Exact Equivariance

Learning-based motion planners pay at training what classical planners pay per query. Trained in world coordinates, they relearn the same motion at every position and orientation. Existing work restores the missing rigid-body equivariance in the training data, in the inference operator, or in the weights, and each carries a cost. We ask how much of that equivariance the planning query supplies for free. A start s and a goal g determine a frame in closed form, with origin at their midpoint and first axis along g-s. Expressing trajectory and obstacles in that frame removes three translations and two rotations of SE(3), at initialisation, for one cross product per query and with no constraint on the architecture. A single rotation about the start-goal axis remains, and no continuous rule removes it. On a cluttered 3D benchmark, holding architecture, data and budget fixed, the frame raises the held-out collision-free rate from 14.60% to 51.10%, where a straight segment from start to goal scores 15.6% and the world-frame model does not beat it. We build all three mechanisms for the residual rotation and each is worth under a point, though the equivariant backbone reaches any given level two to three times sooner. What the representation supplies therefore dominates what any mechanism enforces, and the standard diagnostic does not see the difference: two models with indistinguishable non-equivariance residuals differ by 28 points. Calibrated against a non-symmetry intervention, the frame is not even the largest effect available, since local geometry is worth +40.0 where the frame is worth +36.5.

Rotational symmetry is one of the most important structural principles in machine learning on 3D data. In applications ranging from physics and materials science to 3D computer vision, predictions should not depend on an arbitrary choice of coordinate frame. Rotational equivariance captures this requirement mathematically by enforcing that a rotation of the input induces a corresponding transformation of the model output. This tutorial provides a comprehensive introduction to rotational equivariance, starting from the physical and geometric intuition behind coordinate independence and building up the necessary machinery from geometric deep learning, group theory, and representation theory. We introduce message passing on Euclidean graphs, group actions and representations, spherical harmonics, Wigner matrices, tensor products, and Clebsch-Gordan decomposition, and explain how these ingredients give rise to modern equivariant architectures. We then survey the principal strategies for incorporating rotational equivariance in deep learning, including group convolutions, internal tensorial representations, and canonicalization-based methods, and discuss their practical strengths and limitations. The tutorial aims to lower the barrier to the subject by connecting the underlying mathematics to practical model design, by unifying ideas that are often expressed in different formal languages, and by helping practitioners choose among competing approaches through a clear discussion of their trade-offs.

We give a complete characterization of equivariant multi-head self-attention (MHSA): if an MHSA layer is equivariant to a symmetry group $G$, then $G$ can only act by permuting head-clusters, with QK and OV matrices satisfying an equivariance constraint tied to the group action. As a consequence, we prove that any fixed MHSA architecture that achieves exact equivariance by polynomially parameterizing unconstrained MHSA parameters inevitably leads to expressivity loss within the class of equivariant maps: the equivariance locus of unconstrained MHSA forms a union of extremely many Zariski-irreducible components in a reduced parameter space, and any single architecture covers at most one. For $G=D_4$ acting on $C$ copies of the regular representation as the token feature space, we show that there are $Ω(C^{64})$ components for eight attention heads.

Multi-oriented text is ubiquitous in real-world scenes and remains a major challenge for scene text recognition (STR). Existing rotation-aware methods explicitly estimate text orientation. However, due to the lack of theoretical guarantees, they are prone to error accumulation, increased computational cost, and strong reliance on data. In this work, we incorporate rotation invariance into the STR framework to address these limitations. Specifically, we adopt an encoder-decoder architecture, embedding rotation equivariance in the encoder and rotation invariance in the decoder to construct a fully rotation-invariant network. On the decoder side, we first identify and prove the rotation-invariant property of the cross-attention mechanism and use it to formulate a rotation-invariant text decoder that maps visual features to output text in a rotation-invariant manner. On the encoder side, we propose a rotation-equivariant local-global extraction network that integrates deep equivariant convolutions with self-attention, enabling rotation-equivariant feature extraction while modeling inter-character dependencies and preserving fine-grained visual details. By integrating the encoder and decoder, we obtain an end-to-end Rotation-Invariant Scene Text Recognition network (RISTER). RISTER provides rotation invariance with theoretical guarantees, enhancing robustness on multi-oriented samples without introducing additional inference computation or relying on data-driven orientation correction. Experiments show that RISTER achieves state-of-the-art performance on both standard and multi-oriented benchmarks, surpassing the second-best model by 4.0 percent in accuracy on the general multi-oriented dataset.

