The application of graph Laplacian eigenvectors has been quite popular in the graph signal processing field: one can use them as ingredients to design smooth multiscale basis. Our long-term goal is to study and understand the dual geometry of graph Laplacian eigenvectors. In order to do that, it is necessary to define a certain metric to measure the behavioral differences between each pair of the eigenvectors. Saito (2018) considered the ramified optimal transportation (ROT) cost between the square of the eigenvectors as such a metric. Clonginger and Steinerberger (2018) proposed a way to measure the affinity (or `similarity') between the eigenvectors based on their Hadamard (HAD) product. In this article, we propose a simplified ROT metric that is more computational efficient and introduce two more ways to define the distance between the eigenvectors, i.e., the time-stepping diffusion (TSD) metric and the difference of absolute gradient (DAG) pseudometric. The TSD metric measures the cost of "flattening" the initial graph signal via diffusion process up to certain time, hence it can be viewed as a time-dependent version of the ROT metric. The DAG pseudometric is the l2-distance between the feature vectors derived from the eigenvectors, in particular, the absolute gradients of the eigenvectors. We then compare the performance of ROT, HAD and the two new "metrics: on different kinds of graphs. Finally, we investigate their relationship as well as their pros and cons.
Keywords: Graph Laplacian eigenvectors, metrics between orthonormal vectors, dual geometry of graph Laplacian eigenvectors, multiscale basis dictionaries on graphs, heat diffusion on graphs, Wasserstein distance, optimal transport
more »
« less
This content will become publicly available on May 1, 2024
Sign and Basis Invariant Networks for Spectral Graph Representation Learning
We introduce SignNet and BasisNet---new neural architectures that are invariant to two key symmetries displayed by eigenvectors: (i) sign flips, since if v is an eigenvector then so is -v; and (ii) more general basis symmetries, which occur in higher dimensional eigenspaces with infinitely many choices of basis eigenvectors. We prove that under certain conditions our networks are universal, i.e., they can approximate any continuous function of eigenvectors with the desired invariances. When used with Laplacian eigenvectors, our networks are provably more expressive than existing spectral methods on graphs; for instance, they subsume all spectral graph convolutions, certain spectral graph invariants, and previously proposed graph positional encodings as special cases. Experiments show that our networks significantly outperform existing baselines on molecular graph regression, learning expressive graph representations, and learning neural fields on triangle meshes. Our code is available at https://github.com/cptq/SignNet-BasisNet.
more »
« less
- Award ID(s):
- 2134108
- NSF-PAR ID:
- 10468315
- Publisher / Repository:
- International Conference on Learning Representations (ICLR)
- Date Published:
- Subject(s) / Keyword(s):
- ["Graph Neural Networks, positional encodings, machine learning"]
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Learning representations of sets of nodes in a graph is crucial for applications ranging from node-role discovery to link prediction and molecule classification. Graph Neural Networks (GNNs) have achieved great success in graph representation learning. However, expressive power of GNNs is limited by the 1-Weisfeiler-Lehman (WL) test and thus GNNs generate identical representations for graph substructures that may in fact be very different. More powerful GNNs, proposed recently by mimicking higher-order-WL tests, only focus on representing entire graphs and they are computationally inefficient as they cannot utilize sparsity of the underlying graph. Here we propose and mathematically analyze a general class of structure related features, termed Distance Encoding (DE). DE assists GNNs in representing any set of nodes, while providing strictly more expressive power than the 1-WL test. DE captures the distance between the node set whose representation is to be learned and each node in the graph. To capture the distance DE can apply various graph-distance measures such as shortest path distance or generalized PageRank scores. We propose two ways for GNNs to use DEs (1) as extra node features, and (2) as controllers of message aggregation in GNNs. Both approaches can utilize the sparse structure of the underlying graph, which leads to computational efficiency and scalability. We also prove that DE can distinguish node sets embedded in almost all regular graphs where traditional GNNs always fail. We evaluate DE on three tasks over six real networks: structural role prediction, link prediction, and triangle prediction. Results show that our models outperform GNNs without DE by up-to 15% in accuracy and AUROC. Furthermore, our models also significantly outperform other state-of-the-art methods especially designed for the above tasks.more » « less
-
Inspired by the extensive success of deep learning, graph neural networks (GNNs) have been proposed to learn expressive node representations and demonstrated promising performance in various graph learning tasks. However, existing endeavors predominately focus on the conventional semi-supervised setting where relatively abundant gold-labeled nodes are provided. While it is often impractical due to the fact that data labeling is unbearably laborious and requires intensive domain knowledge, especially when considering the heterogeneity of graph-structured data. Under the few-shot semi-supervised setting, the performance of most of the existing GNNs is inevitably undermined by the overfitting and oversmoothing issues, largely owing to the shortage of labeled data. In this paper, we propose a decoupled network architecture equipped with a novel meta-learning algorithm to solve this problem. In essence, our framework Meta-PN infers high-quality pseudo labels on unlabeled nodes via a meta-learned label propagation strategy, which effectively augments the scarce labeled data while enabling large receptive fields during training. Extensive experiments demonstrate that our approach offers easy and substantial performance gains compared to existing techniques on various benchmark datasets. The implementation and extended manuscript of this work are publicly available at https://github.com/kaize0409/Meta-PN.more » « less
-
null (Ed.)From the perspective of expressive power, this work compares multi-layer Graph Neural Networks (GNNs) with a simplified alternative that we call Graph-Augmented Multi-Layer Perceptrons (GA-MLPs), which first augments node features with certain multi-hop operators on the graph and then applies an MLP in a node-wise fashion. From the perspective of graph isomorphism testing, we show both theoretically and numerically that GA-MLPs with suitable operators can distinguish almost all non-isomorphic graphs, just like the Weifeiler-Lehman (WL) test. However, by viewing them as node-level functions and examining the equivalence classes they induce on rooted graphs, we prove a separation in expressive power between GA-MLPs and GNNs that grows exponentially in depth. In particular, unlike GNNs, GA-MLPs are unable to count the number of attributed walks. We also demonstrate via community detection experiments that GA-MLPs can be limited by their choice of operator family, as compared to GNNs with higher flexibility in learning.more » « less
-
null (Ed.)From the perspective of expressive power, this work compares multi-layer Graph Neural Networks (GNNs) with a simplified alternative thatwe call Graph-Augmented Multi-Layer Perceptrons (GA-MLPs), which first augments node features with certain multi-hop operators on the graph and then applies an MLP in a node-wise fashion. From the perspective of graph isomorphism testing,we showboth theoretically and numerically that GA-MLPs with suitable operators can distinguish almost all non-isomorphic graphs, just like the Weifeiler-Lehman (WL) test. However, by viewing them as node-level functions and examining the equivalence classes they induce on rooted graphs, we prove a separation in expressive power between GA-MLPs and GNNs that grows exponentially in depth. In particular, unlike GNNs, GA-MLPs are unable to count the number of attributed walks. We also demonstrate via community detection experiments that GA-MLPs can be limited by their choice of operator family, as compared to GNNs with higher flexibility in learning.more » « less
