论文检索

General Machine Learning · Representation Learning

Xiaoyuan Cheng, Wenxuan Yuan, Yiming Yang, Yuanzhao Zhang, Sibo Cheng, Yi He, Zhuo Sun

The Koopman operator provides a powerful framework for modeling dynamical systems and has attracted growing interest from the machine learning community. However, its infinite-dimensional nature makes identifying suitable finite-dimensional subspaces challenging, especially for deep architectures. We argue that these difficulties come from suboptimal representation learning, where latent variables fail to balance expressivity and simplicity. This tension is closely related to the information bottleneck (IB) dilemma: constructing compressed representations that are both compact and predictive. Rethinking Koopman learning through this lens, we demonstrate that latent mutual information promotes simplicity, yet an overemphasis on simplicity may cause latent space to collapse onto a few dominant modes. In contrast, expressiveness is sustained by the von Neumann entropy, which prevents such collapse and encourages mode diversity. This insight leads us to propose an information-theoretic Lagrangian formulation that explicitly balances this tradeoff. Furthermore, we propose a new algorithm based on the Lagrangian formulation that encourages both simplicity and expressiveness, leading to a stable and interpretable Koopman representation. Beyond quantitative evaluations, we further visualize the learned manifolds under our representations, observing empirical results consistent with our theoretical predictions. Finally, we validate our approach across a diverse range of dynamical systems, demonstrating improved performance over existing Koopman learning methods.

Computer Vision · Image and Video Generation

Ziyun Zeng, David Junhao Zhang, Wei Li, Mike Zheng Shou

In recent years, integrating multimodal understanding and generation into a single unified model has emerged as a promising paradigm. While this approach achieves strong results in text-to-image (T2I) generation, it still struggles with precise image editing. We attribute this limitation to an imbalanced division of responsibilities. The understanding module primarily functions as a translator that encodes user instructions into semantic conditions, while the generation module must simultaneously act as designer and painter, inferring the original layout, identifying the target editing region, and rendering the new content. This imbalance is counterintuitive because the understanding module is typically trained with several times more data on complex reasoning tasks than the generation module. To address this issue, we introduce *Draw-In-Mind* (DIM), a dataset comprising two complementary subsets: (**i**) DIM-T2I, containing 14M long-context image-text pairs to enhance complex instruction comprehension; and (**ii**) DIM-Edit, consisting of 233K chain-of-thought imaginations generated by GPT-4o, serving as explicit design blueprints for image edits. We connect a frozen Qwen2.5-VL-3B with a trainable SANA1.5-1.6B via a lightweight two-layer MLP, and train it on the proposed DIM dataset, resulting in DIM-4.6B-T2I/Edit. Despite its modest parameter scale, DIM-4.6B-Edit achieves SOTA or competitive performance on the ImgEdit and GEdit-Bench benchmarks, outperforming much larger models such as UniWorld-V1 and Step1X-Edit. These findings demonstrate that explicitly assigning the design responsibility to the understanding module provides significant benefits for image editing. Our dataset and models are available at https://github.com/showlab/DIM.

Deep Learning · Generative Models and Autoencoders

Min Xue, Nikolai Bolik, Lennart Stöpler, Erik Imgrund, Janik Schmid, Artur Andrzejak

Large Language Models (LLMs) have demonstrated outstanding capabilities in various code-related tasks, including code completion, translation, or summarization. However, these pretrained models are static, posing a challenge to incorporate new knowledge into an LLM to correct erroneous behavior. Approaches such as retraining or fine-tuning demand extensive labeled datasets and might be computationally expensive, while prompt engineering fails to change models permanently. Knowledge Editing (KE) techniques offer a more efficient alternative, enabling model updates with minimal data, even just a single example. Nevertheless, existing KE methods often manipulate parameters within the Transformer's multi-layer perceptrons (MLPs), where neuronal polysemanticity hinders both the precision and interpretability of the edits. To address these limitations, we exploit TransCoder, an MLP-like model component with a wide and sparsely activated hidden feature vector. Specifically, we introduce **TransCoder-based Precise Editing** (**TCPE**), a novel method that leverages the sparsity and monosemanticity of the TransCoder’s neurons for highly localized knowledge editing. TCPE exhibits neuron-level mechanistic interpretability characteristics, revealing the correspondence between the edited neurons and the specific code-related knowledge. Furthermore, we present KECode, a new evaluation benchmark for code-to-code translation based on functional equivalence. Using KECode, we conduct a systematic evaluation of representative KE methods in the context of code-to-code translation. Our experimental results demonstrate that TCPE outperforms existing KE methods, achieving a substantial improvement of translation accuracy of CodeLlama-7b-Instruct from 57.5% to 64.0% in a low-resource scenario of Java-to-D translation.