Brain tissue microstructure estimation with machine learning provides higher computational efficiency than conventional fitting. However, machine learning still presents important limitations that hamper its clinical utility. Specifically, current models typically lack generalisation across diffusion MRI acquisition protocols and require retraining whenever b-vectors or b-values change. Moreover, the recent machine learning methods that were developed to address protocol generalisation lack rotational equivariance. Particularly suitable for dMRI parameter estimation is a geometric deep learning model known as spherical convolutional neural network (SCNN), which guarantees rotational equivariance and b-vector generalisation. However, this architecture currently does not account for b-values. Therefore, obtaining a model that combines protocol generalisation and rotational equivariance remains an open challenge. In this paper, we directly address this issue by incorporating explicit b-value dependence into an SCNN architecture via a hypernetwork. This new approach is illustrated using NODDI as an example forward model for estimating brain tissue microstructure. To evaluate b-value generalisation, the original and newly proposed SCNN architectures are trained on synthetic data and tested on both synthetic and real data across different b-value pairs. Results demonstrate that the proposed method achieves reduced RMSE and bias on synthetic data, as well as higher agreement with conventional NODDI fitting on real data, indicating improved robustness to unseen b-values and a reduced need for retraining. By combining generalisation across b-values with generalisation across b-vectors and rotational equivariance, the proposed framework enhances the applicability of deep learning to clinical diffusion MRI parameter estimation. Code available at https://github.com/aerdnairo/arXiv\_generalisedSCNN.

Reflection symmetry detection remains challenging due to interference from asymmetric regions and arbitrary orientations of symmetric patterns. Asymmetric regions introduce background clutter that disrupts symmetric pattern matching, whereas conventional convolutional neural networks lack rotation equivariance, leading to inconsistent feature representations under rotational transformations. To address these issues, we propose an Asymmetric Region Denoising (ARD) module and a Rotation Equivariant Feature Similarity Matching (REFSM) module. The ARD module suppresses asymmetric interference to refine symmetric patterns, while the REFSM module enhances rotation equivariance through feature similarity matching between original and rotated images. Specifically, our dual-input REFSM framework leverages rotation loss to maximize consistency between the score maps of original and rotated images, thereby enabling precise prediction of rotation-equivariant symmetry axes. Furthermore, we introduce GMSYM, a new benchmark dataset that categorizes images into diverse scenarios and incorporates various interferences to address the limitations of existing reflection symmetry detection benchmarks. Extensive experiments on four standard datasets (DENDI, NYU, LDRS, SDRW) and our proposed GMSYM dataset demonstrate that our method achieves state-of-the-art performance in both accuracy and robustness.

Exact-equivariant architectures typically encode prescribed group actions in specialized operators, which can complicate their reuse with generic backbones and across data modalities. We introduce the Generator-Aligned Representation Interface (GARI), a representation-level design principle that exposes selected transformation generators to a generic sequence backbone through aligned canonical and generator-induced views. We formalize the resulting behavior using a probe-specific soft-equivariance residual defined over declared data and transformation distributions. This framework distinguishes representation consistency from task robustness and exact equivariance, and localizes residual mismatch to interface construction, shared stream processing, and terminal fusion. We instantiate the interface as GARI-Net, which constructs generator-indexed streams, converts them into a common interaction frame, processes them with shared parameters, repairs ordering-induced context mismatch, enables cross-stream information exchange, and aggregates them using inter-stream discrepancy. Direct Equivariance Error (DEE) provides a frozen-checkpoint diagnostic of the prescribed representation relation under known token or voxel actions. Experiments on genomic sequences, images, and three-dimensional point clouds examine sequence reversal, planar rotations and reflections, and controlled axial transfer. Across these settings, the same interface principle supports task-relevant transformation consistency and generalization to declared held-out probes without requiring group-specific redesign of the sequence backbone. GARI therefore provides a portable diagnostic complement to hard-equivariant architectures: it makes generator structure accessible, learnable, and measurable, while finite-probe evidence remains distinct from certification of exact equivariance over a continuous group.

Equivariant networks embed geometric symmetries as structural priors through weight sharing, achieving remarkable parameter efficiency across vision tasks. However, this parameter efficiency does not translate into compute efficiency: existing implementations unroll the structured weights into dense matrices and dispatch them to generic dense kernels, so the FLOPs of an equivariant layer are no smaller than those of a non-equivariant counterpart. In this paper, we observe that the equivariant linear (EQ-Linear) layer---the most fundamental and frequently used module in modern equivariant architectures---is essentially a circular convolution along the group dimension composed with a linear transform along the channel dimension. Building on this observation, we propose Flash EQ-Linear, an exact acceleration algorithm that reduces the complexity from $\mathcal{O}(NDC)$ to $\mathcal{O}(NDC/T)$ by combining the Fourier convolution theorem along the group dimension with the conjugate symmetry of the real DFT. We further provide dedicated CUDA kernels for Flash EQ-Linear, covering both forward and backward passes and both FP32 and FP16 precision. At the operator level, Flash EQ-Linear achieves up to ${2\times}$ forward speedup over PyTorch's F.linear; at the network level, Flash EQ-ViT and Flash EQ-Swin achieve up to ${1.7\times}$ end-to-end speedup over both equivariant and non-equivariant baselines. To our knowledge, this is the first time equivariant networks strictly dominate their non-equivariant counterparts along all three axes simultaneously: accuracy, parameter efficiency, and inference speed.Code is available at https://github.com/zhongchenzhao/FlashEQLinear.

