overlapping segmentation

This paper addresses the lack of geometric characterization for the implicit spatial partitioning induced by machine learning models. We propose a modeling framework based on Riemannian simplicial complexes: model decision regions are represented as metric-bearing simplicial complexes, enabling systematic quantification of geometric features—including volumes, facet areas, and dihedral angles. To track geometric evolution across neural network layers, we introduce pullbacks of differential forms and an extended Laplacian operator. Furthermore, we define vertex-wise discrete curvature and edge-wise statistical Ricci curvature to explicitly link model geometry with underlying data distributions. The resulting geometric regularization method directly constrains spatial configurations, enhancing both generalization and interpretability. Empirically and theoretically, it demonstrates consistency in regularizer design and efficacy in diagnosing learning dynamics, offering a computationally tractable and principled approach to geometric deep learning.

Under standard training, neural networks generate far fewer piecewise affine regions in the vicinity of data points than theoretically possible, and existing theory offers little guidance on how optimization shapes data-dependent partitions. This work addresses this gap by analyzing piecewise affine networks from a geometric perspective, establishing a sufficient condition under which neuron switching hyperplanes positioned near data points increase the number of local affine regions. Building on this insight, the authors propose a plug-and-play “region seeding” regularizer that encourages the formation of more effective partitions early in training. Combining switching hyperplane localization theory with differentiable regularization, the method demonstrably increases the number of affine regions on both synthetic datasets and ImageNet-1k, yielding improved early-stage accuracy and final performance that matches or slightly exceeds baseline models.

This paper addresses the poor interpretability of black-box models in spatial prediction. We propose a graph-partitioning-based spatial segmentation method that minimizes the sum of intra-segment prediction variance. Innovatively, we formulate interpretability as a variance-constrained graph partitioning problem and introduce, for the first time, a mixed-integer quadratic programming (MIQP) formulation to capture this objective. To tackle the prohibitive computational complexity on large-scale data, we design an efficient approximation algorithm that exploits intrinsic graph structural properties, ensuring high segmentation quality while drastically improving runtime efficiency. Experiments demonstrate that our method achieves 1–2 orders of magnitude speedup over exact MIQP solvers while reducing intra-segment variance by up to 37%. The approach thus establishes a new paradigm for scalable, interpretable spatial modeling—balancing fidelity, transparency, and computational tractability.

This work addresses the challenge of scaling pixel-level segmentation of wind turbine blades for automated inspection, which typically relies on extensive manual annotations. To overcome this limitation, the authors propose a fully unsupervised and interpretable modular approach that reformulates the segmentation task as region generation followed by binary classification. The method leverages adaptive region growing—incorporating adaptive thresholding and region merging—and introduces RegionMix, a novel region-mixing augmentation strategy, to significantly enhance model generalization. Evaluated across multiple wind farm datasets, the proposed approach achieves state-of-the-art segmentation accuracy and demonstrates strong cross-site generalization capabilities without requiring any labeled data.

This work investigates how to reliably infer the accuracy of clustering structures from the suboptimality of clustering objective functions, particularly in challenging scenarios involving cluster imbalance and ambiguous boundaries. To this end, the authors introduce a geometric framework termed the “clustering condition number,” which quantitatively links objective values to classification error across a broad class of loss functions, independent of any specific algorithm. Through geometric analysis, local margin theory, and non-asymptotic error bounds, the framework reveals a fundamental trade-off between robustness and sensitivity to cluster imbalance. It provides deterministic, non-asymptotic guarantees for structure recovery: small suboptimality necessarily implies low misclassification rates, with errors concentrated near cluster boundaries, and exact recovery of cluster cores becomes possible under strengthened local margin conditions.

This work uncovers the local mechanism underlying memorization in diffusion models—specifically, how a single model can simultaneously generate both memorized and novel samples. Building on the theoretical connection between diffusion processes and kernel density estimation, the study proposes that memorization is governed by the local coverage density of training data: regions with low coverage tend to memorize isolated samples, whereas densely covered regions facilitate generalization. The paper establishes the first theoretical link between memorization and the local geometric structure of data, introducing a “local coverage criterion” to predict where memorization occurs. It further elucidates how intra-class sparsity in multi-class settings modulates memorization strength. Experiments confirm that memorization intensifies with increasing local sparsity and demonstrate the coexistence of memorized and novel generation within the same model.

Current medical image segmentation methods lack a unified geometric prior capable of generalizing across organs and imaging modalities, often failing to simultaneously preserve structural consistency and fine anatomical details. This work proposes HadBalance, the first plug-and-play framework that introduces a unified geometric prior grounded in the near-convex shape assumption. Leveraging the Hadwiger theorem, it formulates global geometric constraints based on area, perimeter, and Euler characteristic, and incorporates a conflict-aware objective balancing mechanism to adaptively fuse geometric priors with task-specific gradients. Evaluated across diverse organs and imaging modalities, HadBalance significantly improves segmentation accuracy while avoiding over-regularization, effectively enhancing both structural coherence and fidelity to anatomical detail.

This work addresses the challenge that combining multiple pre-trained diffusion models often leads to missing objects or attribute mismatches due to dominance or conflict among the models. To mitigate this, the authors propose a dynamic task allocation mechanism grounded in fair division game theory, which dynamically partitions responsibility regions at each denoising step to guide collaborative denoising within assigned areas. By deeply integrating fair division algorithms from game theory with the diffusion sampling process, the method enables efficient and equitable co-evolutionary collaboration among models. Experimental results demonstrate that the approach significantly outperforms existing baselines on conditional image generation tasks, effectively alleviating object omission and attribute misalignment while achieving state-of-the-art performance on metrics such as GenEval.

Existing localized hypergraph flow diffusion (HFD) methods struggle to preserve computational locality during iteration, leading to inefficiency. This work proposes Thresholded Local Hypergraph Flow Diffusion (TL-HFD), which maintains an active region around seed vertices and performs projected subgradient updates exclusively within this region and its boundary. New vertices are dynamically activated based on a top-k thresholding criterion, enabling gradual expansion. TL-HFD is the first method to achieve strictly local HFD updates while being provably equivalent to global updates. Theoretical analysis establishes finite-time bounds on dual suboptimality and the total activated volume. Experiments on general submodular hypergraphs demonstrate that TL-HFD matches or exceeds the clustering performance of standard HFD with significantly less activated volume, exhibiting notably enhanced robustness on noisy data.

This study addresses the p-regions problem in geographic zoning, which involves partitioning a region into p connected and attribute-homogeneous subregions—a task known to be NP-hard. To tackle this challenge, we propose a novel integer linear programming model, ER-S, enhanced with specialized subtour elimination inequalities that strengthen connectivity constraints. Building upon structural insights from the k-partitioning problem, we further develop the ER-S-Tree model, which exhibits superior polyhedral strength. Computational experiments demonstrate that our approach significantly outperforms existing methods in both computational efficiency and provable optimality, successfully solving large-scale instances for several European countries for the first time.