Reinforcement Learning · Deep RL

Max Rudolph, Nathan Lichtlé, Sobhan Mohammadpour, Alexandre M Bayen, Zico Kolter, Amy Zhang, Gabriele Farina, Eugene Vinitsky, Samuel Sokota

In the past decade, motivated by the putative failure of naive self-play deep reinforcement learning (DRL) in adversarial imperfect-information games, researchers have developed numerous DRL algorithms based on fictitious play (FP), double oracle (DO), and counterfactual regret minimization (CFR). In light of recent results of the magnetic mirror descent algorithm, we hypothesize that simpler generic policy gradient methods like PPO are competitive with or superior to these FP-, DO-, and CFR-based DRL approaches. To facilitate the resolution of this hypothesis, we implement and release the first broadly accessible exact exploitability computations for five large games. Using these games, we conduct the largest-ever exploitability comparison of DRL algorithms for imperfect-information games. Over 7000 training runs, we find that FP-, DO-, and CFR-based approaches fail to outperform generic policy gradient methods.

General Machine Learning · Transfer, Multitask and Meta-learning

Siqi Wang, Aoming Liu, Bryan Plummer

Methods addressing Learning with Noisy Labels (LNL) and multi-source Domain Generalization (DG) use training techniques to improve downstream task performance in the presence of label noise or domain shifts, respectively. Prior work often explores these tasks in isolation, and the limited work that does investigate their intersection, which we refer to as Noise-Aware Generalization (NAG), only benchmarks existing methods without also proposing an approach to reduce its effect. We find that this is likely due, in part, to the new challenges that arise when exploring NAG, which does not appear in LNL or DG alone. For example, we show that the effectiveness of DG methods is compromised in the presence of label noise, making them largely ineffective. Similarly, LNL methods often overfit to easy-to-learn domains as they confuse domain shifts for label noise. Instead, we propose Domain Labels for Noise Detection (DL4ND), the first direct method developed for NAG which uses our observation that noisy samples that may appear indistinguishable within a single domain often show greater variation when compared across domains. We find DL4ND outperforms DG and LNL methods, including their combinations, even when simplifying the NAG challenge by using domain labels to isolate domain shifts from noise. Performance gains up to 12.5% over seven diverse datasets with three noise types demonstrates DL4ND’s ability to generalize to a wide variety of settings.

Computer Vision · Everything Else

Minyoung Oh, Jae-Young Sim

Dataset distillation (DD) attempts to construct a compact synthetic dataset that serves as a proxy for a large real dataset under a fixed storage budget, thereby reducing the storage burden and training costs. Prior works assume the full dataset is available upfront which is distilled at once, although real datasets are collected incrementally over time in practice. To alleviate this gap, we introduce a new problem setting, *Domain Incremental Dataset Distillation*, that continually distills datasets from different domains into a single synthetic dataset. The conventional DD sequentially processes arriving datasets in order, overwriting the old knowledge with new one, causing catastrophic forgetting problem. To overcome this drawback, we propose *Asymmetric Synthetic Data Update* strategy that adjusts the per-sample update rates for synthetic dataset while balancing the stability-plasticity trade-off. Specifically, we design a bi-level optimization method based on meta-learning framework to estimate the optimal update rates, which allows each sample to focus on either stability or plasticity, thereby striking a balance between them. Experimental results demonstrate that our approach effectively mitigates the catastrophic forgetting and achieves superior performance of DD across continually incoming datasets compared with existing methods.