3D Gaussian Splatting (3DGS) captures scenes by coupling explicit geometry (position, covariance) with view-dependent photometry (Spherical Harmonics). However, building $\mathrm{SE}(3)$-equivariant architectures on these primitives presents a fundamental representation bottleneck. Color has been treated as a signal rather than a geometric entity, making it nontrivial to unify symmetry across geometry and appearance as the camera frame changes. While translations are handled by relative coordinates, rotations act heterogeneously across attributes: $μ\mapsto Rμ$, $Σ\mapsto RΣR^\top$, and $f_\ell\mapsto D^\ell(R)f_\ell$. This mismatch complicates strict equivariance, leading existing methods to either discard or flatten SH coefficients, thereby breaking symmetry. We propose a unified solution rooted in representation theory: for SH degrees $\ell\le2$, photometry is algebraically isomorphic to a rank-2 geometric tensor. We prove that the Wigner-$D$ action on these SH coefficients can be exactly reformulated as the conjugation action on $3\times3$ matrices. Leveraging this, we introduce the Unified Matrix Embedding, a lifting that maps all Gaussian attributes into a unified carrier space, $\mathfrak{gl}(3)$. Building on the "Color-as-Geometry" formulation, we present E3DGS, a rigid-body ($\mathrm{SE}(3)$) equivariant architecture that processes 3D Gaussians without Clebsch-Gordan tensor products. Evaluations on object vision and action-conditioned Gaussian world modeling demonstrate that our unified approach yields strong robustness under camera-frame changes and improved data efficiency.

Rich internal representations of musical structure are essential for music understanding tasks such as machine-assisted music co-writing, yet self-supervised approaches for symbolic music representation remain underexplored, particularly those that encode the hierarchical multiscale nature of musical structures. We present MIDI-RAE-JEPA, combining a pitch- and time-shift equivariance objective with LeJEPA and a Swin Transformer V2 encoder to learn such hierarchical representations of symbolic music encoded as piano roll images. The time-shift equivariance objective encourages the model to internalize temporal musical relationships. The encoder is trained purely on self-supervised objectives -- including a masked embedding predictor (MEP) -- with collapse prevented via SIGReg. A separate decoder trained on the frozen encoder embeddings achieves reconstruction F1 of 0.995, and a flow matching generative model conditioned on those embeddings produces generations that closely match the pitch register and rhythmic density of the conditioning excerpt, while mismatched conditioning yields unrelated but musically plausible output. Learned representations outperform a Haar scattering transform baseline on a downstream emotion classification task, and embedding distances increase monotonically with pitch and time shift magnitude, confirming measurable equivariance. These results suggest that equivariance-based SSL objectives, combined with sufficient fine-level encoder capacity, provide a viable path toward semantically rich, generatively useful representations of symbolic music.

Circular data, representing angles or directions, are frequently encountered in computer vision, biology, geology, and meteorology. Traditional regression targets the conditional mean, which is often geometrically misleading for circular responses under multimodal, skewed, or asymmetric data structures. To address these limitations, a lightweight deep generative framework, namely ANGLE, is introduced for non-parametric distributional regression on the circle. The full conditional distribution of an angular response, given Euclidean and circular covariates, is learned through a generative map optimized via a generalized circular energy score (GCES) loss. Desirable theoretical properties, including the strict propriety of the loss and the rotational equivariance of the estimators, are established. Furthermore, both pre- and post-additive noise models are accommodated. A unified toolbox is provided for advancing previously underexplored challenges in circular statistics: extrapolation, sufficient dimension reduction, and conditional distribution equality testing. The framework's efficacy is demonstrated through extensive simulations and real-world applications. Specifically, the proposal is utilized for object pose estimation from imagery and wind direction prediction, which are integral to surveillance, autonomous vehicles, and energy systems, respectively. Superior predictive performance and robust uncertainty quantification of the proposed method in these tasks are revealed.

Generating signals on graphs requires permutation-equivariant models that exhibit stability with respect to relative structural perturbations. While favorable stability properties of Graph Neural Networks (GNNs) have been well documented, it is unclear how structural errors propagate through the dynamics of continuous generative flow models that are gaining traction for graph signal generation. In this paper, we analyze continuous normalized flow models parameterized by GNNs and show that permutation equivariance is preserved for both the resulting continuous-time ordinary differential equations and their discrete numerical approximations used as graph signal samplers. Our primary contribution is to derive explicit stability bounds on the generated probability distributions, which quantify how relative graph perturbations affect the final sampled signals. Motivated by these theoretical bounds, we introduce a stability-promoting regularized flow matching strategy that actively penalizes the spatial Lipschitz constant of the vector field during model training. Experiments using synthetic smooth signals on stochastic block model graphs and real-world fMRI signals on brain connectomes demonstrate that this bound-oriented approach yields generative models that are more robust to structural noise, without sacrificing output quality.

Scientists often seek to draw causal inferences from structured data that is not independently and identically distributed, such as spatial data, network data, or molecular data. We develop geometric causal models (GCMs), a framework for causal inference from dependent data that exploits underlying symmetries of the data generating process. For example, in spatial data, we consider processes that are symmetric under translations, or in graph data, symmetric under permutations of the nodes. We show how symmetries, formalized with group theory, can enable causal identification and estimation. We deploy ergodic theory for amenable groups to establish identification, and combine geometric deep learning with scalable Bayesian inference for estimation. We recover i.i.d. causal models and do-calculus when the data is a sequence and the symmetry is permutation equivariance, and find novel types of causal models when we use alternate structures and symmetries. As an example, we construct a causal model that satisfies the symmetries of DNA. This GCM enables new estimators for the effects of genetic variation, combining deep functional genomics models to describe outcomes and DNA language models to describe propensities. We illustrate on semisynthetic data.

