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Probabilistic Methods

Stanislas Strasman, Sobihan Surendran, Claire Boyer, Sylvain Le Corff, Vincent Lemaire, Antonio Ocello

Score-based Generative Models (SGMs) have achieved impressive performance in data generation across a wide range of applications and benefit from strong theoretical guarantees. Recently, methods inspired by statistical mechanics, in particular, Hamiltonian dynamics, have introduced Critically-damped Langevin Diffusions (CLDs), which define diffusion processes on extended spaces by coupling the data with auxiliary variables. These approaches, along with their associated score-matching and sampling procedures, have been shown to outperform standard diffusion-based samplers numerically. In this paper, we analyze a generalized dynamic that extends classical CLDs by introducing an additional hyperparameter controlling the noise applied to the data coordinate, thereby better exploiting the extended space. We further derive a novel upper bound on the sampling error of CLD-based generative models in the Wasserstein metric. This additional hyperparameter influences the smoothness of sample paths, and our discretization error analysis provides practical guidance for its tuning, leading to improved sampling performance.

Probabilistic Methods

Yinjie Min, Chuchen Zhang, Liuhua Peng, Changliang Zou

Personalized federated learning addresses data heterogeneity across distributed agents but lacks uncertainty quantification that is both agent-specific and instance-specific, which is a critical requirement for risk-sensitive applications. We propose personalized federated conformal prediction (PFCP), a novel framework that combines personalized federated learning with conformal prediction to provide statistically valid agent-personalized prediction sets with instance-localization. By leveraging privacy-preserving knowledge transfer from other source agents, PFCP ensures marginal coverage guarantees for target agents while significantly improving conditional coverage performance on individual test instances, which has been validated by extensive experiments.

Deep Learning · Algorithms

Shuli Zhang, Hao Zhou, Jiaqi Zheng, Guibin Jiang, Cheng Bing, Wei Lin, Guihai Chen

Online Internet platforms require sophisticated marketing strategies to optimize user retention and platform revenue — a classical resource allocation problem. Traditional solutions adopt a two-stage pipeline: machine learning (ML) for predicting individual treatment effects to marketing actions, followed by operations research (OR) optimization for decision-making. This paradigm presents two fundamental technical challenges. First, the prediction-decision misalignment: Conventional ML methods focus solely on prediction accuracy without considering downstream optimization objectives, leading to improved predictive metrics that fail to translate to better decisions. Second, the bias-variance dilemma: Observational data suffers from multiple biases (e.g., selection bias, position bias), while experimental data (e.g., randomized controlled trials), though unbiased, is typically scarce and costly --- resulting in high-variance estimates. We propose **Bi**-level **D**ecision-**F**ocused **C**ausal **L**earning (**Bi-DFCL**) that systematically addresses these challenges. First, we develop an unbiased estimator of OR decision quality using experimental data, which guides ML model training through surrogate loss functions that bridge discrete optimization gradients. Second, we establish a bi-level optimization framework that jointly leverages observational and experimental data, solved via implicit differentiation. This novel formulation enables our unbiased OR estimator to correct learning directions from biased observational data, achieving optimal bias-variance tradeoff. Extensive evaluations on public benchmarks, industrial marketing datasets, and large-scale online A/B tests demonstrate the effectiveness of Bi-DFCL, showing statistically significant improvements over state-of-the-art. Currently, Bi-DFCL has been deployed across several marketing scenarios at Meituan, one of the largest online food delivery platforms in the world.