Computer Vision · Image and Video Generation

Kaiyuan Deng, Gen Li, Yang Xiao, Bo Hui, Xiaolong Ma

While multi-concept unlearning has shown progress, extending to large-scale scenarios remains difficult, as existing methods face three persistent challenges: **(i)** they often introduce conflicting weight updates, making some targets difficult to unlearn or causing degradation of generative capability; **(ii)** they lack precise mechanisms to keep unlearning strictly confined to target concepts, resulting in collateral damage on similar content; **(iii)** many approaches rely on additional data or auxiliary modules, causing scalability and efficiency bottlenecks as the number of concepts grows. To simultaneously address these challenges, we propose **Scalable-Precise Concept Unlearning (ScaPre)**, a unified and lightweight framework tailored for scalable and precise large-scale unlearning. ScaPre introduces a *conflict-aware stable design*, which integrates the spectral trace regularizer and geometry alignment to stabilize the optimization space, suppress conflicting updates, and preserve the pretrained global structure. Furthermore, the *Informax Decoupler* identifies concept-relevant parameters and adaptively reweights updates, ensuring that unlearning is confined to the target subspace without collateral damage. ScaPre yields an efficient closed-form solution, requiring no additional data or auxiliary sub-models, while maintaining both scalability and precision. Comprehensive experiments across large-scale objects, styles, and explicit content benchmarks demonstrate that ScaPre effectively removes target concepts while maintaining generation quality. It can forget up to **×5** more concepts than the best baseline within the limits of acceptable generative quality, and outperforms existing multi-concept approaches in precision and efficiency, achieving a new state of the art for large-scale unlearning.

Applications · Physics

Alexander Rudikov, Vladimir Fanaskov, Sergei Stepanov, Buzheng Shan, Ekaterina Muravleva, Yalchin Efendiev, Ivan Oseledets

We propose GMsFEM-NO, a novel hybrid framework that combines the robustness of the Generalized Multiscale Finite Element Method (GMsFEM) with the computational speed of neural operators (NOs) to create an efficient method for solving heterogeneous partial differential equations (PDEs). GMsFEM builds localized spectral basis functions on coarse grids, allowing it to capture important multiscale features and solve PDEs accurately with less computational effort. However, computing these basis functions is costly. While NOs offer a fast alternative by learning the solution operator directly from data, they can lack robustness. Our approach trains a NO to instantly predict the GMsFEM basis by using a novel subspace-informed loss that learns the entire relevant subspace, not just individual functions. This strategy significantly accelerates the costly offline stage of GMsFEM while retaining its foundation in rigorous numerical analysis, resulting in a solution that is both fast and reliable. On standard multiscale benchmarks—including a linear elliptic diffusion problem and the nonlinear, steady-state Richards equation—our GMsFEM-NO method achieves a reduction in solution error compared to standalone NOs and other hybrid methods. The framework demonstrates effective performance for both 2D and 3D problems. A key advantage is its discretization flexibility: the NO can be trained on a small computational grid and evaluated on a larger one with minimal loss of accuracy, ensuring easy scalability. Furthermore, the resulting solver remains independent of forcing terms, preserving the generalization capabilities of the original GMsFEM approach. Our results prove that combining NO with GMsFEM creates a powerful new type of solver that is both fast and accurate.

Deep Learning · Generative Models and Autoencoders

Sergei Kholkin, Grigoriy Ksenofontov, David Li, Nikita Kornilov, Nikita Gushchin, Alexandra Suvorikova, Alexey Kroshnin, Evgeny Burnaev, Aleksandr Korotin

The Iterative Markovian Fitting (IMF) procedure, which iteratively projects onto the space of Markov processes and the reciprocal class, successfully solves the Schrödinger Bridge (SB) problem. However, an efficient practical implementation requires a heuristic modification-alternating between fitting forward and backward time diffusion at each iteration. This modification is crucial for stabilizing training and achieving reliable results in applications such as unpaired domain translation. Our work reveals a close connection between the modified version of IMF and the Iterative Proportional Fitting (IPF) procedure-a foundational method for the SB problem, also known as Sinkhorn’s algorithm. Specifically, we demonstrate that the heuristic modification of the IMF effectively integrates both IMF and IPF procedures. We refer to this combined approach as the Iterative Proportional Markovian Fitting (IPMF) procedure. Through theoretical and empirical analysis, we establish the convergence of the IPMF procedure under various settings, contributing to developing a unified framework for solving SB problems. Moreover, from a practical standpoint, the IPMF procedure enables a flexible trade-off between image similarity and generation quality, offering a new mechanism for tailoring models to specific tasks.

