Permutation-Equivariant Neural Networks

Neural networks that respect the symmetries of point clouds commonly compute high-order representations of the rotation group to improve their expressivity. Such models are known to be universal, and thus able to separate any pair of point clouds, when using representations of sufficiently high order, yet no precise bound on this order, as a function of the number of points, was previously known. We establish the first quadratic lower and upper degree bounds for polynomials that separate point clouds of n points up to rotations and permutations. By showing Tensor Field Networks and MACE can express all invariant polynomials of a given degree with sufficient representation order, we additionally derive the first explicit, quadratic bound on the representation order sufficient for these architectures to be complete, replacing previous existence results with concrete guarantees. We further show that, under local regularity assumptions, point cloud separation can be achieved with a message-passing-based model using invariants whose degree depends only on the size of a local neighborhood, rather than on the size of the full point cloud. Finally, we provide counterexamples showing MACE must compute high order irreducible representations to separate locally regular point clouds.

Weight-space learning across architectures requires representations that respect parameter symmetries and scale to large networks. Existing equivariant tensor encoders require architecture-specific symmetry specifications, while neuron graphs make global attention costly. We introduce a typed graph organized around shared coordinate-permutation actions. Its nodes, called interfaces, represent coordinate spaces whose bound tensor axes must be permuted jointly; operation occurrences, called modules, form typed hyperedges. A compiler derives this graph from supported PyTorch models, with exact recovery of the permutation factors defined by its constraints. The Universal Weight Encoder (UWE) combines local equivariant updates to parameter features with global attention over invariant interface summaries. The small number of interfaces makes global attention practical, and a single encoder processes different supported architectures. UWE is competitive with architecture-specific state-of-the-art methods on established benchmarks. On a new zoo of 1,200 transformers spanning 100 architectures, it outperforms the evaluated baselines at ranking networks within unseen architectures, including architectures larger than those used for training, while using substantially less memory.

Correlation matrices combine nonlinear positive-definiteness constraints with a simultaneous row-column permutation symmetry. Their generation therefore involves two distinct problems. First, every output must remain on the correlation elliptope, irrespective of how its variables are ordered. Second, the generator must learn the domain-specific distribution occupied by empirical data inside that valid space. Existing financial image generators mix these problems: they learn ordered matrix entries and repair invalid draws afterward. We introduce CorrFlow, which separates them. A generalized Fisher transform maps full-rank correlation matrices bijectively to unconstrained coordinates, guaranteeing valid outputs without projection. A graph flow-matching network learns the empirical distribution in those coordinates and is permutation equivariant: reordering the variables only reorders the output. We compare a conditional CorrGAN reimplementation, image diffusion, and CorrFlow on $80\times80$ Taiwan equity matrices using five seeds per neural method. CorrFlow produces 100% valid draws without repair and obtains feature sliced-Wasserstein distance $.712\pm.141$, versus $4.339\pm.252$ for CorrGAN and $4.610\pm.191$ for image diffusion after their invalid outputs are projected back to the correlation space. It also improves this distance over historical lookup, while lookup retains higher local coverage. These results support the combined construction under the evaluated training budget; they do not establish superiority over fully tuned image models or isolate the contribution of each component.

A trained neural network can be represented by a parameter vector in high dimensions. Learning distributions over these vectors enables the generation of new models across various tasks and architectures. A central challenge is permutation symmetry: permuting hidden neurons can produce distant parameter vectors representing the same function. This introduces variations that a generative model must account for when learning from trained networks. Existing methods typically address this using networks derived from a common base model or costly approximate neuron alignment. We instead parameterize a flow-matching velocity field with a permutation-equivariant Graph Meta Network, enabling direct learning from independently trained networks without alignment. Extensive experiments show that our method closely reproduces the joint statistics of accuracy, functional similarity, and weight similarity of independently trained collections, providing evidence of generation beyond checkpoint memorization. A single conditional model also generates task-specific networks on heterogeneous architectures and generalizes to unseen hidden-width configurations. On a tabular domain-shift task, intermediate conditioning produces individual networks with performance comparable to logit ensembles across both domains. Taken together, our results show how permutation equivariance enables learning from diverse collections of independently trained networks without permutation alignment.

We investigate message-passing graph neural networks with random node features. Random node features are known to enhance the expressiveness of graph neural networks (GNNs) both theoretically and empirically. Here, we establish a novel universality result focusing on permutation-equivariant neural networks (PENNs), a class of GNNs built from feedforward neural network components that subsumes many prominent GNN architectures. We show that PENNs, combined with partially random node features, can approximate arbitrarily well in probability any measurable permutation-invariant or permutation-equivariant function on directed graphs of fixed size with multidimensional node and edge features. For $k$-times continuously differentiable functions, $k\geq 2$, we also derive upper bounds on the approximation rates, relating the complexity of the feedforward components of a PENN in terms of layer depth and number of nonzero weights to the desired approximation accuracy.

Existing Stochastic 3D Human Motion Prediction models are fundamentally constrained by hard-coding the skeleton kinematics, severely limiting generalization, preventing cross-dataset training, and requiring complex data retargeting. We introduce EquiFusion, the first kinematics-agnostic model to solve this bottleneck, implementing a latent diffusion model with a permutation equivariant architecture. EquiFusion treats the kinematics' connectivity as an explicit input parameter, ensuring its internal computations are inherently agnostic to joint ordering and graph structure. This novel design enables truly cross-dataset generalization to unseen kinematics and unlocks novel zero-shot directions, such as motion prediction from partial or occluded observations and targeted limb generation. EquiFusion achieves state-of-the-art results on major benchmarks, being up to 75% more compact than previous kinematics-specific methods, while achieving faster training and inference. EquiFusion thus establishes a new, flexible standard for robust human motion prediction. Model and training code are available at https://ceveloper.github.io/publications/equifusion/.