General Machine Learning · Everything Else

Jiawei Gu, Yidi Wang, Qingqiang Sun, Xinming Li, Xiao Luo, Ziyue Qiao

Pseudo-label-based Semi-Supervised Learning (SSL) often suffers from classifier bias, particularly under class imbalance, as inaccurate pseudo-labels tend to exacerbate existing biases towards majority classes. Existing methods, such as \textit{CDMAD}\cite{cdmad}, utilize simplistic reference inputs—typically uniform or blank-colored images—to estimate and correct this bias. However, such simplistic references fundamentally ignore realistic statistical information inherent to real datasets, specifically typical color distributions, texture details, and frequency characteristics. This lack of \emph{statistical representativeness} can lead the model to inaccurately estimate its inherent bias, limiting the effectiveness of bias correction, particularly under severe class imbalance or substantial distribution mismatches between labeled and unlabeled datasets. To overcome these limitations, we introduce the \textbf{FARAD} (Fourier-Adapted Reference for Accurate Debiasing) System. This system utilizes random-phase images, constructed by preserving the amplitude spectrum of real data while randomizing the phase spectrum. This strategy ensures two critical properties: (1) \textbf{Semantic Irrelevance}, as randomizing phase removes any structural or recognizable semantic cues, and (2) \textbf{Statistical Representativeness}, as preserving the amplitude spectrum maintains realistic textures, color distributions, and frequency characteristics. Grounded theoretically in classical Fourier analysis, the FARAD System provides a robust, accurate estimation of per-class biases. Furthermore, computational efficiency is enhanced through optimized real-to-complex (R2C) batched Fast Fourier Transforms (FFTs). Comprehensive experiments demonstrate that our approach, significantly improves minority-class accuracy and overall SSL performance, particularly under challenging imbalance scenarios, compared with existing reference-based bias correction methods.

Applications · Chemistry and Drug Discovery

Jigang Fan, Quanlin Wu, Shengjie Luo, Liwei Wang

The detection of ligand binding sites for proteins is a fundamental step in Structure-Based Drug Design. Despite notable advances in recent years, existing methods, datasets, and evaluation metrics are confronted with several key challenges: (1) current datasets and methods are centered on individual protein–ligand complexes and neglect that diverse binding sites may exist across multiple complexes of the same protein, introducing significant statistical bias; (2) ligand binding site detection is typically modeled as a discontinuous workflow, employing binary segmentation and subsequent clustering algorithms; (3) traditional evaluation metrics do not adequately reflect the actual performance of different binding site prediction methods. To address these issues, we first introduce UniSite-DS, the first UniProt (Unique Protein)-centric ligand binding site dataset, which contains 4.81 times more multi-site data and 2.08 times more overall data compared to the previously most widely used datasets. We then propose UniSite, the first end-to-end ligand binding site detection framework supervised by set prediction loss with bijective matching. In addition, we introduce Average Precision based on Intersection over Union (IoU) as a more accurate evaluation metric for ligand binding site prediction. Extensive experiments on UniSite-DS and several representative benchmark datasets demonstrate that IoU-based Average Precision provides a more accurate reflection of prediction quality, and that UniSite outperforms current state-of-the-art methods in ligand binding site detection. The dataset and codes will be made publicly available at https://github.com/quanlin-wu/unisite.

Deep Learning · Generative Models and Autoencoders

Zeqi Ye, Minshuo Chen

Imputation methods play a critical role in enhancing the quality of practical time-series data, which often suffer from pervasive missing values. Recently, diffusion-based generative imputation methods have demonstrated remarkable success compared to autoregressive and conventional statistical approaches. Despite their empirical success, the theoretical understanding of how well diffusion-based models capture complex spatial and temporal dependencies between the missing values and observed ones remains limited. Our work addresses this gap by investigating the statistical efficiency of conditional diffusion transformers for imputation and quantifying the uncertainty in missing values. Specifically, we derive statistical sample complexity bounds based on a novel approximation theory for conditional score functions using transformers, and, through this, construct tight confidence regions for missing values. Our findings also reveal that the efficiency and accuracy of imputation are significantly influenced by the missing patterns. Furthermore, we validate these theoretical insights through simulation and propose a mixed-masking training strategy to enhance the imputation performance.

Computer Vision · Segmentation

Haijing Liu, Zhiyuan Song, Hefeng Wu, Tao Pu, Keze Wang, Liang Lin

Egocentric Referring Video Object Segmentation (Ego-RVOS) aims to segment the specific object actively involved in a human action, as described by a language query, within first-person videos. This task is critical for understanding egocentric human behavior. However, achieving such segmentation robustly is challenging due to ambiguities inherent in egocentric videos and biases present in training data. Consequently, existing methods often struggle, learning spurious correlations from skewed object-action pairings in datasets and fundamental visual confounding factors of the egocentric perspective, such as rapid motion and frequent occlusions. To address these limitations, we introduce Causal Ego-REferring Segmentation (CERES), a plug-in causal framework that adapts strong, pre-trained RVOS backbones to the egocentric domain. CERES implements dual-modal causal intervention: applying backdoor adjustment principles to counteract language representation biases learned from dataset statistics, and leveraging front-door adjustment concepts to address visual confounding by intelligently integrating semantic visual features with geometric depth information guided by causal principles, creating representations more robust to egocentric distortions. Extensive experiments demonstrate that CERES achieves state-of-the-art performance on Ego-RVOS benchmarks, highlighting the potential of applying causal reasoning to build more reliable models for broader egocentric video understanding.

