论文检索

General Machine Learning · Representation Learning

Yiwen Wang, Diana Benavides-Prado, Yun Sing Koh

Continual learning (CL) aims to continuously learn new tasks from data streams. While most CL research focuses on mitigating catastrophic forgetting, memorizing outdated knowledge can cause negative transfer, where irrelevant prior knowledge interferes with new task learning and impairs adaptability. Inspired by how the human brain selectively unlearns unimportant information to prioritize learning and to recall relevant knowledge, we explore the intuition that effective CL should not only preserve but also selectively unlearn prior knowledge that hinders adaptation. We introduce DEtect, Decide, Unlearn in Continual lEarning (DEDUCE), a novel CL framework that dynamically detects negative transfer and mitigates it by a hybrid unlearning mechanism. Specifically, we investigate two complementary negative transfer detection strategies: transferability bound and gradient conflict analysis. Based on this detection, the model decides whether to activate a Local Unlearning Module (LUM) to filter outdated knowledge before learning new task. Additionally, a Global Unlearning Module (GUM) periodically reclaims model capacity to enhance plasticity. Our experiments demonstrate that DEDUCE effectively mitigates task interference and improves overall accuracy with an average gain of up to 4.55\% over state-of-the-art baselines.

Theory · Interpretability and Visualization

Shahaf Bassan, Yizhak Elboher, Tobias Ladner, Volkan Şahin, Jan Kretinsky, Matthias Althoff, Guy Katz

Despite significant progress in post-hoc explanation methods for neural networks, many remain heuristic and lack provable guarantees. A key approach for obtaining explanations with provable guarantees is by identifying a cardinally-minimal subset of input features which by itself is provably sufficient to determine the prediction. However, for standard neural networks, this task is often computationally infeasible, as it demands a worst-case exponential number of verification queries in the number of input features, each of which is NP-hard. In this work, we show that for Neural Additive Models (NAMs), a recent and more interpretable neural network family, we can efficiently generate explanations with such guarantees. We present a new model-specific algorithm for NAMs that generates provably cardinally-minimal explanations using only a logarithmic number of verification queries in the number of input features, after a parallelized preprocessing step with logarithmic runtime in the required precision is applied to each small univariate NAM component. Our algorithm not only makes the task of obtaining cardinally-minimal explanations feasible, but even outperforms existing algorithms designed to find the relaxed variant of subset-minimal explanations - which may be larger and less informative but easier to compute - despite our algorithm solving a much more difficult task. Our experiments demonstrate that, compared to previous algorithms, our approach provides provably smaller explanations than existing works and substantially reduces the computation time. Moreover, we show that our generated provable explanations offer benefits that are unattainable by standard sampling-based techniques typically used to interpret NAMs.

Theory · Interpretability and Visualization

Lukas Koller, Tobias Ladner, Matthias Althoff

Neural networks are ubiquitous. However, they are often sensitive to small input changes. Hence, to prevent unexpected behavior in safety-critical applications, their formal verification -- a notoriously hard problem -- is necessary. Many state-of-the-art verification algorithms use reachability analysis or abstract interpretation to enclose the set of possible outputs of a neural network. Often, the verification is inconclusive due to the conservatism of the enclosure. To address this problem, we propose a novel specification-driven input refinement procedure, i.e., we iteratively enclose the preimage of a neural network for all unsafe outputs to reduce the set of possible inputs to only enclose the unsafe ones. For that, we transfer output specifications to the input space by exploiting a latent space, which is an artifact of the propagation of a projection-based set representation through a neural network. A projection-based set representation, e.g., a zonotope, is a "shadow" of a higher-dimensional set -- a latent space -- that does not change during a set propagation through a neural network. Hence, the input set and the output enclosure are "shadows" of the same latent space that we can use to transfer constraints. We present an efficient verification tool for neural networks that uses our iterative refinement to significantly reduce the number of subproblems in a branch-and-bound procedure. Using zonotopes as a set representation, unlike many other state-of-the-art approaches, our approach can be realized by only using matrix operations, which enables a significant speed-up through efficient GPU acceleration. We demonstrate that our tool achieves competitive performance compared to the top-ranking tools of the international neural network verification competition.

