We propose a hyperbolic set-to-set distance measure for computing dissimilarity between sets in hyperbolic space. While point-to-point distances in hyperbolic space effectively capture hierarchical relationships between data points, many real-world applications require comparing sets of hyperbolic data points, where the local structure and the global structure of the sets carry crucial semantic information. The proposed the \underline{h}yperbolic \underline{s}et-\underline{to}-\underline{s}et \underline{d}istance measure (HS2SD) integrates both global and local structural information: global structure through geodesic distances between Einstein midpoints of hyperbolic sets, and local structure through topological characteristics of the two sets. To efficiently compute topological differences, we prove that using a finite Thue-Morse sequence of degree and adjacency matrices can serve as a robust approximation to capture the topological structure of a set. In this case, by considering the topological differences, HS2SD provides a more nuanced understanding of the relationships between two hyperbolic sets. Empirical evaluation on entity matching, standard image classification, and few-shot image classification demonstrates that our distance measure outperforms existing methods by effectively modeling the hierarchical and complex relationships inherent in hyperbolic sets.

This paper proposes a novel framework for multi-label image recognition without any training data, called data-free framework, which uses knowledge of pre-trained Large Language Model (LLM) to learn prompts to adapt pretrained Vision-Language Model (VLM) like CLIP to multilabel classification. Through asking LLM by well-designed questions, we acquire comprehensive knowledge about characteristics and contexts of objects, which provides valuable text descriptions for learning prompts. Then we propose a hierarchical prompt learning method by taking the multi-label dependency into consideration, wherein a subset of category-specific prompt tokens are shared when the corresponding objects exhibit similar attributes or are more likely to co-occur. Benefiting from the remarkable alignment between visual and linguistic semantics of CLIP, the hierarchical prompts learned from text descriptions are applied to perform classification of images during inference. Our framework presents a new way to explore the synergies between multiple pre-trained models for novel category recognition. Extensive experiments on three public datasets (MS-COCO, VOC2007, and NUS-WIDE) demonstrate that our method achieves better results than the state-of-the-art methods, especially outperforming the zero-shot multi-label recognition methods by 4.7% in mAP on MS-COCO.