Computer Vision · Image and Video Generation

Yuxiang Bao, Huijie Liu, xun gao, Huan Fu, Guoliang Kang

Naive DDIM inversion process usually suffers from a trajectory deviation issue, i.e., the latent trajectory during reconstruction deviates from the one during inversion. To alleviate this issue, previous methods either learn to mitigate the deviation or design cumbersome compensation strategy to reduce the mismatch error, exhibiting substantial time and computation cost. In this work, we present a nearly free-lunch method (named FreeInv) to address the issue more effectively and efficiently. In FreeInv, we randomly transform the latent representation and keep the transformation the same between the corresponding inversion and reconstruction time-step. It is motivated from a statistical perspective that an ensemble of DDIM inversion processes for multiple trajectories yields a smaller trajectory mismatch error on expectation. Moreover, through theoretical analysis and empirical study, we show that FreeInv performs an efficient ensemble of multiple trajectories. FreeInv can be freely integrated into existing inversion-based image and video editing techniques. Especially for inverting video sequences, it brings more significant fidelity and efficiency improvements. Comprehensive quantitative and qualitative evaluation on PIE benchmark and DAVIS dataset shows that FreeInv remarkably outperforms conventional DDIM inversion, and is competitive among previous state-of-the-art inversion methods, with superior computation efficiency.

Computer Vision · Classification and Understanding

Menghao Zhang, Huazheng Wang, Pengfei Ren, Kangheng Lin, Qi Qi, Haifeng Sun, Zirui Zhuang, Lei Zhang, Jianxin Liao, Jingyu Wang

Large Vision-Language Models (LVLMs) pretrained on large-scale multimodal data have shown promising capabilities in Video Anomaly Detection (VAD). However, their ability to reason about abnormal events based on scene semantics remains underexplored. In this paper, we investigate LVLMs’ behavior in VAD from a visual-textual co-occurrence perspective, focusing on whether their decisions are driven by statistical shortcuts between visual instances and textual phrases. By analyzing visual-textual co-occurrence in pretraining data and conducting experiments under different data settings, we reveal a hallucination phenomenon: LVLMs tend to rely on co-occurrence patterns between visual instances and textual phrases associated with either normality or abnormality, leading to incorrect predictions when these high-frequency objects appear in semantically mismatched contexts. To address this issue, we propose VAD-DPO, a direct preference optimization method supervised with counter-example pairs. By constructing visually similar but semantically contrasting video clips, VAD-DPO encourages the model to align its predictions with the semantics of scene rather than relying on co-occurrence patterns. Extensive experiments on six benchmark datasets demonstrate the effectiveness of VAD-DPO in enhancing both anomaly detection and reasoning performance, particularly in scene-dependent scenarios.

General Machine Learning · Representation Learning

Reuben Dorent, Polina Golland, William (Sandy) Wells

Mutual Information (MI) is a fundamental measure of statistical dependence widely used in representation learning. While direct optimization of MI via its definition as a Kullback-Leibler divergence (KLD) is often intractable, many recent methods have instead maximized alternative dependence measures, most notably, the Jensen-Shannon divergence (JSD) between joint and product of marginal distributions via discriminative losses. However, the connection between these surrogate objectives and MI remains poorly understood. In this work, we bridge this gap by deriving a new, tight, and tractable lower bound on KLD as a function of JSD in the general case. By specializing this bound to joint and marginal distributions, we demonstrate that maximizing the JSD-based information increases a guaranteed lower bound on mutual information. Furthermore, we revisit the practical implementation of JSD-based objectives and observe that minimizing the cross-entropy loss of a binary classifier trained to distinguish joint from marginal pairs recovers a known variational lower bound on the JSD. Extensive experiments demonstrate that our lower bound is tight when applied to MI estimation. We compared our lower bound to state-of-the-art neural estimators of variational lower bound across a range of established reference scenarios. Our lower bound estimator consistently provides a stable, low-variance estimate of a tight lower bound on MI. We also demonstrate its practical usefulness in the context of the Information Bottleneck framework. Taken together, our results provide new theoretical justifications and strong empirical evidence for using discriminative learning in MI-based representation learning.

