentally ill-posed: push-forward mappings are generally non-identifiable, as measure-preserving automorphisms (MPAs) preserve marginals while altering cross-domain correspondences, leading to content-misaligned translation. Recent work shows that MPAs can be eliminated by jointly transferring multiple corresponding source/target conditional distributions, but supervision signals labeling such conditionals are not always available in practice. We develop an alternative route to DT identifiability. Under a structural sparsity condition on the Jacobian support pattern, we show that distribution matching together with a single paired anchor sample suffices to identify the ground-truth transfer---requiring substantially less supervision than prior approaches. To enable practical high-dimensional learning, we further propose an efficient Jacobian sparsity regularizer based on randomized masked finite differences, yielding a scalable surrogate without explicit Jacobian evaluation. Empirical results on synthetic and real-world DT tasks validate the theory.

Poster
Robust-U1: Can MLLMs Self-Recover Corrupted Visual Content for Robust Understanding?

Jiaqi Tang, Jianmin Chen, Youyang Zhai, Wei Wei, Runtao Liu, Mengjie Zhao, Xiangyu Wu, Qingfa Xiao, Qifeng Chen

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Multimodal Large Language Models (MLLMs) have demonstrated remarkable success in visual understanding, yet their performance degrades significantly under real-world visual corruptions. While existing robustness enhancement approaches exist, they are limited: black-box feature alignment lacks interpretability, and white-box text-based reasoning cannot restore lost pixel-level details. This work investigates a fundamental research question: Can MLLMs recover corrupted visual content by themselves? To address this, we propose Robust-U1, a novel framework that equips MLLMs with explicit visual self-recovery capability for robust understanding. The approach comprises three core stages: supervised fine-tuning for initial reconstruction, reinforcement learning with dual rewards (pixel-level SSIM and semantic-level CLIP similarity) for aligning high visual quality, and multimodal reasoning that jointly considers both the corrupted input and the recovered image. Extensive experiments demonstrate that Robust-U1 achieves state-of-the-art robustness on the real-world corruption benchmark and maintains superior performance under adversarial corruptions on general VQA benchmarks. Analysis confirms that high-quality visual recovery directly enhances reasoning performance, establishing self-recovery as a critical mechanism for robust visual understanding. The source code is available at https://github.com/jqtangust/Robust-U1.

Poster
Scheduling LLM Inference with Uncertainty-Aware Output Length Predictions

Haoyu Zheng, Yongqiang Zhang, Fangcheng Fu, Xiaokai Zhou, Hao Luo, Hongchao Zhu, Yuanyuan Zhu, Hao Wang, Xiao Yan, Jiawei Jiang

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To schedule LLM inference, the \textit{shortest job first} (SJF) principle is favorable by prioritizing requests with short output lengths to avoid head-of-line (HOL) blocking. Existing methods usually predict a single output length for each request to facilitate scheduling. We argue that such a \textit{point estimate} does not match the \textit{stochastic} decoding process of LLM inference, where output length is \textit{uncertain} by nature and determined by when the end-of-sequence (EOS) token is sampled. Hence, the output length of each request should be fitted with a distribution rather than a single value. With an in-depth analysis of empirical data and the stochastic decoding process, we observe that output length follows a heavy-tailed distribution and can be fitted with the log-t distribution. On this basis, we propose a simple metric called Tail Inflated Expectation (TIE) to replace the output length in SJF scheduling, which adjusts the expectation of a log-t distribution with its tail probabilities to account for the risk that a request generates long outputs. To evaluate our TIE scheduler, we compare it with three strong baselines, and the results show that TIE reduces the per-token latency by $2.31\times$ for online inference and improves throughput by $1.42\times$ for offline data generation.