While recent advancements like the Poincaré ResNet have demonstrated the potential of learning visual representations directly in hyperbolic space, their optimisation remains hampered by the computationally intensive nature of Riemannian gradients and the strict boundaries of the manifold. Furthermore, standard hyperbolic networks treat spatial transformations of the same object as distinct hierarchical concepts, leading to redundant parameter usage and vanishing signals. We propose Equivariant Poincaré ResNets, combining hyperbolic geometry with discrete symmetry groups ($C_4$ and $D_4$). We identify critical roadblocks in applying Euclidean equivariance to hyperbolic space and propose geometrically safe tensor reshaping, left-regular permutations for hyperbolic group convolutions, and joint-orientation Poincaré Midpoint Batch normalisation. Empirically, embedding equivariance drastically reduces the optimisation space, accelerating convergence while accelerating convergence while respecting the boundary constraints of the Poincaré ball and preserving spatial-group equivariance.

$\mathrm{Cl}(3,0)$ interatomic potentials, despite their algebraic elegance, predict force magnitudes accurately but force directions poorly. Across ten rMD17 molecules, every $L \leq 1$ baseline in our twelve-model study attains aggregate force-cosine similarity below $0.25$. The cause is structural. The geometric product of two vectors in $\mathbb{R}^3$ realises only the $L=0$ and $L=1$ components of its irreducible representation content, leaving the symmetric-traceless rank-2 component absent from the per-edge bilinear that drives each message-passing layer. We address this with CliffordSTF, which couples the Clifford multivector to closed-form symmetric-traceless tensor tracks at ranks two and three through bilinear cross-track contractions, using a single learned bilinear and no Clebsch--Gordan tables, Wigner-$D$ matrices, or e3nn calls. On rMD17, CliffordSTF raises aggregate force-cosine similarity from $0.055$ (base Clifford) to $0.551$, an order-of-magnitude relative directional gain, alongside improved magnitude accuracy (force MAE $15.8\%$ lower; energy MAE $10.9\%$ lower). It outperforms all CG-free or body-ordered baselines in our study (all $\leq 0.17$). On catalysis benchmarks, CliffordSTF achieves the best out-of-distribution S2EF energy MAE on OC22 in our experiments, and the best in-distribution energy MAE among $L \geq 2$ methods on OC22 IS2RE. An eleven-variant ablation shows the two tracks are complementary: neither alone matches the combined model.

Symmetries are important for many deep learning tasks, ranging from applications in the sciences to medical imaging. However, there is an ongoing debate about whether to impose symmetry constraints on the neural network architecture (yielding equivariant neural networks) or learn them from augmented training data. Although equivariant networks are well-studied theoretically, much less is known about data augmentation, since analyzing augmentation requires control over the training dynamics. Inspired by recent results that show that augmented infinite deep ensembles are exactly equivariant, we study data augmentation for Bayesian neural networks (BNNs) trained with variational inference. We focus on variational distributions in the exponential family and derive conditions under which exact equivariance is reached. We furthermore obtain bounds on the equivariance error and introduce three novel symmetrization techniques which boost the effect of data augmentation in this setting. We conduct extensive numerical experiments which show that one of our symmetrization methods (orbit expansion) outperforms the baseline in both equivariance and overall performance. Our code is available at github.com/dmw1998/augment-BNNs

In this paper, we propose a discrete roto-reflection group equivariant vision transformer with convolutional attention. Roto-reflection equivariant networks preserve the rotational, flip and positional symmetry in feature maps, making them useful for tasks where orientation of the inputs is relevant to the model outputs. In image classification and object detection, most of the studies on roto-reflection equivariant models have focused on using convolutional neural networks rather than vision transformers. In this paper, we examine the challenges involved in achieving equivariance in vision transformers, and we propose a simpler way to implement a discretized roto-reflection group equivariant vision transformer. The experimental results demonstrate that our approach outperforms the existing approaches for developing discrete roto-reflection group equivariant neural networks for image classification.

Object pose estimation is crucial to many industrial applications, with one example being automated spray painting using a robot. However, confidentiality concerns often limit access to high-quality 3D models, posing a significant challenge for point-cloud-based pose estimation. In such scenarios, rotational symmetry, a readily accessible characteristic of many industrial objects, can provide valuable prior information to facilitate pose estimation.In this paper, we propose a method that leverages the rotational symmetry commonly found in industrial objects to address the challenge caused by the absence of 3D models. The object pose is jointly estimated with point cloud refinement through an iterative optimization process. This optimization relies on a rotational symmetry constraint loss. To construct this loss, each 3D point is rotated according to the currently estimated pose, and multiple correspondences are identified using nearest-neighbor search by exploiting the rotational symmetry property. These correspondences are then used to compute the rotational symmetry constraint loss, which iteratively refines both the pose and the point cloud.By explicitly incorporating rotational symmetry into the optimization process, the proposed method achieves robust pose estimation and generalizes well across diverse object types. The proposed method is evaluated on a dataset specifically created for point clouds without known 3D models, consisting of four categories of synthetic objects and one real wheel hub collected from a production line. Experimental results demonstrate that the proposed method achieves performance comparable to methods that rely on known 3D models.