Applications · Neuroscience, Cognitive Science

Christian Internò, Robert Geirhos, Markus Olhofer, Sunny Liu, Barbara Hammer, David Klindt

The rapid advancement of generative AI enables highly realistic synthetic video, posing significant challenges for content authentication and raising urgent concerns about misuse. Existing detection methods often struggle with generalization and capturing subtle temporal inconsistencies. We propose $ReStraV$ ($Re$presentation $Stra$ightening for $V$ideo), a novel approach to distinguish natural from AI-generated videos. Inspired by the ``perceptual straightening'' hypothesis—which suggests real-world video trajectories become more straight in neural representation domain—we analyze deviations from this expected geometric property. Using a pre-trained self-supervised vision transformer (DINOv2), we quantify the temporal curvature and stepwise distance in the model's representation domain. We aggregate statistical and signals descriptors of these measures for each video and train a classifier. Our analysis shows that AI-generated videos exhibit significantly different curvature and distance patterns compared to real videos. A lightweight classifier achieves state-of-the-art detection performance (e.g., $97.17$ % accuracy and $98.63$ % AUROC on the VidProM benchmark, substantially outperforming existing image- and video-based methods. ReStraV is computationally efficient, it is offering a low-cost and effective detection solution. This work provides new insights into using neural representation geometry for AI-generated video detection.

Computer Vision · Image and Video Generation

Alexandros Graikos, Nebojsa Jojic, Dimitris Samaras

Large denoising diffusion models, such as Stable Diffusion, have been trained on billions of image-caption pairs to perform text-conditioned image generation. As a byproduct of this training, these models have acquired general knowledge about image statistics, which can be useful for other inference tasks. However, when confronted with sampling an image under new constraints, e.g. generating the missing parts of an image, using large pre-trained text-to-image diffusion models is inefficient and often unreliable. Previous approaches either utilized backpropagation through the denoiser network, making them significantly slower and more memory-demanding than simple text-to-image generation, or only enforced the constraint locally, failing to capture critical long-range correlations in the sampled image. In this work, we propose an algorithm that enables fast, high-quality generation under arbitrary constraints. We show that in denoising diffusion models, we can employ an approximation to Newton’s optimization method that allows us to speed up inference and avoid the expensive backpropagation operations. Our approach produces results that rival or surpass the state-of-the-art training-free inference methods while requiring a fraction of the time. We demonstrate the effectiveness of our algorithm under both linear (inpainting, super-resolution) and non-linear (style-guided generation) constraints. An implementation is provided at https://github.com/cvlab-stonybrook/fast-constrained-sampling.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Siyuan Wu, Yongzhe Jia, Haolong Xiang, Xiaolong Xu, Xuyun Zhang, Lianyong Qi, Wanchun Dou

Federated learning (FL) is a distributed machine learning paradigm that enables multiple clients to collaboratively train a shared model without exposing their raw data. However, existing FL research has primarily focused on optimizing learning performance based on the assumption of uniform client participation, with few studies delving into performance fairness under inconsistent client participation, particularly in model-heterogeneous FL environments. In view of this challenge, we propose PHP-FL, a novel model-heterogeneous FL method that explicitly addresses scenarios with varying client participation probabilities to enhance both model accuracy and performance fairness. Specifically, we introduce a Dual-End Aligned ensemble Learning (DEAL) module, where small auxiliary models on clients are used for dual-end knowledge alignment and local ensemble learning, effectively tackling model heterogeneity without a public dataset. Furthermore, to mitigate update conflicts caused by inconsistent participation probabilities, we propose an Importance-driven Selective Parameter Update (ISPU) module, which accurately updates critical local parameters based on training progress. Finally, we implement PHP-FL on a lightweight FL platform with heterogeneous clients across three different client participation patterns. Extensive experiments under heterogeneous settings and diverse client participation patterns demonstrate that PHP-FL achieves state-of-the-art performance in both accuracy and fairness. Our code is available at: https://github.com/Siyuan01/PHP-FL-main.

