Spectral embedding is a powerful graph embedding technique that has received a lot of attention recently due to its effectiveness on Graph Transformers. However, from a theoretical perspective, the universal expressive power of spectral embedding comes at the price of losing two important invariance properties of graphs, sign and basis invariance, which also limits its effectiveness on graph data. To remedy this issue, many previous methods developed costly approaches to learn new invariants and suffer from high computation complexity. In this work, we explore a minimal approach that resolves the ambiguity issues by directly finding canonical directions for the eigenvectors, named Laplacian Canonization (LC). As a pure pre-processing method, LC is light-weighted and can be applied to any existing GNNs. We provide a thorough investigation, from theory to algorithm, on this approach, and discover an efficient algorithm named Maximal Axis Projection (MAP) that works for both sign and basis invariance and successfully canonizes more than 90% of all eigenvectors. Experiments on real-world benchmark datasets like ZINC, MOLTOX21, and MOLPCBA show that MAP consistently outperforms existing methods while bringing minimal computation overhead.

Spectral embedding is a powerful graph embedding technique that has received a lot of attention recently due to its effectiveness on Graph Transformers.

In many applications, we desire neural networks to exhibit invariance or equivariance to certain groups due to symmetries inherent in the data. Recently, frame-averaging methods emerged to be a unified framework for attaining symmetries efficiently by averaging over input-dependent subsets of the group, i.e., frames. What we currently lack is a principled understanding of the design of frames. In this work, we introduce a canonization perspective that provides an essential and complete view of the design of frames. Canonization is a classic approach for attaining invariance by mapping inputs to their canonical forms. We show that there exists an inherent connection between frames and canonical forms. Leveraging this connection, we can efficiently compare the complexity of frames as well as determine the optimality of certain frames. Guided by this principle, we design novel frames for eigenvectors that are strictly superior to existing methods -- some are even optimal -- both theoretically and empirically. The reduction to the canonization perspective further uncovers equivalences between previous methods. These observations suggest that canonization provides a fundamental understanding of existing frame-averaging methods and unifies existing equivariant and invariant learning methods.

Spectral embedding is a powerful graph embedding technique that has received a lot of attention recently due to its effectiveness on Graph Transformers. However, from a theoretical perspective, the universal expressive power of spectral embedding comes at the price of losing two important invariance properties of graphs, sign and basis invariance, which also limits its effectiveness on graph data. To remedy this issue, many previous methods developed costly approaches to learn new invariants and suffer from high computation complexity. In this work, we explore a minimal approach that resolves the ambiguity issues by directly finding canonical directions for the eigenvectors, named Laplacian Canonization (LC). As a pure pre-processing method, LC is light-weighted and can be applied to any existing GNNs. We provide a thorough investigation, from theory to algorithm, on this approach, and discover an efficient algorithm named Maximal Axis Projection (MAP) that works for both sign and basis invariance and successfully canonizes more than 90% of all eigenvectors. Experiments on real-world benchmark datasets like ZINC, MOLTOX21, and MOLPCBA show that MAP consistently outperforms existing methods while bringing minimal computation overhead.

Message passing graph neural networks iteratively compute node embeddings by aggregating messages from all neighbors. This procedure can be viewed as a neural variant of the Weisfeiler-Leman method, which limits their expressive power. Moreover, oversmoothing and oversquashing restrict the number of layers these networks can effectively utilize. The repeated exchange and encoding of identical information in message passing amplifies oversquashing. We propose a novel aggregation scheme based on neighborhood trees, which allows for controlling the redundancy by pruning branches of the unfolding trees underlying standard message passing. We prove that reducing redundancy improves expressivity and experimentally show that it alleviates oversquashing. We investigate the interaction between redundancy in message passing and redundancy in computation and propose a compact representation of neighborhood trees, from which we compute node and graph embeddings via a neural tree canonization technique. Our method is provably more expressive than the Weisfeiler-Leman method, less susceptible to oversquashing than message passing neural networks, and provides high classification accuracy on widely-used benchmark datasets.

Lifted inference scales to large probability models by exploiting symmetry. However, existing exact lifted inference techniques do not apply to general factor graphs, as they require a relational representation. In this work we provide a theoretical framework and algorithm for performing exact lifted inference on symmetric factor graphs by computing colored graph automorphisms, as is often done for approximate lifted inference. Our key insight is to represent variable assignments directly in the colored factor graph encoding. This allows us to generate representatives and compute the size of each orbit of the symmetric distribution. In addition to exact inference, we use this encoding to implement an MCMC algorithm that explores the space of orbits quickly by uniform orbit sampling.