Group-Equivariant Poincaré Convolutional Networks

Equivariant neural networks encode geometric symmetries by construction, yet they are often difficult to optimize and can underperform less constrained architectures. A growing body of work addresses this through architectural modifications such as constraint relaxation or approximate equivariance, while the role of the optimizer remains comparatively underexplored. We study this direction by comparing Muon and Adam across several equivariant and geometric architectures under pointcloud and molecular learning settings. On ModelNet40, where the comparison is clearest, Muon consistently improves over Adam across all architectures considered. We then analyze the trained ModelNet40 checkpoints through Hessian estimates, loss surface visualizations, and spectral properties of learned weights and intermediate representations. The checkpoints reached by Muon have larger Hessian curvature summaries but more regular loss surfaces, and their learned weights and representations have higher stable and effective ranks. These observations suggest that the interaction between optimizer design and geometric inductive bias deserves further attention from the community.

Open-vocabulary semantic segmentation requires adapting image-level vision-language models such as CLIP to dense pixel-level prediction, which is challenging due to the mismatch between hierarchical structure and semantic alignment in the embedding space. While recent works leverage hyperbolic geometry to model hierarchical relationships, they align embeddings across hierarchical levels but overlook semantic misalignment among embeddings within the same level. In this work, we propose HyRo, a hyperbolic fine-tuning framework that decouples hierarchical and semantic alignment in the Poincaré ball model. HyRo aligns hierarchical levels by adjusting the hyperbolic radius and refines semantic relationships through angular alignment using an orthogonal transformation that theoretically preserves the hyperbolic radius. Experiments on standard open-vocabulary semantic segmentation benchmarks demonstrate that HyRo achieves state-of-the-art performance over prior methods.

Semantic segmentation in hyperbolic space enables compact modeling of hierarchical structure while providing inherent uncertainty quantification. Prior approaches predominantly rely on the Poincaré 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.