Deep Learning · Algorithms

HAO SUN, Zhongyi Han, Hao Chen, Jindong Wang, Xin Gao, Yilong Yin

Foundation models pretrained on web-scale data drive contemporary transfer learning in vision, language, and multimodal tasks. Recent work shows that mild label noise in these corpora may lift in-distribution accuracy yet sharply reduce out-of-distribution generalization, an effect known as catastrophic inheritance. Medical data is especially sensitive because annotations are scarce, domain shifts are large, and pretraining sources are noisy. We present the first systematic analysis of catastrophic inheritance in medical models. Controlled label-corruption experiments expose a clear structural collapse: as noise rises, the skewness and kurtosis of feature and logit distributions decline, signaling a flattened representation space and diminished discriminative detail. These higher-order statistics form a compact, interpretable marker of degradation in fine-grained tasks such as histopathology. Guided by this finding, we introduce a fine-tuning objective that restores skewness and kurtosis through two scalar regularizers added to the task loss. The method leaves the backbone unchanged and incurs negligible overhead. Tests on PLIP models trained with Twitter pathology images, as well as other large-scale vision and language backbones, show consistent gains in robustness and cross-domain accuracy under varied noise levels.

Deep Learning · Everything Else

Frank Shih, Zhenghao Jiang, Faming Liang

Uncertainty quantification (UQ) in scientific machine learning is increasingly critical as neural networks are widely adopted to tackle complex problems across diverse scientific disciplines. For physics-informed neural networks (PINNs), a prominent model in scientific machine learning, uncertainty is typically quantified using Bayesian or dropout methods. However, both approaches suffer from a fundamental limitation: the prior distribution or dropout rate required to construct honest confidence sets cannot be determined without additional information. In this paper, we propose a novel method within the framework of extended fiducial inference (EFI) to provide rigorous uncertainty quantification for PINNs. The proposed method leverages a narrow-neck hyper-network to learn the parameters of the PINN and quantify their uncertainty based on imputed random errors in the observations. This approach overcomes the limitations of Bayesian and dropout methods, enabling the construction of honest confidence sets based solely on observed data. This advancement represents a significant breakthrough for PINNs, greatly enhancing their reliability, interpretability, and applicability to real-world scientific and engineering challenges. Moreover, it establishes a new theoretical framework for EFI, extending its application to large-scale models, eliminating the need for sparse hyper-networks, and significantly improving the automaticity and robustness of statistical inference.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Haoyuan Liang, Shilei Cao, Li, Zhiyu Ye, Haohuan Fu, Juepeng Zheng

Federated learning (FL) has recently emerged as the primary approach to overcoming data silos, enabling collaborative model training without sharing sensitive or proprietary data. Parallel federated learning (PFL) aggregates models trained independently on each client’s local data, which can lead to suboptimal convergence due to limited data exposure. In contrast, Sequential Federated Learning (SFL) allows models to traverse client datasets sequentially, enhancing data utilization. However, SFL effectiveness is limited in real-world non-IID scenarios characterized by category shift (inconsistent class distributions) and domain shift (distribution discrepancies). These shifts cause two critical issues: update order sensitivity, where model performance varies significantly with the sequence of client updates, and catastrophic forgetting, where the model forgets previously learned features when trained on new client data. We propose SPFL, a novel updating method that can be integrated into existing FL methods, integrating sequential updates with parallel aggregation to enhance data utilization and ease update order sensitivity. At the same time, we give the convergence analysis of SPFL under strong convex, general convex, and non-convex conditions, proving that this update scheme is significantly better than PFL and SFL. Additionally, we introduce the Global-Local Alignment Module to mitigate catastrophic forgetting by aligning the predictions of the global model with those of the local and previous models during training. Our extensive experiments demonstrate that integrating SPFL into existing PFL methods significantly improves performance under category and domain shifts.