Deep learning surrogates for 3D Partial Differential Equations (PDEs) often fail to generalize across geometric transformations because they depend heavily on specific coordinate systems. While equivariant networks offer a solution, they typically rely on local operations in the spatial domain, making the global receptive field, which is essential for PDE dynamics, computationally expensive. Conversely, Fourier Neural Operators (FNOs) efficiently capture global interactions, yet establishing 3D equivariance within them remains impractical due to the prohibitive cost of spectral group convolutions. To bridge this gap, we introduce EqGINO, a geometrically robust framework that enforces isotropy in the spectral domain. By design, EqGINO guarantees exact equivariance to the discrete symmetries inherent to the discretized computational domain. Beyond this discrete guarantee, our structural prior enables effective generalization to arbitrary continuous orientations even with a limited number of SE(3)-transformed training samples. Consequently, our method robustly models coordinate-invariant physical laws on complex irregular 3D geometries. Our code is available at https://github.com/sung-won-kim/EqGINO

A latent world model built from an equivariant encoder and predictor inherits a provable symmetry of its training loss: when the dynamics carries a group $G$ acting on latents by an orthogonal representation $ρ(g)$, the one-step prediction relMSE is exactly invariant across the whole group, so fitting a restricted slice of orientations mathematically determines it on the entire orbit. The symmetry survives a real Muon/AdamW+EMA+VICReg run -- composed residual $\sim 10^{-6}$ after training, under any optimiser (intrinsic Vector-Neuron/e3nn parametrisation) -- and one-step error is flat across the group (5-seed medians: equivariant $\times 1.00$ vs a higher-capacity non-equivariant baseline $\times 12.7$ in 2D, $\times 17.2$ in 3D), while that baseline fits the slice but breaks out-of-distribution. The flatness is not a synthetic artefact: on real-robot DROID end-effector trajectories the equivariant model stays flat across the orbit ($\times 1.000$, rotation residual $1.5\times 10^{-16}$) while a $4.5\times$-larger baseline degrades $\times 11$. One caution is load-bearing: flatness is necessary, not sufficient -- the theorem transports the in-distribution error level unchanged but does not lower it (3D relMSE $\approx 0.43$): across-group error is constant, not low. The same isometry lifts to a closed-loop corollary: under a matching equivariant planner the control error is invariant across the group -- float-floor-exact in 2D/SO(2), statistically flat in 3D/SE(3). Stress-tested against Sutton's Bitter Lesson (augmentation, scale, soft-equivariance), each closes at most the across-group task metric, never the float-floor exactness. This is the generalisation-side foundation of a certified-world-models programme (arXiv:2606.13092, 2606.24945, 2606.24946): flatness transports competence, and the trust bounds built on it are downstream products.

Diffusion-based Vision-Language-Action policies achieve remarkable success in robotic manipulation, yet commit a fundamental geometric error we term the $\textbf{Euclidean Fallacy}$: representing SE(3) poses as flat $\mathbb{R}^{12}$ vectors. This approximation induces (1) manifold drift violating SO(3) constraints, (2) broken equivariance under coordinate transformations, and (3) non-geodesic trajectories with excessive kinematic cost. We introduce $\textbf{Lie Diffuser Actor (LDA)}$, a diffusion framework operating intrinsically on SE(3). Our method injects noise through left-invariant SDEs, predicts scores in the tangent space, and retracts samples via the exponential map. This formulation eliminates manifold drift by construction while guaranteeing coordinate-frame equivariance and geodesic optimality. On CALVIN ABC$\rightarrow$D, LDA improves average task length from $3.27$ to $3.51$ ($+7.3\%$). We further validate our method on real robot and the results show that our methodology outperforms the baseline on majority tasks.

Combinatorial routing problems such as the Traveling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) are fundamental NP-hard problems with broad real-world applications. While recent deep reinforcement learning methods have shown promising performance, they typically handle geometric symmetries only through data augmentation, resulting in inconsistent decisions and limited generalization. To address this issue, we propose MViewRouter, a multi-view framework that internalizes geometric equivariance as a structural inductive bias to achieve invariant decision-making across routing problem variants. Our approach introduces a Multi-view Alternating Attention (MAA) mechanism that enables parallel processing over the $D_4$ symmetry group, alternating between intra-view relational modeling and inter-view feature alignment. Furthermore, we optimize the policy via Collective Policy Gradient Aggregation (CPGA), leveraging consensus gradients from multiple symmetric views to stabilize training and accelerate convergence. Experiments on TSP and CVRP benchmarks, as well as real-world TSPLIB instances, demonstrate that MViewRouter achieves competitive solution quality and strong zero-shot generalization.

