The Empirical Impact of Neural Parameter Symmetries, or Lack Thereof

Many algorithms and observed phenomena in deep learning appear to be affected by parameter symmetries -- transformations of neural network parameters that do not change the underlying neural network function. These include linear mode connectivity, model merging, Bayesian neural network inference, metanetworks, and several other characteristics of optimization or loss-landscapes. However, theoretical analysis of the relationship between parameter space symmetries and these phenomena is difficult. In this work, we empirically investigate the impact of neural parameter symmetries by introducing new neural network architectures that have reduced parameter space symmetries. We develop two methods, with some provable guarantees, of modifying standard neural networks to reduce parameter space symmetries. With these new methods, we conduct a comprehensive experimental study consisting of multiple tasks aimed at assessing the effect of removing parameter symmetries. Our experiments reveal several interesting observations on the empirical impact of parameter symmetries; for instance, we observe linear mode connectivity between our networks without alignment of weight spaces, and we find that our networks allow for faster and more effective Bayesian neural network training.

GRANOLA: Adaptive Normalization for Graph Neural Networks

In recent years, significant efforts have been made to refine the design of Graph Neural Network (GNN) layers, aiming to overcome diverse challenges, such as limited expressive power and oversmoothing. Despite their widespread adoption, the incorporation of off-the-shelf normalization layers like BatchNorm or InstanceNorm within a GNN architecture may not effectively capture the unique characteristics of graph-structured data, potentially reducing the expressive power of the overall architecture. Moreover, existing graph-specific normalization layers often struggle to offer substantial and consistent benefits. In this paper, we propose GRANOLA, a novel graph-adaptive normalization layer. Unlike existing normalization layers, GRANOLA normalizes node features by adapting to the specific characteristics of the graph, particularly by generating expressive representations of its neighborhood structure, obtained by leveraging the propagation of Random Node Features (RNF) in the graph. We present theoretical results that support our design choices. Our extensive empirical evaluation of various graph benchmarks underscores the superior performance of GRANOLA over existing normalization techniques. Furthermore, GRANOLA emerges as the top-performing method among all baselines within the same time complexity of Message Passing Neural Networks (MPNNs).

Learning Priors for Non Rigid SfM from Casual Videos

We tackle the long-standing challenge of reconstructing 3D structures and camera positions from videos. The problem is particularly hard when objects are transformed in a non-rigid way. Current approaches to this problem make unrealistic assumptions or require a long optimization time. We present TracksTo4D, a novel deep learning-based approach that enables inferring 3D structure and camera positions from dynamic content originating from in-the-wild videos using a single feed-forward pass on a sparse point track matrix. To achieve this, we leverage recent advances in 2D point tracking and design an equivariant neural architecture tailored for directly processing 2D point tracks by leveraging their symmetries. TracksTo4D is trained on a dataset of in-the-wild videos utilizing only the 2D point tracks extracted from the videos, without any 3D supervision. Our experiments demonstrate that TracksTo4D generalizes well to unseen videos of unseen semantic categories at inference time, producing equivalent results to state-of-the-art methods while significantly reducing the runtime compared to other baselines.

Subgraphormer: Unifying Subgraph GNNs and Graph Transformers via Graph Products

In the realm of Graph Neural Networks (GNNs), two exciting research directions have recently emerged: Subgraph GNNs and Graph Transformers. In this paper, we propose an architecture that integrates both approaches, dubbed Subgraphormer, which combines the enhanced expressive power, message-passing mechanisms, and aggregation schemes from Subgraph GNNs with attention and positional encodings, arguably the most important components in Graph Transformers. Our method is based on an intriguing new connection we reveal between Subgraph GNNs and product graphs, suggesting that Subgraph GNNs can be formulated as Message Passing Neural Networks (MPNNs) operating on a product of the graph with itself. We use this formulation to design our architecture: first, we devise an attention mechanism based on the connectivity of the product graph. Following this, we propose a novel and efficient positional encoding scheme for Subgraph GNNs, which we derive as a positional encoding for the product graph. Our experimental results demonstrate significant performance improvements over both Subgraph GNNs and Graph Transformers on a wide range of datasets.

Improved Generalization of Weight Space Networks via Augmentations

Learning in deep weight spaces (DWS), where neural networks process the weights of other neural networks, is an emerging research direction, with applications to 2D and 3D neural fields (INRs, NeRFs), as well as making inferences about other types of neural networks. Unfortunately, weight space models tend to suffer from substantial overfitting. We empirically analyze the reasons for this overfitting and find that a key reason is the lack of diversity in DWS datasets. While a given object can be represented by many different weight configurations, typical INR training sets fail to capture variability across INRs that represent the same object. To address this, we explore strategies for data augmentation in weight spaces and propose a MixUp method adapted for weight spaces. We demonstrate the effectiveness of these methods in two setups. In classification, they improve performance similarly to having up to 10 times more data. In self-supervised contrastive learning, they yield substantial 5-10% gains in downstream classification.