General Machine Learning · Everything Else

Louis Allain, Sébastien Da Veiga, Brian Staber

Conformal Prediction (CP) is a popular framework for constructing prediction bands with valid coverage in finite samples, while being free of any distributional assumption. A well-known limitation of conformal prediction is the lack of adaptivity, although several works introduced practically efficient alternate procedures. In this work, we build upon recent ideas that rely on recasting the CP problem as a statistical learning problem, directly targeting coverage and adaptivity. This statistical learning problem is based on reproducible kernel Hilbert spaces (RKHS) and kernel sum-of-squares (SoS) methods. First, we extend previous results with a general representer theorem and exhibit the dual formulation of the learning problem. Crucially, such dual formulation can be solved efficiently by accelerated gradient methods with several hundreds or thousands of samples, unlike previous strategies based on off-the-shelf semidefinite programming algorithms. Second, we introduce a new hyperparameter tuning strategy tailored specifically to target adaptivity through bounds on test-conditional coverage. This strategy, based on the Hilbert-Schmidt Independence Criterion (HSIC), is introduced here to tune kernel lengthscales in our framework, but has broader applicability since it could be used in any CP algorithm where the score function is learned. Finally, extensive experiments are conducted to show how our method compares to related work. All figures can be reproduced with the accompanying code at https://www.gitlab.com/drti/ksos-bands.

Social Aspects · Accountability, Transparency and Interpretability

Hidde Fokkema, Tim van Erven, Sara Magliacane

Machine learning is a vital part of many real-world systems, but several concerns remain about the lack of interpretability, explainability and robustness of black-box AI systems. Concept Bottleneck Models (CBM) address some of these challenges by learning interpretable concepts from high-dimensional data, e.g. images, which are used to predict labels. An important issue in CBMs are spurious correlation between concepts, which effectively lead to learning “wrong” concepts. Current mitigating strategies have strong assumptions, e.g., they assume that the concepts are statistically independent of each other, or require substantial interaction in terms of both interventions and labels provided by annotators. In this paper, we describe a framework that provides theoretical guarantees on the correctness of the learned concepts and on the number of required labels, without requiring any interventions. Our framework leverages causal representation learning (CRL) methods to learn latent causal variables from high-dimensional observations in a unsupervised way, and then learns to align these variables with interpretable concepts with few concept labels. We propose a linear and a non-parametric estimator for this mapping, providing a finite-sample high probability result in the linear case and an asymptotic consistency result for the non-parametric estimator. We evaluate our framework in synthetic and image benchmarks, showing that the learned concepts have less impurities and are often more accurate than other CBMs, even in settings with strong correlations between concepts.

General Machine Learning · Representation Learning

Zhichao Zhu, YANG QI, Hengyuan Ma, Wenlian Lu, Jianfeng Feng

The Forward-Forward (FF) learning algorithm provides a bottom-up alternative to backpropagation (BP) for training neural networks, relying on a layer-wise "goodness" function with well-designed negative samples for contrastive learning. Existing goodness functions are typically defined as the sum of squared postsynaptic activations, neglecting correlated variability between neurons. In this work, we propose a novel goodness function termed dimensionality compression that uses the effective dimensionality (ED) of fluctuating neural responses to incorporate second-order statistical structure. Our objective minimizes ED for noisy copies of individual inputs while maximizing it across the sample distribution, promoting structured representations without the need to prepare negative samples. We demonstrate that this formulation achieves competitive performance compared to other non-BP methods. Moreover, we show that noise plays a constructive role that can enhance generalization and improve inference when predictions are derived from the mean of squared output, which is equivalent to making predictions based on an energy term. Our findings contribute to the development of more biologically plausible learning algorithms and suggest a natural fit for neuromorphic computing, where stochasticity is a computational resource rather than a nuisance. The code is available at https://github.com/ZhichaoZhu/StochasticForwardForward.

Applications · Language, Speech and Dialog

Zhimeng Guo, Minhao Cheng

The rapid advancement of Large Language Models (LLMs) in code generation has raised significant attribution and intellectual property concerns. Code watermarking offers a potential solution but faces unique challenges due to programming languages' strict syntactic constraints and semantic requirements. To address these challenges, we introduce ACW (AST-guided Code Watermarking), a novel adaptive framework that leverages Abstract Syntax Tree (AST) analysis during training to learn watermark embedding strategies. Our framework identifies substitutable code components and strategically biases token selections to embed watermarks. We also propose a novel sampling scheme that distributes tokens between green/red lists according to semantic context, ensuring statistical distinguishability while preserving code functionality. Extensive experiments demonstrate that ACW achieves a significant improvement in watermark detection accuracy compared to existing methods, with negligible impact on code functionality. This adaptive framework offers a promising solution for effective and practical code watermarking in the age of LLMs. Our code is available at: https://github.com/TimeLovercc/code-watermark.