Fourier Neural Operators are often assumed to generalize across spatial resolutions, enabling training on a coarse grid and deployment on a finer grid. We test this assumption by contrasting two inference-time choices when moving from training resolution $s$ to test resolution $S>s$: running FNO directly at $S$, or running at $s$ and upsampling the prediction to $S$ via Fourier zero-padding. On Darcy flow, we observe that direct fine-grid inference is not reliably beneficial and can be worse than the low-grid-plus-upsampling baseline. We further analyze layerwise spectra and find that, under Fourier truncation, intermediate representations increasingly concentrate energy in low frequencies, with high-frequency output produced mainly by late nonlinear/decoder stages. This offers a mechanistic explanation for why FNO can perform well while retaining few modes, yet remain sensitive under resolution shifts. Our findings highlight a simple but strong baseline for cross-resolution evaluation and point to nonlinear aliasing as a key obstacle to zero-shot resolution equivariance.

Geometry-aware generative models and novel view synthesis approaches have shown strong potential in visual fidelity and consistency. In parallel, equivariant representation learning has emerged as a powerful framework for constructing latent spaces where analytically known group transformations could act directly, capturing geometric structure in data and enhancing both interpretability and generalization in novel view synthesis. However, we identify that existing approaches often suffer from latent misalignment, a discrepancy between the intended group action and the actually required transformations in the latent space. Consequently, the learned latents often fail to consistently preserve the equivariant relations imposed by the underlying group symmetry. To address this, we propose Residual Latent Flow, a flow-based framework that corrects the misaligned latents, thereby improving compliance with the underlying equivariance relation. Our comprehensive experiments show that our method significantly reduces latent misalignment and improves novel view synthesis quality, under rotation groups SO(n).

Formal theorem provers based on large language models (LLMs) are highly sensitive to superficial variations in problem representation: semantically equivalent statements can exhibit drastically different proof success rates, revealing a failure to respect structural symmetries inherent in formal mathematics. This raises a central question: what are the right symmetries for formal theorem proving? We introduce rewriting categories, a category-theoretic framework capturing the compositional, generally non-invertible transformations induced by proof tactics, and use it to formalize two symmetry notions: proof equivariance, governing how proof distributions transform under rewrites, and success invariance (i.e., invariance of success probability), requiring equivalent statements to be solved with the same probability. We observe that state-based next-tactic provers naturally satisfy proof equivariance by operating on proof states. In contrast, state-of-the-art LLM-based provers satisfy neither property, exhibiting large performance variation across equivalent formulations. To mitigate this, we propose test-time methods that aggregate over equivalent rewritings of the input, showing theoretically that they recover success invariance in the sampling limit, and empirically, that they improve robustness and performance under fixed inference budgets. Our results highlight symmetry as a key missing inductive bias in LLM-based theorem proving and suggest test-time computation as a practical route to approximate it.

Cone-beam CT (CBCT) is routinely acquired during radiotherapy for patient setup, but its quantitative reliability is degraded by scatter, noise, and reconstruction artifacts, limiting Hounsfield Unit (HU) accuracy. We propose EPC-3D-Diff, a novel conditional 3D latent diffusion framework for volumetric CBCT to CT synthesis that introduces a projection domain equivariance loss derived from acquisition physics. Unlike common image domain equivariance, we exploit the fact that an in plane rotation of the volume corresponds to an angular shift in its projections. During training, we enforce this relationship by forward projecting rotated synthesized CT volumes and matching them to appropriately angle shifted projections of the paired target CT, yielding a physics consistent equivariance constraint integrated into the diffusion objective. To capture full 3D context efficiently, conditional diffusion is performed in a compact latent space learnt by a lightweight 3D autoencoder, preserving axial depth while downsampling in plane resolution for stable training. We validate on a paired head CBCT/CT phantom dataset, including repeat scans, and paired clinical data using patient wise splits, and perform single and mixed domain training, ablations, and comparisons with diffusion and CycleGAN. EPC-3D-Diff generalizes well and achieved substantial improvements, +7.4 dB (phantom) and +1.8 dB (clinical data) in PSNR compared to state of the art methods, alongside improved SSIM and HU accuracy, within tissue boundaries. Overall, EPC-3D-Diff improves robustness and physics consistency, supporting HU aware synthesis for downstream radiotherapy workflows. The open source code for EPC-3D-Diff is available at https://github.com/ALZAHRAALTALIB/EPC-3D-Diff.

We introduce the $\star_G$ tensor algebra, in which any finite group $G$ defines the multiplication rule, making equivariance an intrinsic algebraic property rather than an architectural constraint. The framework rests on three machine-verified theoretical pillars: (i)~an Eckart-Young optimality guarantee for the $\star_G$-SVD: the first such result for symmetry-preserving tensor approximation, exact and polynomial-time; (ii)~a Kronecker factorization that composes multiple symmetries by replacing $F_G$ with $F_{G_1} \otimes F_{G_2}$ with no architectural redesign; and (iii)a 600-line Lean4 formalization of the $\star_G$ algebra. The framework provides capabilities that equivariant neural networks (ENNs) structurally cannot: a closed-form per-irreducible-representation decomposition of every prediction, and data-driven discovery of the symmetry group that best fits a dataset. As a non-trivial empirical demonstration, decomposing QM9 molecular geometry over the chiral octahedral subgroup of SO(3) recovers the Wigner--Eckart selection rules of angular momentum from data alone, with no quantum mechanical input: scalar properties are A$_1$-dominated, dipole components are T$_1$-dominated, the isotropic polarizability is uniquely insensitive to $l\!=\!1$ as the rank-2-trace decomposition $l\!=\!0 \oplus l\!=\!2$ requires, and the T$_1$/A$_1$ predictive-power ratio separates vector observables from scalar observables by a factor of five. On full QM9 (130{,}831 molecules), $\star_G$-SVD with ridge regression provides closed form predictions at $\sim50-90\times$ fewer parameters than parameter-matched MLPs. Algebraic equivariance thus complements architectural equivariance not as a faster-better-cheaper alternative but as a different mathematical affordance: provably-optimal symmetry-preserving compression, per-irrep interpretability, and data-driven physical discovery.