Poster
Feature Resemblance: Towards a Theoretical Understanding of Analogical Reasoning in Transformers

Ruichen Xu, Wenjing Yan, Ying-Jun Angela Zhang

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Understanding reasoning in large language models is complicated by evaluations that conflate multiple reasoning types. We isolate analogical reasoning, where a model transfers an attribute between entities that share known properties, and study when such transfer can emerge from training. To make the problem analytically tractable, we study a minimal transformer-style abstraction that isolates how learned representations support analogical reasoning. Within this setting, we prove three key results. First, joint training on similarity and attribution premises enables analogical reasoning through aligned representations. Second, sequential training succeeds only when similarity structure is learned before specific attributes, revealing a curriculum asymmetry. Third, in our stylized setting, two-hop reasoning $(a \to b, b \to c \Rightarrow a \to c)$ can be viewed as analogical reasoning with identity bridges $(b=b)$, which appear explicitly in training data. Together, these results reveal a unified mechanism: entities with shared properties become aligned in representation space, enabling property transfer through feature resemblance. Experiments with architectures up to 8B parameters show qualitative agreement with the theory and suggest that representational geometry plays an important role in analogical reasoning beyond the stylized model.

Poster
Decoupling The "What" and "Where" With Polar Coordinate Positional Embedding

Anand Gopalakrishnan, Róbert Csordás, Jürgen Schmidhuber, Michael Curtis Mozer

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The attention mechanism in a Transformer architecture matches key to query based on both content—the what—and position in a sequence—the where. We present an analysis indicating that what and where are entangled in the popular rotary position embedding (RoPE). This entanglement can impair performance particularly when decisions require independent matches on these two factors. We propose an improvement to RoPE, which we call Polar Coordinate Position Embedding or PoPE, that eliminates the what-where confound. PoPE is far superior on a diagnostic task requiring indexing solely by position or by content. On autoregressive sequence modeling in music, genomic, and natural language domains, Transformers using PoPE as the positional encoding scheme outperform baselines using RoPE with respect to evaluation loss (perplexity) and downstream task performance. On language modeling, these gains persist across model scale, from 124M to 774M parameters. Crucially, PoPE shows strong zero-shot length extrapolation capabilities compared not only to RoPE but even a method designed for extrapolation, YaRN, which requires additional fine tuning and frequency interpolation.

Spotlight
Speculative Sampling For Faster Molecular Dynamics

Arthur Kosmala, Stephan Günnemann, Meng Gao, Brandon M. Wood

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Molecular dynamics (MD) is a key tool for simulating the dynamical behavior of atomic systems. However, MD is inherently serial, which makes it difficult to increase single-system throughput with concurrent compute. To address this, we introduce **L**angevin **S**peculative **D**ynamics (**LSD**), a distributed and model-agnostic speculative sampler for accelerating MD *without adding relative error*. Inspired by speculative methods in language and diffusion modeling, LSD uses a draft model to propose fast simulation steps and verifies them in parallel with a slower target model, applying a transport map from the draft to the target distribution. We extend speculative sampling to second-order Langevin dynamics, derive the achievable speedup as a function of physical parameters, show that LSD generalizes across different systems and draft-target combinations with a 3-9x speedup, and confirm theoretically and empirically that LSD samples trajectories from the same distribution as its target model.

Poster
Mobility-Embedded POIs: Learning What A Place Is and How It Is Used from Human Movement

Maria Despoina Siampou, Shushman Choudhury, Shang-Ling Hsu, Neha Arora, Cyrus Shahabi

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Recent progress in geospatial foundation models highlights the importance of learning general-purpose representations for real-world locations, particularly points-of-interest (POIs) where human activity concentrates. Existing approaches, however, focus primarily on place identity derived from static textual metadata, or learn representations tied to trajectory context, which capture movement regularities rather than how places are actually used (i.e., POI's function). We argue that POI function is a missing but essential signal for general POI representations. We introduce Mobility-Embedded POIs (ME-POIs), a framework that augments POI embeddings derived, from language models with large-scale human mobility data to learn POI-centric, context-independent representations grounded in real-world usage. ME-POIs encodes individual visits as temporally contextualized embeddings and aligns them with learnable POI representations via contrastive learning to capture usage patterns across users and time. To address long-tail sparsity, we propose a novel mechanism that propagates temporal visit patterns from nearby, frequently visited POIs across multiple spatial scales. We evaluate ME-POIs on five newly proposed map enrichment tasks, testing its ability to capture both the identity and function of POIs. Across all tasks, augmenting text-based embeddings with ME-POIs consistently outperforms both text-only and mobility-only baselines. Notably, ME-POIs trained on mobility data alone can surpass text-only models on certain tasks, highlighting that POI function is a critical component of accurate and generalizable POI representations.

Poster