Zhichao Wang, Dongyang Ma, Xinting Huang, Deng Cai, Tian Lan, Jiahao Xu, Haitao Mi, Xiaoying Tang, Yan Wang

The "end-to-end" label for LLMs is a misnomer. In practice, they depend on a non-differentiable decoding process that requires laborious, hand-tuning of hyperparameters like temperature and top-p. This paper introduces AutoDeco, a novel architecture that enables truly "end-to-end'' generation by learning to control its own decoding strategy. We augment the standard transformer with lightweight heads that, at each step, dynamically predict context-specific temperature and top-p values alongside the next-token logits. This approach transforms decoding into a parametric, token-level process, allowing the model to self-regulate its sampling strategy within a single forward pass. Through extensive experiments on eight benchmarks, we demonstrate that AutoDeco not only significantly outperforms common decoding strategies but also achieves performance comparable to an oracle-tuned baseline derived from "hacking the test set"—a practical upper bound for any static method. Besides, we demonstrate an emergent capability for instruction-based decoding control: the model learns to interpret natural language commands (e.g., ''generate with low randomness'') and adjusts its predicted temperature and top-p on a token-by-token basis, which may open a new paradigm for steerable and interactive LLM decoding.

Deep Learning · Graph Neural Networks

Moshe Eliasof, Krishna Sri Ipsit Mantri, Beatrice Bevilacqua, Bruno Ribeiro, Carola-Bibiane Schönlieb

Unlike vision and language domains, graph learning lacks a shared input space, as input features differ across graph datasets not only in semantics, but also in value ranges and dimensionality. This misalignment prevents graph models from generalizing across datasets, limiting their use as foundation models. In this work, we propose ALL-IN, a simple and theoretically grounded method that enables transferability across datasets with different input features. Our approach projects node features into a shared random space and constructs representations via covariance-based statistics, thus eliminating dependence on the original feature space. We show that the computed node-covariance operators and the resulting node representations are invariant in distribution to permutations of the input features. We further demonstrate that the expected operator exhibits invariance to general orthogonal transformations of the input features. Empirically, ALL-IN achieves strong performance across diverse node- and graph-level tasks on unseen datasets with new input features, without requiring architecture changes or retraining. These results point to a promising direction for input-agnostic, transferable graph models.

Deep Learning · Algorithms

Xiangyang Yin, Xingyu Liu, Tianhua Xia, BO BAO, Vithursan Thangarasa, Valavan Manohararajah, Eric Sather, Sai Qian Zhang

Outliers have emerged as a fundamental bottleneck in preserving accuracy for low-precision large models, particularly within Mixture-of-Experts (MoE) architectures that are increasingly central to large-scale language modeling. Under post-training quantization (PTQ), these outliers induce substantial quantization errors, leading to severe accuracy degradation. While recent rotation-based smoothing techniques alleviate the problem by redistributing outlier magnitudes, residual errors remain and continue to impede reliable low-precision deployment. In this work, we tackle this challenge by introducing CodeQuant, a unified quantization-and-clustering scheme that contains smoothing activation outliers via learnable rotation and absorbing weight outliers into fine-tuned cluster centroids for MoE. This design reduces the influence of extreme values by fitting them within cluster centroids, thereby lowering quantization error while maintaining expressive capacity. Coupled with a dedicated kernel design for GPU and CPU, CodeQuant achieves up to $4.15\times$ speedup while delivering significantly higher accuracy than state-of-the-art quantization approaches across diverse MoE models. Our results highlight CodeQuant as a promising direction for efficient and accurate deployment of MoE-based large language models under low-precision constraints. Our code is available at https://github.com/SAI-Lab-NYU/CodeQuant.

Reinforcement Learning · Deep RL

Dan Haramati, Carl Qi, Tal Daniel, Amy Zhang, Aviv Tamar, George D Konidaris

We propose a hierarchical entity-centric framework for offline Goal-Conditioned Reinforcement Learning (GCRL) that combines subgoal decomposition with factored structure to solve long-horizon tasks in domains with multiple entities. Achieving long-horizon goals in complex environments remains a core challenge in Reinforcement Learning (RL). Domains with multiple entities are particularly difficult due to their combinatorial complexity. GCRL facilitates generalization across goals and the use of subgoal structure, but struggles with high-dimensional observations and combinatorial state-spaces, especially under sparse reward. We employ a two-level hierarchy composed of a value-based GCRL agent and a factored subgoal-generating conditional diffusion model. The RL agent and subgoal generator are trained independently and composed post hoc through selective subgoal generation based on the value function, making the approach modular and compatible with existing GCRL algorithms. We introduce new variations to benchmark tasks that highlight the challenges of multi-entity domains, and show that our method consistently boosts performance of the underlying RL agent on image-based long-horizon tasks with sparse rewards, achieving over $150$% higher success rates on the hardest task in our suite and generalizing to increasing horizons and numbers of entities. Rollout videos are provided at: https://sites.google.com/view/hecrl.