General Machine Learning · Representation Learning

Jiehui Luo, Yuguo Yin, Yuxin Xie, Jinghan Ru, Xianwei Zhuang, Minghua He, Aofan Liu, Zihan Xiong, Dongchao Yang

Contrastive language-audio pretraining, which aims to unify multimodal representations in a shared embedding space, serves as a cornerstone for building a wide range of applications, from cross-modal retrieval to cutting-edge multimodal large language models. However, we find that the perpendicular component of the pushing force from negative samples in contrastive learning is a double-edged sword: it contains rich supplementary information from negative samples, yet its unconstrained nature causes optimization trajectory drift and training instability. To address this, we propose Support Vector Regularization (SVR), a method that introduces an auxiliary support vector to control this perpendicular component, aiming to harness its rich information while mitigating the associated trajectory drift. The efficacy of SVR is critically governed by its semantic radius, for which we explore two unsupervised modeling strategies: direct parameterization and an adaptive radius predictor module enhanced with constraints to improve its predicting accuracy. Extensive experimental results demonstrate that our method surpasses widely used baselines like InfoNCE and SigLIP loss across classification, monolingual retrieval, and multilingual retrieval on standard audio-text datasets. Both the theoretical analysis and the experimental results on optimizing trajectory drift validate the correctness and effectiveness of our SVR method. Notably, our method is highly efficient, it operates without the need for extra training data or inference computation, and adds only a negligible overhead to the training.

General Machine Learning · Causality

Haoyue Dai, Immanuel Albrecht, Peter Spirtes, Kun Zhang

Causal discovery with latent variables is a fundamental task. Yet most existing methods rely on strong structural assumptions, such as enforcing specific indicator patterns for latents or restricting how they can interact with others. We argue that a core obstacle to a general, structural-assumption-free approach is the lack of an equivalence characterization: without knowing what can be identified, one generally cannot design methods for how to identify it. In this work, we aim to close this gap for linear non-Gaussian models. We establish the graphical criterion for when two graphs with arbitrary latent structure and cycles are distributionally equivalent, that is, they induce the same observed distribution set. Key to our approach is a new tool, edge rank constraints, which fills a missing piece in the toolbox for latent-variable causal discovery in even broader settings. We further provide a procedure to traverse the whole equivalence class and develop an algorithm to recover models from data up to such equivalence. To our knowledge, this is the first equivalence characterization with latent variables in any parametric setting without structural assumptions, and hence the first structural-assumption-free discovery method. Code and an interactive demo are available at https://equiv.cc.

General Machine Learning · Representation Learning

Panagiotis Alimisis, Christos Diou

Causal representation learning has attracted significant research interest during the past few years, as a means for improving model generalization and robustness. Causal representations of interventional image pairs (also called ``actionable counterfactuals'' in the literature), have the property that only variables corresponding to scene elements affected by the intervention / action are changed between the start state and the end state. While most work in this area has focused on identifying and representing the variables of the scene under a causal model, fewer efforts have focused on representations of the interventions themselves. In this work, we show that an effective strategy for improving out of distribution (OOD) robustness is to focus on the representation of actionable counterfactuals in the latent space. Specifically, we propose that an intervention can be represented by a Causal Delta Embedding that is invariant to the visual scene and sparse in terms of the causal variables it affects. Leveraging this insight, we propose a method for learning causal representations from image pairs, without any additional supervision. Experiments in the Causal Triplet challenge demonstrate that Causal Delta Embeddings are highly effective in OOD settings, significantly exceeding baseline performance in both synthetic and real-world benchmarks.

Bo Liu, Simon Yu, Zichen Liu, Leon Guertler, Penghui Qi, Daniel Balcells, Mickel Liu, Cheston Tan, Weiyan Shi, Min Lin 等

Recent advances in reinforcement learning have shown that language models can develop sophisticated reasoning through training on tasks with verifiable rewards, but these approaches depend on human-curated problem-answer pairs and domain-specific reward engineering. We introduce SPIRAL, a self-play framework where models learn by playing multi-turn, zero-sum games against continuously improving versions of themselves, generating an automatic curriculum of stronger opponents, and eliminating the need for human supervision. To enable this self-play training at scale, we implement a fully online, multi-turn, multi-agent reinforcement learning system for LLMs and propose role-conditioned advantage estimation (RAE) to stabilize multi-agent training. SPIRAL produces reasoning capabilities that transfer broadly, improving performance by up to 10\% across a suite of 8 reasoning benchmarks on 4 different models spanning Qwen and Llama model families, outperforming supervised fine-tuning on 25,000 expert game trajectories. Multi-game training (TicTacToe, Kuhn Poker, Simple Negotiation) yields the strongest results, with improvements observed across both base and instruction-tuned models. Analysis of chain-of-thought traces reveals that games develop distinct cognitive patterns that transfer to improve reasoning performance, with different games developing complementary strengths. Even models which have already been trained on reasoning tasks using RLVR, like DeepSeek-R1-Distill-Qwen-7B, still benefit from our approach. These results demonstrate that zero-sum games naturally develop transferable reasoning capabilities across diverse model architectures and training stages, highlighting a promising direction for autonomous reasoning development.

