Vincent Létourneau

Semih Cantürk, Renming Liu, Olivier Lapointe-Gagné et al. · mila

Positional and structural encodings (PSE) enable better identifiability of nodes within a graph, rendering them essential tools for empowering modern GNNs, and in particular graph Transformers. However, designing PSEs that work optimally for all graph prediction tasks is a challenging and unsolved problem. Here, we present the Graph Positional and Structural Encoder (GPSE), the first-ever graph encoder designed to capture rich PSE representations for augmenting any GNN. GPSE learns an efficient common latent representation for multiple PSEs, and is highly transferable: The encoder trained on a particular graph dataset can be used effectively on datasets drawn from markedly different distributions and modalities. We show that across a wide range of benchmarks, GPSE-enhanced models can significantly outperform those that employ explicitly computed PSEs, and at least match their performance in others. Our results pave the way for the development of foundational pre-trained graph encoders for extracting positional and structural information, and highlight their potential as a more powerful and efficient alternative to explicitly computed PSEs and existing self-supervised pre-training approaches. Our framework and pre-trained models are publicly available at https://github.com/G-Taxonomy-Workgroup/GPSE. For convenience, GPSE has also been integrated into the PyG library to facilitate downstream applications.

Gabriela Moisescu-Pareja, Gavin McCracken, Harley Wiltzer et al.

The Clock and Pizza interpretations, associated with architectures differing in either uniform or learnable attention, were introduced to argue that different architectural designs can yield distinct circuits for modular addition. In this work, we show that this is not the case, and that both uniform attention and trainable attention architectures implement the same algorithm via topologically and geometrically equivalent representations. Our methodology goes beyond the interpretation of individual neurons and weights. Instead, we identify all of the neurons corresponding to each learned representation and then study the collective group of neurons as one entity. This method reveals that each learned representation is a manifold that we can study utilizing tools from topology. Based on this insight, we can statistically analyze the learned representations across hundreds of circuits to demonstrate the similarity between learned modular addition circuits that arise naturally from common deep learning paradigms.

Laurent Cheret, Vincent Létourneau, Isar Nejadgholi et al.

We study a simple unsupervised regularization scheme for autoencoders called Manifold-Matching (MMAE): we align the pairwise distances in the latent space to those of the input data space by minimizing mean squared error. Because alignment occurs on pairwise distances rather than coordinates, it can also be extended to a lower-dimensional representation of the data, adding flexibility to the method. We find that this regularization outperforms similar methods on metrics based on preservation of nearest-neighbor distances and persistent homology-based measures. We also observe that MMAE provides a scalable approximation of Multi-Dimensional Scaling (MDS).

Gavin McCracken, Gabriela Moisescu-Pareja, Vincent Letourneau et al.

We propose a testable universality hypothesis, asserting that seemingly disparate neural network solutions observed in the simple task of modular addition are unified under a common abstract algorithm. While prior work interpreted variations in neuron-level representations as evidence for distinct algorithms, we demonstrate - through multi-level analyses spanning neurons, neuron clusters, and entire networks - that multilayer perceptrons and transformers universally implement the abstract algorithm we call the approximate Chinese Remainder Theorem. Crucially, we introduce approximate cosets and show that neurons activate exclusively on them. Furthermore, our theory works for deep neural networks (DNNs). It predicts that universally learned solutions in DNNs with trainable embeddings or more than one hidden layer require only O(log n) features, a result we empirically confirm. This work thus provides the first theory-backed interpretation of multilayer networks solving modular addition. It advances generalizable interpretability and opens a testable universality hypothesis for group multiplication beyond modular addition.

Devin Kreuzer, Dominique Beaini, William L. Hamilton et al.

In recent years, the Transformer architecture has proven to be very successful in sequence processing, but its application to other data structures, such as graphs, has remained limited due to the difficulty of properly defining positions. Here, we present the $\textit{Spectral Attention Network}$ (SAN), which uses a learned positional encoding (LPE) that can take advantage of the full Laplacian spectrum to learn the position of each node in a given graph. This LPE is then added to the node features of the graph and passed to a fully-connected Transformer. By leveraging the full spectrum of the Laplacian, our model is theoretically powerful in distinguishing graphs, and can better detect similar sub-structures from their resonance. Further, by fully connecting the graph, the Transformer does not suffer from over-squashing, an information bottleneck of most GNNs, and enables better modeling of physical phenomenons such as heat transfer and electric interaction. When tested empirically on a set of 4 standard datasets, our model performs on par or better than state-of-the-art GNNs, and outperforms any attention-based model by a wide margin, becoming the first fully-connected architecture to perform well on graph benchmarks.

Dominique Beaini, Saro Passaro, Vincent Létourneau et al.

The lack of anisotropic kernels in graph neural networks (GNNs) strongly limits their expressiveness, contributing to well-known issues such as over-smoothing. To overcome this limitation, we propose the first globally consistent anisotropic kernels for GNNs, allowing for graph convolutions that are defined according to topologicaly-derived directional flows. First, by defining a vector field in the graph, we develop a method of applying directional derivatives and smoothing by projecting node-specific messages into the field. Then, we propose the use of the Laplacian eigenvectors as such vector field. We show that the method generalizes CNNs on an $n$-dimensional grid and is provably more discriminative than standard GNNs regarding the Weisfeiler-Lehman 1-WL test. We evaluate our method on different standard benchmarks and see a relative error reduction of 8% on the CIFAR10 graph dataset and 11% to 32% on the molecular ZINC dataset, and a relative increase in precision of 1.6% on the MolPCBA dataset. An important outcome of this work is that it enables graph networks to embed directions in an unsupervised way, thus allowing a better representation of the anisotropic features in different physical or biological problems.