Many natural and engineered time series -- equity returns, climate anomalies, turbulent velocities, neural recordings, packet-level network traffic -- are approximately self-similar: their horizon-$T$ distribution is tied to the horizon-$1$ distribution by one scaling exponent $H$. Standard deep generative sequence models (transformers, dilated TCNs, the WaveNet family) ignore this. Their receptive fields are wide, but kernel parameters live independently at every dilation level, yielding a multi-scale architecture, not a scale-equivariant one. We make three contributions. First, we give a precise definition of discrete scale equivariance for 1D causal networks and prove that dyadic dilation commutes (up to boundary effects) with any dilated-convolution stack whose kernel weights are shared across levels. Tying the kernel shrinks the convolutional parameter budget by an $L$-fold factor (where $L$ is depth) and hard-wires self-similarity in as an inductive bias. Second, we wrap this Scale-Equivariant WaveNet (SE-WaveNet) backbone in three components that carry the same prior: a one-level Daubechies-4 wavelet input, a Hurst-FiLM block exposing the local scaling exponent, and a spectral-consistency training term targeting the $|f|^{-(2H+1)}$ power-law spectrum. The head is a conditional normalising flow, chosen to preserve equivariance. Third, on 30 years of S&P 500 daily log-returns, SE-WaveNet samples reproduce the empirical scaling-collapse diagnostic on the Allan-Variance top-25 universe (median $\mathcal{C}^\star = 0.020$), while a vanilla WaveNet at matched capacity does not ($\geq 0.06$). NLL, KS-calibration, and tail energy distance tie or beat the baseline, with $L\times$ fewer convolutional parameters.

Image restoration is an inherently ill posed inverse problem. Equivariant networks that embed geometric symmetry priors can mitigate this ill posedness and improve performance. However, current understanding of the relationship between network equivariance and data symmetry remains largely heuristic. Particularly for real world data with imperfect symmetry, existing research lacks a systematic theoretical framework to quantify symmetry, select transformation groups, or evaluate model data alignment. To bridge this gap, we conduct an analysis from an optimization perspective and formalize the intrinsic relationship among data symmetry priors, model equivariance, and generalization capability. Specifically, we propose for the first time a quantifiable definition of non strict symmetry at the dataset level (rather than sample level) and use it as a constraint to formulate the restoration inverse problem. We then show that the equivariance for restoration models can be naturally derived from this inverse problems incorporated the proposed symmetry constraints, and that the equivariance error of the optimal restoration operator is strictly bounded by the data symmetry error and the discretization mesh size. Furthermore, by analyzing the network's empirical risk, we demonstrate that aligning equivariance with data symmetry optimizes the bias variance trade off, minimizing the total expected risk. Guided by these insights, we propose a Sample Adaptive Equivariant Network that uses a hypernetwork and transformation learnable equivariant convolutions to dynamically align with each sample's inherent symmetry. Extensive experiments on super resolution, denoising, and deraining validate our theoretical findings and show significant superiority over standard baselines and traditional equivariant models. Our code and supplementary material are available at https://github.com/tanfy929/SA-Conv.

Weight matrices in deep networks exhibit geometric continuity -- principal singular vectors of adjacent layers point in similar directions. While this property has been widely observed, its origin remains unexplained. Through experiments on toy MLPs and small transformers, we identify two mechanisms: residual connections create cross-layer gradient coherence that aligns weight updates across layers, and symmetry-breaking nonlinearities constrain all layers to a shared coordinate frame, preventing the rotation drift that would otherwise destabilize weight structure. Crucially, a nonlinear but rotation-preserving activation fails to retain continuity, isolating symmetry breaking -- not nonlinearity itself -- as the active ingredient. Activation and normalization play distinct roles: activation concentrates continuity in the leading singular direction, while normalization distributes it across multiple directions. In transformers, continuity is projection-specific: Q, K, Gate, and Up (which read from the residual stream) develop input-space ($\mathbf{v}_1$) continuity; O and Down (which write to it) develop output-space ($\mathbf{u}_1$) continuity; V alone, lacking an adjacent nonlinearity, develops only low continuity.

Normalization Equivariance (NE) is a structural prior that improves robustness to distribution shift in image-to-image tasks. A function $f$ is normalization equivariant iff $f(a y + b\mathbf{1}) = a f(y) + b\mathbf{1}$ for all $a>0$ and $b\in\mathbb{R}$. Existing NE methods constrain every internal layer to NE-compatible operations. These constraints add runtime cost and exclude standard transformer components such as softmax attention and LayerNorm. We introduce Wrapped Normalization Equivariance (WNE), a parameter-free wrapper that normalizes the input, applies any backbone, and denormalizes the output. We prove every NE function admits this factorization, so the wrapper exactly parameterizes the class of NE functions. On blind denoising, wrapping CNN and transformer architectures improves robustness under noise-level mismatch with no measurable GPU overhead, while architectural NE baselines are up to $1.6\times$ slower.