Reliable work zone mapping is important for connected and autonomous vehicles (CAVs) to navigate safely and smoothly through work zone areas. Cone-mounted ultra-wideband (UWB) roadside units (RSU) offer a cost-effective way for work zone layout inference, as roadside anchors and vehicle tags provide direct vehicle-to-infrastructure (V2I) range constraints for work zone geometry reconstruction. However, UWB range estimation is degraded by bursty outliers, non-line-of-sight (NLOS) errors, arbitrary anchor-ordering issues, and vehicle pose uncertainties in practical field deployments. To address these challenges, this study proposes a pose-conditioned, permutation-equivariant predictive denoiser for multi-anchor UWB ranging. The model employs shared anchor-wise temporal prediction to capture range dynamics, symmetric set aggregation to handle unordered and missing anchors, and pose-conditioned residual decoding to incorporate vehicle motion as a geometric prior. A two-stage training strategy first learns prediction from observed ranges, and then fine-tunes the denoiser with NLOS-weighted supervision. The method is evaluated on rare real-world V2I UWB field data collected with a CAV, as well as on controlled large-scale simulation benchmarks for ablative insights. Results show that the proposed method substantially improves range accuracy, cone localization, and work zone geometry reconstruction in challenging NLOS-dominated regimes, remains robust to anchor re-indexing and moderate anchor dropout, and reduces measurement-weighted field MSE by 66.9% relative to the raw input.

We introduce a novel framework for constructing scalable and flexible covariance kernels for Gaussian processes (GPs) by directly learning the covariance structure under a regression-type parameterization induced by Vecchia approximations, using deep neural architectures. Specifically, we model kriging coefficients and conditional standard deviations, deterministic quantities that uniquely characterize the covariance, providing stable and informative learning targets. Exploiting the permutation-equivariant structure of conditioning sets in the Vecchia factorization, we derive a universal representation for permutation-preserving functions and design neural architectures that respect this symmetry, leading to improved training stability and data efficiency. The proposed approach enables expressive, non-stationary kernel learning while maintaining computational scalability, thereby bridging classical GP methodology with modern deep learning.

The inclusion of symmetries as an inductive bias, known as equivariance, often improves generalization on geometric data (e.g. grids, sets, and graphs). However, equivariant architectures are usually highly constrained, designed for symmetries chosen a priori, and not applicable to datasets with other symmetries. This precludes the development of flexible, multi-modal foundation models capable of processing diverse data equivariantly. In this work, we build a single model -- the Any-Subgroup Equivariant Network (ASEN) -- that can be simultaneously equivariant to several groups, simply by modulating a certain auxiliary input feature. In particular, we start with a fully permutation-equivariant base model, and then obtain subgroup equivariance by using a symmetry-breaking input whose automorphism group is that subgroup. However, finding an input with the desired automorphism group is computationally hard. We overcome this by relaxing from exact to approximate symmetry breaking, leveraging the notion of 2-closure to derive fast algorithms. Theoretically, we show that our subgroup-equivariant networks can simulate equivariant MLPs, and their universality can be guaranteed if the base model is universal. Empirically, we validate our method on symmetry selection for graph and image tasks, as well as multitask and transfer learning for sequence tasks, showing that a single network equivariant to multiple permutation subgroups outperforms both separate equivariant models and a single non-equivariant model.

Reasoning problems such as Sudoku and ARC-AGI remain challenging for neural networks. The structured problem solving architecture family of Recurrent Reasoning Models (RRMs), including Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM), offer a compact alternative to large language models, but currently handle symbol symmetries only implicitly via costly data augmentation. We introduce Symbol-Equivariant Recurrent Reasoning Models (SE-RRMs), which enforce permutation equivariance at the architectural level through symbol-equivariant layers, guaranteeing identical solutions under symbol or color permutations. SE-RRMs outperform prior RRMs on 9x9 Sudoku and generalize from just training on 9x9 to smaller 4x4 and larger 16x16 and 25x25 instances, to which existing RRMs cannot extrapolate. On ARC-AGI-1 and ARC-AGI-2, SE-RRMs achieve competitive performance with substantially less data augmentation and only 2 million parameters, demonstrating that explicitly encoding symmetry improves the robustness and scalability of neural reasoning. Code is available at https://github.com/ml-jku/SE-RRM.

The universal approximation theorem establishes that neural networks can approximate any continuous function on a compact set. Later works in approximation theory provide quantitative approximation rates for ReLU networks on the class of $α$-Hölder functions $f: [0,1]^N \to \mathbb{R}$. The goal of this paper is to provide similar quantitative approximation results in the context of group equivariant learning, where the learned $α$-Hölder function is known to obey certain group symmetries. While there has been much interest in the literature in understanding the universal approximation properties of equivariant models, very few quantitative approximation results are known for equivariant models. In this paper, we bridge this gap by deriving quantitative approximation rates for several prominent group-equivariant and invariant architectures. The architectures that we consider include: the permutation-invariant Deep Sets architecture; the permutation-equivariant Sumformer and Transformer architectures; joint invariance to permutations and rigid motions using invariant networks based on frame averaging; and general bi-Lipschitz invariant models. Overall, we show that equally-sized ReLU MLPs and equivariant architectures are equally expressive over equivariant functions. Thus, hard-coding equivariance does not result in a loss of expressivity or approximation power in these models.