Hyperbolic Busemann Learning with Ideal Prototypes
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by
Mina Ghadimi Atigh, Martin Keller-Ressel, Pascal Mettes
2021
Abstract
Hyperbolic space has become a popular choice of manifold for representation
learning of arbitrary data, from tree-like structures and text to graphs.
Building on the success of deep learning with prototypes in Euclidean and
hyperspherical spaces, a few recent works have proposed hyperbolic prototypes
for classification. Such approaches enable effective learning in
low-dimensional output spaces and can exploit hierarchical relations amongst
classes, but require privileged information about class labels to position the
hyperbolic prototypes. In this work, we propose Hyperbolic Busemann Learning.
The main idea behind our approach is to position prototypes on the ideal
boundary of the Poincare ball, which does not require prior label knowledge. To
be able to compute proximities to ideal prototypes, we introduce the penalised
Busemann loss. We provide theory supporting the use of ideal prototypes and the
proposed loss by proving its equivalence to logistic regression in the
one-dimensional case. Empirically, we show that our approach provides a natural
interpretation of classification confidence, while outperforming recent
hyperspherical and hyperbolic prototype approaches.
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