20 research outputs found
Spectrally approximating large graphs with smaller graphs
How does coarsening affect the spectrum of a general graph? We provide
conditions such that the principal eigenvalues and eigenspaces of a coarsened
and original graph Laplacian matrices are close. The achieved approximation is
shown to depend on standard graph-theoretic properties, such as the degree and
eigenvalue distributions, as well as on the ratio between the coarsened and
actual graph sizes. Our results carry implications for learning methods that
utilize coarsening. For the particular case of spectral clustering, they imply
that coarse eigenvectors can be used to derive good quality assignments even
without refinement---this phenomenon was previously observed, but lacked formal
justification.Comment: 22 pages, 10 figure
Spectrally approximating large graphs with smaller graphs
How does coarsening affect the spectrum of a general graph? We provide conditions such that the principal eigenvalues and eigenspaces of a coarsened and original graph Laplacian matrices are close. The achieved approximation is shown to depend on standard graph-theoretic properties, such as the degree and eigenvalue distributions, as well as on the ratio between the coarsened and actual graph sizes. Our results carry implications for learning methods that utilize coarsening. For the particular case of spectral clustering, they imply that coarse eigenvectors can be used to derive good quality assignments even without refinement---this phenomenon was previously observed, but lacked formal justification
Low-Rank Projections of GCNs Laplacian
In this work, we study the behavior of standard models for community
detection under spectral manipulations. Through various ablation experiments,
we evaluate the impact of bandpass filtering on the performance of a GCN: we
empirically show that most of the necessary and used information for nodes
classification is contained in the low-frequency domain, and thus contrary to
images, high frequencies are less crucial to community detection. In
particular, it is sometimes possible to obtain accuracies at a state-of-the-art
level with simple classifiers that rely only on a few low frequencies
Unsupervised Learning of Graph Hierarchical Abstractions with Differentiable Coarsening and Optimal Transport
Hierarchical abstractions are a methodology for solving large-scale graph
problems in various disciplines. Coarsening is one such approach: it generates
a pyramid of graphs whereby the one in the next level is a structural summary
of the prior one. With a long history in scientific computing, many coarsening
strategies were developed based on mathematically driven heuristics. Recently,
resurgent interests exist in deep learning to design hierarchical methods
learnable through differentiable parameterization. These approaches are paired
with downstream tasks for supervised learning. In practice, however, supervised
signals (e.g., labels) are scarce and are often laborious to obtain. In this
work, we propose an unsupervised approach, coined OTCoarsening, with the use of
optimal transport. Both the coarsening matrix and the transport cost matrix are
parameterized, so that an optimal coarsening strategy can be learned and
tailored for a given set of graphs. We demonstrate that the proposed approach
produces meaningful coarse graphs and yields competitive performance compared
with supervised methods for graph classification and regression.Comment: AAAI 2021. Code is available at
https://github.com/matenure/OTCoarsenin
SMGRL: Scalable Multi-resolution Graph Representation Learning
Graph convolutional networks (GCNs) allow us to learn topologically-aware
node embeddings, which can be useful for classification or link prediction.
However, they are unable to capture long-range dependencies between nodes
without adding additional layers -- which in turn leads to over-smoothing and
increased time and space complexity. Further, the complex dependencies between
nodes make mini-batching challenging, limiting their applicability to large
graphs. We propose a Scalable Multi-resolution Graph Representation Learning
(SMGRL) framework that enables us to learn multi-resolution node embeddings
efficiently. Our framework is model-agnostic and can be applied to any existing
GCN model. We dramatically reduce training costs by training only on a
reduced-dimension coarsening of the original graph, then exploit
self-similarity to apply the resulting algorithm at multiple resolutions. The
resulting multi-resolution embeddings can be aggregated to yield high-quality
node embeddings that capture both long- and short-range dependencies. Our
experiments show that this leads to improved classification accuracy, without
incurring high computational costs.Comment: 22 page