General Machine Learning · Representation Learning

Guan-Yi Jhang, Jeng-Chung Lien, Yu Hui-Ching, Hsu-Chao Lai, Jiun-Long Huang

We present the TRIplet-based Distance-preserving Embedding Network for Trajectories (TRIDENT), a spatio-temporal representation framework for compressing and retrieving trajectories across scales, from badminton courts to large-scale urban environments. Existing methods often assume smooth, continuous motion, but real trajectories exhibit event-driven annotation, abrupt direction changes, GPS errors, irregular sampling, and domain shifts, exposing the inefficiency, limited generalization, and inability to robustly integrate temporal order with spatial sequence structure of prior models. TRIDENT addresses these challenges by combining Graph Convolutional Network (GCN) spatial embeddings with temporal features in a Dual-Attention Encoder (DAEncoder), along with a Nonlinear Tanh-Projection Attention Pooling (NTAP) module that preserves local order and robustness under noise. For metric learning, we introduce a Distance-preserving Multi-kernel Triplet Loss (DMT) to preserve pairwise spatio-temporal distances in the native feature space and their rank order within the embedding, thereby reducing geometry distortion and improving cross-domain generalization. Experiments on urban mobility and badminton datasets show that TRIDENT outperforms strong baselines in retrieval accuracy, efficiency, and cross-domain generalization. Furthermore, the learned embeddings capture spatio-temporal sequence patterns, facilitating tactical analysis of badminton rallies via silhouette-guided spectral clustering that provides more actionable insights than direct trajectory classification.

Simon Yu, Gang Li, Weiyan Shi, Peng Qi

Large language models (LLMs) are moving beyond static uses and are now powering agents that learn during their interaction with external environments. For example, agents can learn reusable skills while navigating web pages or toggling new tools. However, existing methods for skill learning often create skills that are over-specialized to a single website and fail to generalize. We introduce PolySkill, a new framework that enables agents to learn generalizable and compositional skills. The core idea, inspired by polymorphism in software engineering, is to decouple a skill's abstract goal (*what* it accomplishes) and its concrete implementation (*how* it is executed). Experiments show that our method (1) improves skill reuse by 1.7x on seen websites and (2) boosts success rates by up to 9.4\% on Mind2Web and 13.9\% on unseen websites, while reducing steps by over 20\%. (3) In self-exploration settings without specified tasks, our framework improves the quality of proposed tasks and enables agents to learn generalizable skills that work across different sites. By enabling the agent to identify and refine its own goals, the PolySkill enhance the agent a better curriculum, leading to the acquisition of more generalizable skills compared to baseline methods. Our findings show that separating a skill's goal from its execution is a crucial step toward developing autonomous agents that can learn and generalize across the open web continuously. Our code can be found in \href{https://github.com/simonucl/PolySkill}{\texttt{https://github.com/simonucl/PolySkill}}.

Deep Learning · Generative Models and Autoencoders

Xinpeng Li, Enming Liang, Minghua Chen

Generative models, particularly diffusion and flow-matching approaches, have achieved remarkable success across diverse domains, including image synthesis and robotic planning. However, a fundamental challenge persists: ensuring generated samples strictly satisfy problem-specific constraints — a crucial requirement for physics-informed problems, safety-critical applications, watermark embedding, etc. Existing approaches, such as mirror maps and reflection methods, either have limited applicable constraint sets or introduce significant computational overhead. In this paper, we develop gauge flow matching (GFM), a simple yet efficient framework for constrained generative modeling. Our GFM approach introduces a novel bijective gauge mapping to transform generation over arbitrary compact convex sets into an equivalent process over the unit ball, which allows low-complexity feasibility-ensuring operations such as reflection or projection. The generated samples are then mapped back to the original domain for output. We prove that our GFM framework guarantees strict constraint satisfaction, with low generation complexity and bounded distribution approximation errors. We further extend our GFM framework to two non-convex settings, namely, star-convex and geodesic-convex sets. Extensive experiments demonstrate that GFM outperforms existing methods in both generation speed and quality across multiple benchmarks, including synthetic data, time series, and image generation.

