Lorentz Entailment Cone for Semantic Segmentation

Semantic segmentation in hyperbolic space can capture hierarchical structure in low dimensions with uncertainty quantification. Existing approaches choose the Poincare´ ball model for hyperbolic geometry, which suffers from numerical instabilities, optimization, and computational challenges. We propose a novel, tractable, architecture-agnostic semantic segmentation framework 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 further introduce a novel uncertainty and confidence indicator in Lorentz cone embeddings. Extensive experiments on ADE20K, COCO-Stuff-164k, Pascal-VOC, and Cityscapes with state-of-the-art models (DeepLabV3 and SegFormer) 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.