Iterate convergence of Bregman proximal methods (BPMs) has long remained open, especially for nonconvex objectives. Recently, \citet{chen2026skl} made progress by establishing iterate convergence for a BPM via the so-called scaled Kurdyka-\L{}ojasiewicz (SK\L{}) property, but only for the Shannon entropy kernel and linearly constrained problems. In this paper, we develop a unified iterate convergence framework that applies to a broad group of kernels and composite objective functions. Our approach extends the analytical tools in \cite{chen2026skl}, in particular the SK\L{} property, which plays a central role in ensuring convergence of the generated sequences. By introducing kernel-dependent parameterization functions, we show that the extended SK\L{} property holds for all continuous subanalytic functions, particularly when the kernel has a closed domain. We then verify that the assumptions of the framework are satisfied by standard BPMs under mild regularity conditions, thereby establishing their iterate convergence for a wide range of objective functions. Furthermore, based on the parameterization functions, we show that the continuous-time BPM (mirror flow) converges to a stationary point for o-minimal definable objective functions, yielding the first trajectory convergence result for mirror flow without imposing convexity assumptions on the objective function or isolation assumptions on stationary points. Taken together, these discrete- and continuous-time convergence results provide a unified trajectory convergence theory for BPMs.
He Chen, Jiaming Fan, Anthony Man-Cho So· 1 citation· ⚡1
ABSTRACT Remote sensing image semantic segmentation plays a pivotal role in converting complex image data into quantifiable geographic spatial information, underpinning applications such as disaster assessment, urban planning, and agricultural resource investigation. Fully supervised semantic segmentation methods rely heavily on labour-intensive pixel-level annotations, prompting a shift towards weakly supervised semantic segmentation (WSSS) that utilizes image-level annotations. However, remote sensing images are characterized by dense, intricate targets and the absence of distinct backgrounds, leading to challenges, such as sparse activation of local regions, incomplete localization in class activation maps (CAMs), and noisy, rough boundaries in pseudo-labels generated from image-level supervision. To address these issues, we propose a deep learning method with pixel relationship constraints for WSSS in remote sensing images. Specifically, we design an image reconstruction (IR) loss function to provide pixel-level supervision, enhancing the completeness of CAMs; a pixel relationship constraint (PRC) module to strengthen the global correlation of target regions and improve detailed information extraction; and an intersection optimization strategy (IOS) based on the Segment Anything Model (SAM) to refine pseudo-labels and segmentation results by mitigating noise. Here, we show that our method achieves mean Intersection over Union (mIoU) values of 63.85%, 71.20%, and 40.96% on the Vaihingen, Potsdam, and iSAID datasets, respectively, reaching 89.25%, 89.02%, and 65.66% of the performance of fully supervised methods. This work advances WSSS for remote sensing images by addressing key limitations of CAM-based pseudo-labels generation, offering a cost-effective alternative to fully supervised approaches and facilitating broader applications in geographic information science and earth observation. The code is available at https://github.com/CHENDL-SHEN/PRCCAM.
Jiaming Fan, Dali Chen, Yang Liu et al.· International Journal of Rem...· 0 citations