Lorentz Framework for Semantic Segmentation

Semantic segmentation in hyperbolic space enables compact modeling of hierarchical structure while providing inherent uncertainty quantification. Prior approaches predominantly rely on the Poincar\'e ball model, which suffers from numerical instability, optimization, and computational challenges. We propose a novel, tractable, architecture-agnostic semantic segmentation framework (pixel-wise and mask classification) in the hyperbolic Lorentz model. We employ text embeddings with semantic and visual cues to guide hierarchical pixel-level representations in Lorentz space. This enables stable and efficient optimization without requiring a Riemannian optimizer, and easily integrates with existing Euclidean architectures. Beyond segmentation, our approach yields free uncertainty estimation, confidence map, boundary delineation, hierarchical and text-based retrieval, and zero-shot performance, reaching generalized flatter minima. We introduce a novel uncertainty and confidence indicator in Lorentz cone embeddings. Further, we provide analytical and empirical insights into Lorentz optimization via gradient analysis. Extensive experiments on ADE20K, COCO-Stuff-164k, Pascal-VOC, and Cityscapes, utilizing state-of-the-art per-pixel classification models (DeepLabV3 and SegFormer) and mask classification models (mask2former and maskformer), validate the effectiveness and generality of our approach. Our results demonstrate the potential of hyperbolic Lorentz embeddings for robust and uncertainty-aware semantic segmentation. Code is available at https://github.com/mxahan/Lorentz_semantic_segmentation.