Equivariant Polynomials for Graph Neural Networks

Graph Neural Networks (GNN) are inherently limited in their expressive power. Recent seminal works (Xu et al., 2019; Morris et al., 2019b) introduced the Weisfeiler-Lehman (WL) hierarchy as a measure of expressive power. Although this hierarchy has propelled significant advances in GNN analysis and architecture developments, it suffers from several significant limitations. These include a complex definition that lacks direct guidance for model improvement and a WL hierarchy that is too coarse to study current GNNs. This paper introduces an alternative expressive power hierarchy based on the ability of GNNs to calculate equivariant polynomials of a certain degree. As a first step, we provide a full characterization of all equivariant graph polynomials by introducing a concrete basis, significantly generalizing previous results. Each basis element corresponds to a specific multi-graph, and its computation over some graph data input corresponds to a tensor contraction problem. Second, we propose algorithmic tools for evaluating the expressiveness of GNNs using tensor contraction sequences, and calculate the expressive power of popular GNNs. Finally, we enhance the expressivity of common GNN architectures by adding polynomial features or additional operations / aggregations inspired by our theory. These enhanced GNNs demonstrate state-of-the-art results in experiments across multiple graph learning benchmarks.

MultiDiffusion: Fusing Diffusion Paths for Controlled Image Generation

Recent advances in text-to-image generation with diffusion models present transformative capabilities in image quality. However, user controllability of the generated image, and fast adaptation to new tasks still remains an open challenge, currently mostly addressed by costly and long re-training and fine-tuning or ad-hoc adaptations to specific image generation tasks. In this work, we present MultiDiffusion, a unified framework that enables versatile and controllable image generation, using a pre-trained text-to-image diffusion model, without any further training or finetuning. At the center of our approach is a new generation process, based on an optimization task that binds together multiple diffusion generation processes with a shared set of parameters or constraints. We show that MultiDiffusion can be readily applied to generate high quality and diverse images that adhere to user-provided controls, such as desired aspect ratio (e.g., panorama), and spatial guiding signals, ranging from tight segmentation masks to bounding boxes. Project webpage: https://multidiffusion.github.io

Riemannian Flow Matching on General Geometries

We propose Riemannian Flow Matching (RFM), a simple yet powerful framework for training continuous normalizing flows on manifolds. Existing methods for generative modeling on manifolds either require expensive simulation, inherently cannot scale to high dimensions, or use approximations to limiting quantities that result in biased objectives. Riemannian Flow Matching bypasses these inconveniences and exhibits multiple benefits over prior approaches: It is completely simulation-free on simple geometries, it does not require divergence computation, and its target vector field is computed in closed form even on general geometries. The key ingredient behind RFM is the construction of a simple kernel function for defining per-sample vector fields, which subsumes existing Euclidean cases. Extending to general geometries, we rely on the use of spectral decompositions to efficiently compute kernel functions. Our method achieves state-of-the-art performance on real-world non-Euclidean datasets, and we showcase, for the first time, tractable training on general geometries, including on triangular meshes and maze-like manifolds with boundaries.

Flow Matching for Generative Modeling

We introduce a new paradigm for generative modeling built on Continuous Normalizing Flows (CNFs), allowing us to train CNFs at unprecedented scale. Specifically, we present the notion of Flow Matching (FM), a simulation-free approach for training CNFs based on regressing vector fields of fixed conditional probability paths. Flow Matching is compatible with a general family of Gaussian probability paths for transforming between noise and data samples -- which subsumes existing diffusion paths as specific instances. Interestingly, we find that employing FM with diffusion paths results in a more robust and stable alternative for training diffusion models. Furthermore, Flow Matching opens the door to training CNFs with other, non-diffusion probability paths. An instance of particular interest is using Optimal Transport (OT) displacement interpolation to define the conditional probability paths. These paths are more efficient than diffusion paths, provide faster training and sampling, and result in better generalization. Training CNFs using Flow Matching on ImageNet leads to state-of-the-art performance in terms of both likelihood and sample quality, and allows fast and reliable sample generation using off-the-shelf numerical ODE solvers.

Neural Conservation Laws: A Divergence-Free Perspective

We investigate the parameterization of deep neural networks that by design satisfy the continuity equation, a fundamental conservation law. This is enabled by the observation that solutions of the continuity equation can be represented as a divergence-free vector field. We hence propose building divergence-free neural networks through the concept of differential forms, and with the aid of automatic differentiation, realize two practical constructions. As a result, we can parameterize pairs of densities and vector fields that always satisfy the continuity equation by construction, foregoing the need for extra penalty methods or expensive numerical simulation. Furthermore, we prove these models are universal and so can be used to represent any divergence-free vector field. Finally, we experimentally validate our approaches on neural network-based solutions to fluid equations, solving for the Hodge decomposition, and learning dynamical optimal transport maps the Hodge decomposition, and learning dynamical optimal transport maps.