Deep Learning · Graph Neural Networks

Juan P Amboage, Ernst Roell, Patrick Schnider, Bastian Rieck

Graph neural networks (GNNs) largely rely on the message-passing paradigm, where nodes iteratively aggregate information from their neighbors. Yet, standard message passing neural networks (MPNNs) face well-documented theoretical and practical limitations. Graph positional encoding (PE) has emerged as a promising direction to address these limitations. The Euler Characteristic Transform (ECT) is an efficiently computable geometric–topological invariant that characterizes shapes and graphs. In this work, we combine the differentiable approximation of the ECT (DECT) and its local variant ($\ell$-ECT) to propose LEAP, a new end-to-end trainable local structural PE for graphs. We evaluate our approach on multiple real-world datasets as well as on a synthetic task designed to test its ability to extract topological features. Our results underline the potential of LEAP-based encodings as a powerful component for graph representation learning pipelines.

General Machine Learning · Probabilistic Methods

Anastasia Nastya Krouglova, Hayden Johnson, Basile Confavreux, Michael Deistler, Pedro J Goncalves

Across many domains of science, stochastic models are an essential tool to understand the mechanisms underlying empirically observed data. Models can be of different levels of detail and accuracy, with models of high-fidelity (i.e., high accuracy) to the phenomena under study being often preferable. However, inferring parameters of high-fidelity models via simulation-based inference is challenging, especially when the simulator is computationally expensive. We introduce a multifidelity approach to neural posterior estimation that uses transfer learning to leverage inexpensive low-fidelity simulations to efficiently infer parameters of high-fidelity simulators. Our method applies the multifidelity scheme to both amortized and non-amortized neural posterior estimation. We further improve simulation efficiency by introducing a sequential variant that uses an acquisition function targeting the predictive uncertainty of the density estimator to adaptively select high-fidelity parameters. On established benchmark and neuroscience tasks, our approaches require up to two orders of magnitude fewer high-fidelity simulations than current methods, while showing comparable performance. Overall, our approaches open new opportunities to perform efficient Bayesian inference on computationally expensive simulators.

Wuyang Cong, Junqi Shi, Ming Lu, Xu Zhang, Zhan Ma

Hierarchical coding offers distinct advantages for learned image compression by capturing multi-scale representations to support scale-wise modeling and enable flexible quality scalability, making it a promising alternative to single-scale models. However, its practical performance remains limited. Through spectral analysis of training dynamics, we reveal that existing hierarchical image coding approaches suffer from cross-scale energy dispersion and spectral aliasing, resulting in optimization inefficiency and performance bottlenecks. To address this, we propose explicit spectral regularization schemes for hierarchical image coding, consisting of (i) intra-scale frequency regularization, which encourages a smooth low‑to‑high frequency buildup as scales increase, and (ii) inter-scale similarity regularization, which suppresses spectral aliasing across scales. Both regularizers are applied only during training and impose no overhead at inference. Extensive experiments demonstrate that our method accelerates the training of the vanilla model by 2.3$\times$, delivers an average 20.65\% rate–distortion gain over the latest VTM-22.0 on public datasets, and outperforms existing single-scale approaches, thereby setting a new state of the art in learned image compression.

Applications · Language, Speech and Dialog

Lingjie Jiang, Shaohan Huang, xun wu, Yixia Li, Guanhua Chen, Dongdong Zhang, Furu Wei

Multimodal large language models (MLLMs) have significantly advanced the integration of visual and textual understanding. However, their ability to generate code from multimodal inputs remains limited. In this work, we introduce VisCodex, a unified framework that seamlessly merges vision and coding language models to empower MLLMs with strong multimodal code generation abilities. Leveraging a task vector-based model merging technique, we integrate a state-of-the-art coding LLM into a strong vision-language backbone, while preserving both visual comprehension and advanced coding skills. To support training and evaluation, we introduce the Multimodal Coding Dataset (MCD), a large-scale and diverse collection of 598k samples, including high-quality HTML code, chart image-code pairs, image-augmented StackOverflow QA, and algorithmic problems. Furthermore, we propose InfiBench-V, a novel and challenging benchmark specifically designed to assess models on visually-rich, real-world programming questions that demand a nuanced understanding of both textual and visual contexts. Extensive experiments show that VisCodex achieves state-of-the-art performance among open-source MLLMs and approaches proprietary models like GPT-4o, highlighting the effectiveness of our model merging strategy and new datasets.

