Hyperbolic Concept Bottleneck Models

Problem
This paper addresses the limitations of existing Concept Bottleneck Models (CBMs) that embed concepts in flat Euclidean space, which inadequately represents the hierarchical and structured nature of concepts. The authors propose Hyperbolic Concept Bottleneck Models (HypCBM) to better capture the semantic relationships among concepts. This work is presented as a preprint and has not yet undergone peer review.

Method
HypCBM reformulates the concept activation process by leveraging hyperbolic geometry, which allows for the representation of concepts as asymmetric geometric containment. This approach utilizes entailment cones to encode a natural test-time activation signal, where the margin of inclusion within a concept’s entailment cone results in sparse, hierarchy-aware activations. The model does not require additional supervision or learned modules, making it a post-hoc framework. Furthermore, the authors introduce an adaptive scaling law that facilitates coherent propagation of user corrections through the concept hierarchy, enhancing the model’s interpretability and usability.

Results
Empirical evaluations demonstrate that HypCBM achieves performance comparable to Euclidean models trained on 20 times more data, particularly in sparse regimes critical for human interpretability. The model exhibits stronger hierarchical consistency and improved robustness against input corruptions. Specific metrics and benchmarks were not disclosed in the abstract, but the authors emphasize the effectiveness of HypCBM in maintaining interpretability while leveraging the structured nature of concepts.

Limitations
The authors acknowledge that their approach is a post-hoc framework, which may limit its applicability in scenarios requiring end-to-end training. Additionally, while the model shows improved robustness and interpretability, the paper does not provide extensive details on the computational overhead introduced by hyperbolic embeddings compared to traditional Euclidean approaches. The scalability of the model to larger, more complex datasets and its performance in diverse application domains remain to be fully explored.

Why it matters
The introduction of HypCBM has significant implications for the field of interpretable machine learning. By effectively capturing the hierarchical structure of concepts, this model enhances the interpretability of neural networks, making them more aligned with human cognitive understanding. The ability to perform coherent interventions through the concept tree could facilitate more intuitive user interactions and corrections, potentially leading to more reliable AI systems. This work opens avenues for future research on hyperbolic embeddings in other domains and encourages the development of models that prioritize semantic structure in their design.

Authors: Daniel Uyterlinde, Swasti Shreya Mishra, Pascal Mettes
Source: arXiv:2605.06440
URL: https://arxiv.org/abs/2605.06440v1