Social Aspects · Trustworthy Machine Learning

Amira Guesmi, Muhammad Shafique

Deep neural networks remain highly vulnerable to adversarial examples, and most defenses collapse once gradients can be reliably estimated. We identify \emph{gradient consensus}—the tendency of randomized transformations to yield aligned gradients—as a key driver of adversarial transferability. Attackers exploit this consensus to construct perturbations that remain effective across transformations. We introduce \textbf{DRIFT} (Divergent Response in Filtered Transformations), a stochastic ensemble of lightweight, learnable filters trained to actively disrupt gradient consensus. Unlike prior randomized defenses that rely on gradient masking, DRIFT enforces \emph{gradient dissonance} by maximizing divergence in Jacobian- and logit-space responses while preserving natural predictions. Our contributions are threefold: (i) we formalize gradient consensus and provide a theoretical analysis linking consensus to transferability; (ii) we propose a consensus-divergence training strategy combining prediction consistency, Jacobian separation, logit-space separation, and adversarial robustness; and (iii) we show that DRIFT achieves substantial robustness gains on ImageNet across CNNs and Vision Transformers, outperforming state-of-the-art preprocessing, adversarial training, and diffusion-based defenses under adaptive white-box, transfer-based, and gradient-free attacks. DRIFT delivers these improvements with negligible runtime and memory cost, establishing gradient divergence as a practical and generalizable principle for adversarial defense.

Theory · Domain Adaptation and Transfer Learning

Aël Quélennec, Nour Hezbri, Pavlo Mozharovskyi, Van-Tam Nguyen, Enzo Tartaglione

Memory-efficient training of deep neural networks has become increasingly important as models grow larger while deployment environments impose strict resource constraints. We propose TraDy, a novel transfer learning scheme leveraging two key insights: layer importance for updates is architecture-dependent and determinable a priori, while dynamic stochastic channel selection provides superior gradient approximation compared to static approaches. We introduce a dynamic channel selection approach that stochastically resamples channels between epochs within preselected layers. Extensive experiments demonstrate TraDy achieves state-of-the-art performance across various downstream tasks and architectures while maintaining strict memory constraints, achieving up to 99\% activation sparsity, 95\% weight derivative sparsity, and 97\% reduction in FLOPs for weight derivative computation.

David Ma, Yuanxing Zhang, JinCheng Ren, Jiawei Guo, Yifan Yao, Zhenlin Wei, Zhenzhu Yang, Zhongyuan Peng, Boyu Feng, Jun Ma 等

Current benchmarks for Multimodal Large Language Models (MLLMs) predominantly rely on text-only queries, overlooking the essential role of images as visual context for enhancing video comprehension and facilitating natural human-AI interaction. To bridge this gap, we introduce \textbf{IV-Bench}, the first comprehensive benchmark for evaluating MLLMs on Image-Grounded Video Perception and Reasoning. IV-Bench comprises 966 videos paired with 2,560 meticulously annotated image-text queries across 13 tasks (7 perception and 6 reasoning tasks) spanning 5 distinct categories. We extensively evaluate state-of-the-art MLLMs, including open-source models (e.g., InternVL2.5, Qwen2.5-VL) and closed-source models (e.g., GPT-4o, Gemini2.0 series), revealing substantial performance gaps, with the best-performing model achieving only 28.9\% accuracy. Ablation studies demonstrate that incorporating images significantly enhances video understanding and highlight key model design factors influencing performance. Our findings provide valuable insights and guidance for future research. The code and dataset are available at \url{https://github.com/multimodal-art-projection/IV-Bench}.

