Geometry-aware Distance Measure for Diverse Hierarchical Structures in Hyperbolic Spaces
Keywords: hyperbolic learning, metric learning
Abstract: Learning in hyperbolic spaces has gained increasing attention due to the superior capability of modeling hierarchical structures. Existing hyperbolic learning methods use a fixed distance measure that assumes a uniform hierarchical structure across all data points. However, this assumption does not always hold in real-world scenarios, considering the diversity of the hierarchical structures of data. This work proposes to learn geometry aware distance measures that dynamically adjust to accommodate diverse hierarchical structures in hyperbolic spaces. We derive geometry aware distance measures by generating projections and curvatures for each pair of samples, which maps each pair to a suitable hyperbolic space. We introduce a revised low-rank decomposition scheme and a hard-pair mining mechanism to reduce the computational cost incurred by the pairwise generation without compromising accuracy. Moreover, we derive an upper bound of the low-rank approximation error via Talagrand concentration inequality to guarantee the effectiveness of our low-rank decomposition scheme. Theoretical analysis and experiments on standard image classification and few-shot learning tasks affirm the effectiveness of our method in refining hyperbolic learning through our geometry aware distance measures.
Primary Area: transfer learning, meta learning, and lifelong learning
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Submission Number: 9248
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