Deep Learning · Everything Else

Daniel Israel, Tian Jin, Ellie Cheng, Guy Van den Broeck, Aditya Grover, Suvinay Subramanian, Michael Carbin

Most existing large language models are autoregressive: they generate text one token at a time, and cannot decode any new tokens until they have decoded every token before it. Discrete diffusion language models offer a promising alternative by generating multiple tokens in parallel, but sampling from them requires a _denoising order_, the strategy for deciding which tokens to decode at each step. Determining the right denoising order is difficult, and existing approaches use heuristics that create a steep trade-off between quality and latency. We propose _planned diffusion_, a system that trains the model to determine its own denoising order. Planned diffusion uses a single model that transitions between autoregressive and diffusion-based generation: first, the model autoregressively generates a plan that partitions the response into semantically independent chunks, defining a denoising order that parallelizes sampling across chunks; second, the model executes this plan via diffusion denoising. On AlpacaEval, a suite of 805 instruction-following prompts, planned diffusion achieves Pareto-optimal trade-off between quality and latency, achieving 1.27x to 1.81x speedup over autoregressive generation with only 0.87\% to 5.4\% drop in win rate. Our empirical results show that planned diffusion exhibits superior performance scaling on downstream tasks compared to autoregressive baselines while offering the runtime flexibility to precisely navigate the quality-latency trade-off.

Deep Learning · Everything Else

Anik Pramanik, Murat Kantarcioglu, Vincent Oria, Shantanu Sharma

Federated Learning (FL) enables a group of clients to collaboratively train a model without sharing individual data, but its performance drops when client data are heterogeneous. Clustered FL tackles this by grouping similar clients. However, existing clustered FL approaches rely solely on either data similarity or gradient similarity; however, this results in an incomplete assessment of client similarities. Prior clustered FL approaches also restrict knowledge and representation sharing to clients within the same cluster. This prevents cluster models from benefiting from the diverse client population across clusters. To address these limitations, FEDDAG introduces a clustered FL framework, FEDDAG, that employs a weighted, class-wise similarity metric that integrates both data and gradient information, providing a more holistic measure of similarity during clustering. In addition, FEDDAG adopts a dual-encoder architecture for cluster models, comprising a primary encoder trained on its own clients' data and a secondary encoder refined using gradients from complementary clusters. This enables cross-cluster feature transfer while preserving cluster-specific specialization. Experiments on diverse benchmarks and data heterogeneity settings show that FEDDAG consistently outperforms state-of-the-art clustered FL baselines in accuracy.

Optimization · Non-Convex

Ganzhao Yuan

Nonsmooth composite optimization with orthogonality constraints has a wide range of applications in statistical learning and data science. However, this problem is challenging due to its nonsmooth objective and computationally expensive, non-convex constraints. In this paper, we propose a new approach called \textbf{OBCD}, which leverages Block Coordinate Descent to address these challenges. \textbf{OBCD} is a feasible method with a small computational footprint. In each iteration, it updates $k$ rows of the solution matrix, where $k \geq 2$, by globally solving a small nonsmooth optimization problem under orthogonality constraints. We prove that the limiting points of \textbf{OBCD}, referred to as (global) block-$k$ stationary points, offer stronger optimality than standard critical points. Furthermore, we show that \textbf{OBCD} converges to $\epsilon$-block-$k$ stationary points with an iteration complexity of $\mathcal{O}(1/\epsilon)$. Additionally, under the Kurdyka-Lojasiewicz (KL) inequality, we establish the non-ergodic convergence rate of \textbf{OBCD}. We also demonstrate how novel breakpoint search methods can be used to solve the subproblem in \textbf{OBCD}. Empirical results show that our approach consistently outperforms existing methods.