Social Aspects · Trustworthy Machine Learning

Amira Guesmi, Bassem Ouni, Muhammad Shafique

Quantized Neural Networks (QNNs) are widely deployed in edge and resource-constrained environments for their efficiency in computation and memory. While quantization distorts gradient landscapes and weakens pixel-level attacks, it offers limited robustness against patch-based adversarial attacks—localized, high-saliency perturbations that remain highly transferable across bit-widths. Existing defenses either overfit to specific quantization settings or fail to address this cross-bit vulnerability. We propose \textbf{TriQDef}, a tri-level quantization-aware defense framework that disrupts the transferability of patch-based attacks across QNNs. TriQDef integrates: (1) a \emph{Feature Disalignment Penalty (FDP)} that enforces semantic inconsistency by penalizing perceptual similarity in intermediate features; (2) a \emph{Gradient Perceptual Dissonance Penalty (GPDP)} that misaligns input gradients across quantization levels using structural metrics such as Edge IoU and HOG Cosine; and (3) a \emph{Joint Quantization-Aware Training Protocol} that applies these penalties within a \emph{shared backbone} jointly optimized across multiple quantizers. Extensive experiments on CIFAR-10 and ImageNet show that TriQDef lowers Attack Success Rates (ASR) by over 40\% on unseen patch and quantization combinations while preserving high clean accuracy. These results highlight the importance of disrupting both semantic and perceptual gradient alignment to mitigate patch transferability in QNNs.

Computer Vision · Vision Models & Multimodal

David Ma, Huaqing Yuan, Xingjian Wang, Qianbo ZANG, Tianci Liu, Xinyang He, Yanbin Wei, Jiawei Guo, nijiahui, Zhenzhu Yang 等

Although long-video understanding demands that models capture hierarchical temporal information—from clip and shot to event and story—existing benchmarks either neglect this multi-scale design or scatter scale-specific questions across different videos, preventing direct comparison of model performance across timescales on the same content. To address this, we introduce ScaleLong, the first benchmark to disentangle these factors by embedding questions targeting four hierarchical timescales\textemdash clip, shot, event, and story\textemdash all within the same video content. This within-content multi-timescale questioning design enables direct comparison of model performance across timescales on identical videos. ScaleLong features 269 long videos (avg. 86 min) from 5 main categories and 36 sub-categories, with 4–8 carefully designed questions, with at least one question targeting each timescale. Evaluating 23 MLLMs reveals a distinct U-shaped performance trend: higher accuracy at the shortest (clip) and longest (story) timescales, with a dip at intermediate levels. Furthermore, ablation studies demonstrate that increased visual token capacity consistently enhances reasoning across all timescales. ScaleLong offers a crucial fine-grained, multi-timescale benchmark for advancing MLLM capabilities in long-video understanding. The code and dataset are available at \url{https://github.com/multimodal-art-projection/ScaleLong}

Reinforcement Learning · Multi-agent

Arjun Manoj, Rijul Tandon, Agam Gupta, HARIPRASAD KODAMANA, Manojkumar Charandas Ramteke

Mixed-Integer Programming (MIP) solvers rely heavily on cutting planes to tighten LP relaxations, but traditional approaches generate thousands of cuts that consume gigabytes of memory while providing minimal benefit. We present an intelligent cut selection framework that achieves a 98.1\% reduction in memory usage while maintaining competitive solving with an objective gap of approximately 0.08\%. Within this RL framework, we use Proximal Policy Optimization (PPO) to learn a behavioral model that imitates the expert solver’s decisions. The adversarially imitated behavioral model drives an agent comprising these key innovations: (i) a cut-selection policy trained via curriculum learning; and (ii) adaptive inference that dynamically adjusts computational budgets. Through comprehensive evaluation across SetCover and diverse MIPLIB problems, we demonstrate consistent speedups (3.78$\times$ average on MIPLIB) and achieve a 100\% success rate on instances where traditional SCIP fails 47-53\% of the time. Our method also reduces peak memory consumption from 3.03GB to 46 MB, enabling optimization in previously inaccessible and other resource-constrained environments where traditional solvers face fundamental limitations.