Matching Normalizing Flows and Probability Paths on Manifolds

Continuous Normalizing Flows (CNFs) are a class of generative models that transform a prior distribution to a model distribution by solving an ordinary differential equation (ODE). We propose to train CNFs on manifolds by minimizing probability path divergence (PPD), a novel family of divergences between the probability density path generated by the CNF and a target probability density path. PPD is formulated using a logarithmic mass conservation formula which is a linear first order partial differential equation relating the log target probabilities and the CNF's defining vector field. PPD has several key benefits over existing methods: it sidesteps the need to solve an ODE per iteration, readily applies to manifold data, scales to high dimensions, and is compatible with a large family of target paths interpolating pure noise and data in finite time. Theoretically, PPD is shown to bound classical probability divergences. Empirically, we show that CNFs learned by minimizing PPD achieve state-of-the-art results in likelihoods and sample quality on existing low-dimensional manifold benchmarks, and is the first example of a generative model to scale to moderately high dimensional manifolds.

Weisfeiler and Leman go Machine Learning: The Story so far

In recent years, algorithms and neural architectures based on the Weisfeiler-Leman algorithm, a well-known heuristic for the graph isomorphism problem, emerged as a powerful tool for machine learning with graphs and relational data. Here, we give a comprehensive overview of the algorithm's use in a machine learning setting, focusing on the supervised regime. We discuss the theoretical background, show how to use it for supervised graph- and node representation learning, discuss recent extensions, and outline the algorithm's connection to (permutation-)equivariant neural architectures. Moreover, we give an overview of current applications and future directions to stimulate further research.

Frame Averaging for Equivariant Shape Space Learning

The task of shape space learning involves mapping a train set of shapes to and from a latent representation space with good generalization properties. Often, real-world collections of shapes have symmetries, which can be defined as transformations that do not change the essence of the shape. A natural way to incorporate symmetries in shape space learning is to ask that the mapping to the shape space (encoder) and mapping from the shape space (decoder) are equivariant to the relevant symmetries. In this paper, we present a framework for incorporating equivariance in encoders and decoders by introducing two contributions: (i) adapting the recent Frame Averaging (FA) framework for building generic, efficient, and maximally expressive Equivariant autoencoders; and (ii) constructing autoencoders equivariant to piecewise Euclidean motions applied to different parts of the shape. To the best of our knowledge, this is the first fully piecewise Euclidean equivariant autoencoder construction. Training our framework is simple: it uses standard reconstruction losses and does not require the introduction of new losses. Our architectures are built of standard (backbone) architectures with the appropriate frame averaging to make them equivariant. Testing our framework on both rigid shapes dataset using implicit neural representations, and articulated shape datasets using mesh-based neural networks show state-of-the-art generalization to unseen test shapes, improving relevant baselines by a large margin. In particular, our method demonstrates significant improvement in generalizing to unseen articulated poses.

Frame Averaging for Invariant and Equivariant Network Design

Many machine learning tasks involve learning functions that are known to be invariant or equivariant to certain symmetries of the input data. However, it is often challenging to design neural network architectures that respect these symmetries while being expressive and computationally efficient. For example, Euclidean motion invariant/equivariant graph or point cloud neural networks. We introduce Frame Averaging (FA), a general purpose and systematic framework for adapting known (backbone) architectures to become invariant or equivariant to new symmetry types. Our framework builds on the well known group averaging operator that guarantees invariance or equivariance but is intractable. In contrast, we observe that for many important classes of symmetries, this operator can be replaced with an averaging operator over a small subset of the group elements, called a frame. We show that averaging over a frame guarantees exact invariance or equivariance while often being much simpler to compute than averaging over the entire group. Furthermore, we prove that FA-based models have maximal expressive power in a broad setting and in general preserve the expressive power of their backbone architectures. Using frame averaging, we propose a new class of universal Graph Neural Networks (GNNs), universal Euclidean motion invariant point cloud networks, and Euclidean motion invariant Message Passing (MP) GNNs. We demonstrate the practical effectiveness of FA on several applications including point cloud normal estimation, beyond $2$-WL graph separation, and $n$-body dynamics prediction, achieving state-of-the-art results in all of these benchmarks.

Augmenting Implicit Neural Shape Representations with Explicit Deformation Fields

Implicit neural representation is a recent approach to learn shape collections as zero level-sets of neural networks, where each shape is represented by a latent code. So far, the focus has been shape reconstruction, while shape generalization was mostly left to generic encoder-decoder or auto-decoder regularization. In this paper we advocate deformation-aware regularization for implicit neural representations, aiming at producing plausible deformations as latent code changes. The challenge is that implicit representations do not capture correspondences between different shapes, which makes it difficult to represent and regularize their deformations. Thus, we propose to pair the implicit representation of the shapes with an explicit, piecewise linear deformation field, learned as an auxiliary function. We demonstrate that, by regularizing these deformation fields, we can encourage the implicit neural representation to induce natural deformations in the learned shape space, such as as-rigid-as-possible deformations.