# Hyperbolic Multimodal Continual Learning: A Closest-Admissible Solution

arXiv: https://arxiv.org/abs/2609.29329 · HTML: https://mlsanity.com/paper/2609.29329/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 58 of its 82 references that are in the map.
- First posted on arXiv 24 Sep 2026.

## Builds on (most cited first, 30 of 58)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Describing Textures in the Wild](https://mlsanity.com/paper/1311.3618/) (2013)
- [Gradient Episodic Memory for Continual Learning](https://mlsanity.com/paper/1706.08840/) (2017)
- [Fine-Grained Visual Classification of Aircraft](https://mlsanity.com/paper/1306.5151/) (2013)
- [EuroSAT: A Novel Dataset and Deep Learning Benchmark for Land Use and Land Cover Classification](https://mlsanity.com/paper/1709.00029/) (2017)
- [CLEVR: A Diagnostic Dataset for Compositional Language and Elementary Visual Reasoning](https://mlsanity.com/paper/1612.06890/) (2016)
- [Progressive Neural Networks](https://mlsanity.com/paper/1606.04671/) (2016)
- [Dark Experience for General Continual Learning: a Strong, Simple Baseline](https://mlsanity.com/paper/2004.07211/) (2020)
- [PackNet: Adding Multiple Tasks to a Single Network by Iterative Pruning](https://mlsanity.com/paper/1711.05769/) (2017)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Rotation Equivariant CNNs for Digital Pathology](https://mlsanity.com/paper/1806.03962/) (2018)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Boosting Continual Learning of Vision-Language Models via Mixture-of-Experts Adapters](https://mlsanity.com/paper/2403.11549/) (2024)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [VAST: A Vision-Audio-Subtitle-Text Omni-Modality Foundation Model and Dataset](https://mlsanity.com/paper/2305.18500/) (2023)
- [ImageBind-LLM: Multi-modality Instruction Tuning](https://mlsanity.com/paper/2309.03905/) (2023)
- [Hyperbolic Image-Text Representations](https://mlsanity.com/paper/2304.09172/) (2023)
- [Preventing Zero-Shot Transfer Degradation in Continual Learning of Vision-Language Models](https://mlsanity.com/paper/2303.06628/) (2023)
- [Fully Hyperbolic Neural Networks](https://mlsanity.com/paper/2105.14686/) (2021)
- [Compositional Entailment Learning for Hyperbolic Vision-Language Models](https://mlsanity.com/paper/2410.06912/) (2024)
- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [A Survey of Personalized Large Language Models: Progress and Future Directions](https://mlsanity.com/paper/2502.11528/) (2025)
- [Hyperbolic Contrastive Learning for Visual Representations beyond Objects](https://mlsanity.com/paper/2212.00653/) (2022)
- [The Numerical Stability of Hyperbolic Representation Learning](https://mlsanity.com/paper/2211.00181/) (2022)
- [Evaluation Measures for Hierarchical Classification: a unified view and novel approaches](https://mlsanity.com/paper/1306.6802/) (2013)

## Abstract (arXiv)

Existing continual-learning methods protect parameters, replayed examples, or Euclidean feature subspaces. When applied to hyperbolic multimodal models, they do not explicitly preserve the Lorentz geometry that jointly encodes within-modality similarity, cross-modal correspondence, and semantic hierarchy; sequential updates can therefore retain task scores while still distorting previously learned relations. We address this gap with Hyperbolic Multimodal Continual Learning (HMCL). We show that preserving the old multimodal geometry amounts to restricting all modalities to one shared hyperbolic isometry, which induces a family of admissible first-order parameter changes. We formulate a joint closest-admissible (CA) correction that retains the shared rotation best matching the candidate modal updates; its minimal-rotation (MR) special case fixes this rotation to zero. Both variants correct the displacement realized by AdamW, and task anchoring bounds within-task accumulation while preserving learning freedom. Across a unified 16-task classification-retrieval stream with three hyperbolic backbones, HMCL improves final performance and backward transfer over sequential fine-tuning and four continual-learning baselines; HMCL-CA gives the highest Overall score on every backbone. A modality-extended stream confirms the retrieval gains. Representation analyses find 81.2 to 95.5 percent less radial, angular, cross-modal, and paired-distance drift; ImageNet-WordNet results show better semantic ancestry and radial hierarchy.