Applications · Everything Else

Chunyang Li, Yilun Zheng, Xinting Huang, Tianqing Fang, Jiahao Xu, Lihui Chen, Yangqiu Song, Winston Hu

The paradigm of LLM-as-a-judge is emerging as a scalable and efficient alternative to human evaluation, demonstrating strong performance on well-defined tasks. However, its reliability in open-ended tasks with dynamic environments and complex interactions remains unexplored. To bridge the gap, we introduce WebDevJudge, a systematic benchmark for assessing LLM-as-a-judge performance in web development, with support for both non-interactive evaluation based on static observations and continuous interactive evaluation with a dynamic web environment. WebDevJudge comprises human preference labels over paired web implementations, annotated with structured and query-grounded rubrics to ensure high-quality ground truth. Using this benchmark, we comprehensively evaluate various evaluators, including LLMs, MLLMs, and agentic workflows. We systematically investigate the impact of different paradigms and guidance mechanisms. Our experiments reveal a significant gap between LLM judges and human experts. In-depth analysis indicates this gap stems from fundamental model limitations, including failures in recognizing functional equivalence, verifying task feasibility, and mitigating bias. Overall, WebDevJudge presents a significant challenge to LLM-as-a-judge, offering insights to guide future research toward developing more reliable and capable automated evaluators for complicated scenarios.

Applications · Everything Else

Victor Baillet, Yuanzhang Xiao, Nicolás Astorga, Mihaela van der Schaar

Autoformulation is an emerging field that uses large language models (LLMs) to translate natural-language descriptions of decision-making problems into formal mathematical formulations. Existing works have focused on autoformulating mathematical optimization problems for $\textit{one-shot}$ decision-making. However, many real-world decision-making problems are $\textit{sequential}$, best modeled as $\textit{Markov decision processes}$ (MDPs). MDPs introduce unique challenges for autoformulation, including a significantly larger formulation search space, and for computing and interpreting the optimal policy. In this work, we address these challenges in the context of queueing problems---central to domains such as healthcare and logistics---which often require substantial technical expertise to formulate correctly. We propose a novel operator-theoretic autoformulation framework using LLMs. Our approach captures the underlying decision structure of queueing problems through constructing the Bellman equation as a graph of $\textit{operators}$, where each operator is an $\textit{interpretable}$ transformation of the value function corresponding to certain $\textit{event}$ (e.g., arrival, departure, routing). Theoretically, we prove a universal three-level operator-graph topology covering a broad class of MDPs, significantly shrinking the formulation search space. Algorithmically, we propose customized Monte Carlo tree search to build operator graphs while incorporating self-evaluation, solver feedback, and intermediate syntax checking for early assessment, and present a provably low-complexity algorithm that automatically identifies structures of the optimal policy (e.g., threshold-based), accelerating downstream solving. Numerical results demonstrate the effectiveness of our approach in formulating queueing problems and identifying structural results.

Applications · Robotics

Yicheng Liu, Shiduo Zhang, Zibin Dong, Baijun Ye, Tianyuan Yuan, Xiaopeng Yu, Linqi Yin, Chenhao Lu, Junhao Shi, Luca Jiang-Tao Yu 等

Autoregressive vision-language-action (VLA) models have recently demonstrated strong capabilities in robotic manipulation. However, their core process of action tokenization often involves a trade-off between reconstruction fidelity and inference efficiency. We introduce \textbf{FASTer}, a unified framework for efficient and generalizable robot learning that integrates a learnable tokenizer with an autoregressive policy built upon it. FASTerVQ encodes action chunks as single-channel images, capturing global spatio-temporal dependencies while maintaining a high compression ratio. FASTerVLA builds on this tokenizer with block-wise autoregressive decoding and a lightweight action expert, achieving both faster inference and higher task performance. Extensive experiments across simulated and real-world benchmarks show that FASTerVQ delivers superior reconstruction quality, high token utilization, and strong cross-task and cross-embodiment generalization, while FASTerVLA further improves overall capability, surpassing previous state-of-the-art VLA models in both inference speed and task performance.

Deep Learning · Generative Models and Autoencoders

Hao Luan, Yi Xian Goh, See-Kiong Ng, Chun Kai Ling

Modifications to test-time sampling have emerged as an important extension to diffusion algorithms, with the goal of biasing the generative process to achieve a given objective without having to retrain the entire diffusion model. However, generating jointly correlated samples from multiple pre-trained diffusion models while simultaneously enforcing task-specific constraints without costly retraining has remained challenging. To this end, we propose Projected Coupled Diffusion (PCD), a novel test-time framework for constrained joint generation. PCD introduces a coupled guidance term into the generative dynamics to encourage coordination between diffusion models and incorporates a projection step at each diffusion step to enforce hard constraints. Empirically, we demonstrate the effectiveness of PCD in application scenarios of image-pair generation, object manipulation, and multi-robot motion planning. Our results show improved coupling effects and guaranteed constraint satisfaction without incurring excessive computational costs.