142 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
Capacity Bounds for Hyperbolic Neural Network Representations of Latent Tree Structures
We study the representation capacity of deep hyperbolic neural networks
(HNNs) with a ReLU activation function. We establish the first proof that HNNs
can -isometrically embed any finite weighted tree into a
hyperbolic space of dimension at least equal to with prescribed
sectional curvature (where
being optimal). We establish rigorous upper bounds for the network complexity
on an HNN implementing the embedding. We find that the network complexity of
HNN implementing the graph representation is independent of the representation
fidelity/distortion. We contrast this result against our lower bounds on
distortion which any ReLU multi-layer perceptron (MLP) must exert when
embedding a tree with leaves into a -dimensional Euclidean space,
which we show at least ; independently of the depth, width,
and (possibly discontinuous) activation function defining the MLP.Comment: 22 Pages + References, 1 Table, 4 Figure
Hyperbolic Geometry in Computer Vision: A Novel Framework for Convolutional Neural Networks
Real-world visual data exhibit intrinsic hierarchical structures that can be
represented effectively in hyperbolic spaces. Hyperbolic neural networks (HNNs)
are a promising approach for learning feature representations in such spaces.
However, current methods in computer vision rely on Euclidean backbones and
only project features to the hyperbolic space in the task heads, limiting their
ability to fully leverage the benefits of hyperbolic geometry. To address this,
we present HCNN, the first fully hyperbolic convolutional neural network (CNN)
designed for computer vision tasks. Based on the Lorentz model, we generalize
fundamental components of CNNs and propose novel formulations of the
convolutional layer, batch normalization, and multinomial logistic regression
(MLR). Experimentation on standard vision tasks demonstrates the effectiveness
of our HCNN framework and the Lorentz model in both hybrid and fully hyperbolic
settings. Overall, we aim to pave the way for future research in hyperbolic
computer vision by offering a new paradigm for interpreting and analyzing
visual data. Our code is publicly available at
https://github.com/kschwethelm/HyperbolicCV
Hyperbolic Graph Representation Learning: A Tutorial
Graph-structured data are widespread in real-world applications, such as
social networks, recommender systems, knowledge graphs, chemical molecules etc.
Despite the success of Euclidean space for graph-related learning tasks, its
ability to model complex patterns is essentially constrained by its
polynomially growing capacity. Recently, hyperbolic spaces have emerged as a
promising alternative for processing graph data with tree-like structure or
power-law distribution, owing to the exponential growth property. Different
from Euclidean space, which expands polynomially, the hyperbolic space grows
exponentially which makes it gains natural advantages in abstracting tree-like
or scale-free graphs with hierarchical organizations.
In this tutorial, we aim to give an introduction to this emerging field of
graph representation learning with the express purpose of being accessible to
all audiences. We first give a brief introduction to graph representation
learning as well as some preliminary Riemannian and hyperbolic geometry. We
then comprehensively revisit the hyperbolic embedding techniques, including
hyperbolic shallow models and hyperbolic neural networks. In addition, we
introduce the technical details of the current hyperbolic graph neural networks
by unifying them into a general framework and summarizing the variants of each
component. Moreover, we further introduce a series of related applications in a
variety of fields. In the last part, we discuss several advanced topics about
hyperbolic geometry for graph representation learning, which potentially serve
as guidelines for further flourishing the non-Euclidean graph learning
community.Comment: Accepted as ECML-PKDD 2022 Tutoria
Hyperbolic Graph Neural Networks at Scale: A Meta Learning Approach
The progress in hyperbolic neural networks (HNNs) research is hindered by
their absence of inductive bias mechanisms, which are essential for
generalizing to new tasks and facilitating scalable learning over large
datasets. In this paper, we aim to alleviate these issues by learning
generalizable inductive biases from the nodes' local subgraph and transfer them
for faster learning over new subgraphs with a disjoint set of nodes, edges, and
labels in a few-shot setting. We introduce a novel method, Hyperbolic GRAph
Meta Learner (H-GRAM), that, for the tasks of node classification and link
prediction, learns transferable information from a set of support local
subgraphs in the form of hyperbolic meta gradients and label hyperbolic
protonets to enable faster learning over a query set of new tasks dealing with
disjoint subgraphs. Furthermore, we show that an extension of our meta-learning
framework also mitigates the scalability challenges seen in HNNs faced by
existing approaches. Our comparative analysis shows that H-GRAM effectively
learns and transfers information in multiple challenging few-shot settings
compared to other state-of-the-art baselines. Additionally, we demonstrate
that, unlike standard HNNs, our approach is able to scale over large graph
datasets and improve performance over its Euclidean counterparts.Comment: Accepted to NeurIPS 2023. 14 pages of main paper, 5 pages of
supplementar
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