56 research outputs found
Hyperbolic Interaction Model For Hierarchical Multi-Label Classification
Different from the traditional classification tasks which assume mutual
exclusion of labels, hierarchical multi-label classification (HMLC) aims to
assign multiple labels to every instance with the labels organized under
hierarchical relations. Besides the labels, since linguistic ontologies are
intrinsic hierarchies, the conceptual relations between words can also form
hierarchical structures. Thus it can be a challenge to learn mappings from word
hierarchies to label hierarchies. We propose to model the word and label
hierarchies by embedding them jointly in the hyperbolic space. The main reason
is that the tree-likeness of the hyperbolic space matches the complexity of
symbolic data with hierarchical structures. A new Hyperbolic Interaction Model
(HyperIM) is designed to learn the label-aware document representations and
make predictions for HMLC. Extensive experiments are conducted on three
benchmark datasets. The results have demonstrated that the new model can
realistically capture the complex data structures and further improve the
performance for HMLC comparing with the state-of-the-art methods. To facilitate
future research, our code is publicly available
Adversarial Autoencoders with Constant-Curvature Latent Manifolds
Constant-curvature Riemannian manifolds (CCMs) have been shown to be ideal
embedding spaces in many application domains, as their non-Euclidean geometry
can naturally account for some relevant properties of data, like hierarchy and
circularity. In this work, we introduce the CCM adversarial autoencoder
(CCM-AAE), a probabilistic generative model trained to represent a data
distribution on a CCM. Our method works by matching the aggregated posterior of
the CCM-AAE with a probability distribution defined on a CCM, so that the
encoder implicitly learns to represent data on the CCM to fool the
discriminator network. The geometric constraint is also explicitly imposed by
jointly training the CCM-AAE to maximise the membership degree of the
embeddings to the CCM. While a few works in recent literature make use of
either hyperspherical or hyperbolic manifolds for different learning tasks,
ours is the first unified framework to seamlessly deal with CCMs of different
curvatures. We show the effectiveness of our model on three different datasets
characterised by non-trivial geometry: semi-supervised classification on MNIST,
link prediction on two popular citation datasets, and graph-based molecule
generation using the QM9 chemical database. Results show that our method
improves upon other autoencoders based on Euclidean and non-Euclidean
geometries on all tasks taken into account.Comment: Submitted to Applied Soft Computin
The Numerical Stability of Hyperbolic Representation Learning
Given the exponential growth of the volume of the ball w.r.t. its radius, the
hyperbolic space is capable of embedding trees with arbitrarily small
distortion and hence has received wide attention for representing hierarchical
datasets. However, this exponential growth property comes at a price of
numerical instability such that training hyperbolic learning models will
sometimes lead to catastrophic NaN problems, encountering unrepresentable
values in floating point arithmetic. In this work, we carefully analyze the
limitation of two popular models for the hyperbolic space, namely, the
Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit
arithmetic system, the Poincar\'e ball has a relatively larger capacity than
the Lorentz model for correctly representing points. Then, we theoretically
validate the superiority of the Lorentz model over the Poincar\'e ball from the
perspective of optimization. Given the numerical limitations of both models, we
identify one Euclidean parametrization of the hyperbolic space which can
alleviate these limitations. We further extend this Euclidean parametrization
to hyperbolic hyperplanes and exhibits its ability in improving the performance
of hyperbolic SVM
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