Learning data-efficient object dynamics models for robotic manipulation remains challenging, especially for deformable objects. A popular approach is to model objects as sets of 3D particles and learn their motion using graph neural networks. In practice, this is not enough to maintain physical feasibility over long horizons and may require large amounts of interaction data to learn. We introduce PIEGraph, a novel approach to combining analytical physics and data-driven models to capture object dynamics for both rigid and deformable bodies using limited real-world interaction data. PIEGraph consists of two components: (1) a \textbf{P}hysically \textbf{I}nformed particle-based analytical model (implemented as a spring--mass system) to enforce physically feasible motion, and (2) an \textbf{E}quivariant \textbf{Graph} Neural Network with a novel action representation that exploits symmetries in particle interactions to guide the analytical model. We evaluate PIEGraph in simulation and on robot hardware for reorientation and repositioning tasks with ropes, cloth, stuffed animals and rigid objects. We show that our method enables accurate dynamics prediction and reliable downstream robotic manipulation planning, which outperforms state of the art baselines.

Proximal operators are fundamental across many applications in signal processing and machine learning, including solving ill-posed inverse problems. Recent work has introduced Learned Proximal Networks (LPNs), providing parametric functions that compute exact proximals for data-driven and potentially non-convex regularizers. However, in many settings it is important to include additional structure to these regularizers--and their corresponding proximals--such as shift and scale equivariance. In this work, we show how to obtain learned functions parametrized by neural networks that provably compute exact proximal operators while being equivariant to shifts and scaling, which we dub Affine-Equivariant Learned Proximal Networks (AE-LPNs). We demonstrate our results on synthetic, constructive examples, and then on real data via denoising in out-of-distribution settings. Our equivariant learned proximals enhance robustness to noise distributions and affine shifts far beyond training distributions, improving the practical utility of learned proximal operators

We consider monocular 3D human pose estimation (HPE), where the goal is to predict 3D human skeletal joints from a single 2D image, typically via 2D keypoint detection followed by 2D-to-3D lifting. Despite their success, we find that current lifting models exhibit strong performance degradation under rotations. We systematically study rotation equivariance across lifting architectures with increasing degrees of equivariant inductive bias, and investigate whether equivariance should be enforced by architectural design or learned from data, given the inherent ambiguity of monocular 2D-to-3D lifting. Utilising common HPE benchmarks, we demonstrate that rotation equivariance can be effectively learned via rotation-based data augmentation applied jointly to input and output poses. Compared with non-augmented models, this reduces error on rotated poses by over 30% across the benchmarks, with reductions of up to 72%. Moreover, augmentation-based models achieve up to 15% lower error than methods that are fully equivariant by design, while providing up to 37$\times$ faster inference. We further show that these findings generalise to real-world human pose sequences involving full-body rotations and to a state-of-the-art HPE model.

We present E(3)-Pose, a novel fast pose estimation method that jointly and explicitly models rotation equivariance and object symmetry. Our work is motivated by the challenging problem of accounting for fetal head motion during a diagnostic MRI scan. We aim to enable automatic adaptive prescription of diagnostic 2D MRI slices with 6-DoF head pose estimation, supported by rapid low-resolution 3D MRI volumes acquired before each 2D slice. Existing pose estimation methods struggle to generalize to clinical volumes due to pose ambiguities induced by inherent anatomical symmetries, as well as low resolution, noise, and artifacts. In contrast, E(3)-Pose captures anatomical symmetries and rigid pose equivariance by construction, and yields robust estimates of the fetal head pose. Our experiments on publicly available and representative clinical fetal MRI datasets demonstrate the superior robustness and generalization of our method across domains. Crucially, E(3)-Pose achieves state-of-the-art accuracy on clinical MRI volumes, supporting future clinical translation. Our implementation is publicly available at github.com/MedicalVisionGroup/E3-Pose.

Edge-attributed multigraphs, in which multiple edges with distinct attributes connect the same pair of nodes, arise naturally in many real-world systems. In these graphs, effective learning requires preserving information from repeated interactions while distinguishing contributions from different neighbors. Existing neural network solutions for edge-attributed multigraphs remain limited: some lose information from repeated interactions, while others break permutation equivariance. To address this, we introduce \emph{neighbor-aware aggregation}, an operator that first combines multi-edge features for each neighbor and then aggregates across neighbors. This operator captures per-neighbor statistics that standard single-stage aggregation cannot represent. Building on this operator, we present MEGA-GNN, a model-agnostic message-passing framework for edge-attributed multigraphs. We show that MEGA-GNN is permutation equivariant and has the same asymptotic complexity as standard GNNs with edge updates. We evaluate our approach on datasets from social networks and financial transaction networks. Neighbor-aware aggregation consistently improves GNN performance and matches or surpasses state-of-the-art methods.