Future Directions in Foundations of Graph Machine Learning

Machine learning on graphs, especially using graph neural networks (GNNs), has seen a surge in interest due to the wide availability of graph data across a broad spectrum of disciplines, from life to social and engineering sciences. Despite their practical success, our theoretical understanding of the properties of GNNs remains highly incomplete. Recent theoretical advancements primarily focus on elucidating the coarse-grained expressive power of GNNs, predominantly employing combinatorial techniques. However, these studies do not perfectly align with practice, particularly in understanding the generalization behavior of GNNs when trained with stochastic first-order optimization techniques. In this position paper, we argue that the graph machine learning community needs to shift its attention to developing a more balanced theory of graph machine learning, focusing on a more thorough understanding of the interplay of expressive power, generalization, and optimization.

Graph Metanetworks for Processing Diverse Neural Architectures

Neural networks efficiently encode learned information within their parameters. Consequently, many tasks can be unified by treating neural networks themselves as input data. When doing so, recent studies demonstrated the importance of accounting for the symmetries and geometry of parameter spaces. However, those works developed architectures tailored to specific networks such as MLPs and CNNs without normalization layers, and generalizing such architectures to other types of networks can be challenging. In this work, we overcome these challenges by building new metanetworks - neural networks that take weights from other neural networks as input. Put simply, we carefully build graphs representing the input neural networks and process the graphs using graph neural networks. Our approach, Graph Metanetworks (GMNs), generalizes to neural architectures where competing methods struggle, such as multi-head attention layers, normalization layers, convolutional layers, ResNet blocks, and group-equivariant linear layers. We prove that GMNs are expressive and equivariant to parameter permutation symmetries that leave the input neural network functions unchanged. We validate the effectiveness of our method on several metanetwork tasks over diverse neural network architectures.

Expressive Sign Equivariant Networks for Spectral Geometric Learning

Recent work has shown the utility of developing machine learning models that respect the structure and symmetries of eigenvectors. These works promote sign invariance, since for any eigenvector v the negation -v is also an eigenvector. However, we show that sign invariance is theoretically limited for tasks such as building orthogonally equivariant models and learning node positional encodings for link prediction in graphs. In this work, we demonstrate the benefits of sign equivariance for these tasks. To obtain these benefits, we develop novel sign equivariant neural network architectures. Our models are based on a new analytic characterization of sign equivariant polynomials and thus inherit provable expressiveness properties. Controlled synthetic experiments show that our networks can achieve the theoretically predicted benefits of sign equivariant models. Code is available at https://github.com/cptq/Sign-Equivariant-Nets.

Data Augmentations in Deep Weight Spaces

Learning in weight spaces, where neural networks process the weights of other deep neural networks, has emerged as a promising research direction with applications in various fields, from analyzing and editing neural fields and implicit neural representations, to network pruning and quantization. Recent works designed architectures for effective learning in that space, which takes into account its unique, permutation-equivariant, structure. Unfortunately, so far these architectures suffer from severe overfitting and were shown to benefit from large datasets. This poses a significant challenge because generating data for this learning setup is laborious and time-consuming since each data sample is a full set of network weights that has to be trained. In this paper, we address this difficulty by investigating data augmentations for weight spaces, a set of techniques that enable generating new data examples on the fly without having to train additional input weight space elements. We first review several recently proposed data augmentation schemes %that were proposed recently and divide them into categories. We then introduce a novel augmentation scheme based on the Mixup method. We evaluate the performance of these techniques on existing benchmarks as well as new benchmarks we generate, which can be valuable for future studies.

Efficient Subgraph GNNs by Learning Effective Selection Policies

Subgraph GNNs are provably expressive neural architectures that learn graph representations from sets of subgraphs. Unfortunately, their applicability is hampered by the computational complexity associated with performing message passing on many subgraphs. In this paper, we consider the problem of learning to select a small subset of the large set of possible subgraphs in a data-driven fashion. We first motivate the problem by proving that there are families of WL-indistinguishable graphs for which there exist efficient subgraph selection policies: small subsets of subgraphs that can already identify all the graphs within the family. We then propose a new approach, called Policy-Learn, that learns how to select subgraphs in an iterative manner. We prove that, unlike popular random policies and prior work addressing the same problem, our architecture is able to learn the efficient policies mentioned above. Our experimental results demonstrate that Policy-Learn outperforms existing baselines across a wide range of datasets.