Equivariant Representation Learning

Ask about this topic

Overview

Equivariant representation learning studies how to encode data so that transformations such as rotation or translation produce corresponding, predictable changes in a model’s features. Group-equivariant convolutions, irrep-based graph networks, and equivariant transformers encode these rules directly; data augmentation and self-supervised consistency objectives can encourage them from examples, and some methods learn the transformations themselves. These representations are useful in vision, molecular and physical prediction, robotics, and audio, where they can support data efficiency and generalization across viewpoints or configurations. A central tension is balancing exact symmetry guarantees with flexibility, computational cost, and task accuracy: exact equivariance ensures consistent behavior across transformations but does not guarantee strong predictions, while softer or learned symmetry may be easier to scale but can miss important constraints.

Latest papers

CardsList

54 papers11 papers in the last four weeks, up 22% on the four weeks before. 0.1% of all new papers.

Jul 21, 2026

GEqTrain: A Configuration-Driven Framework for Retargeting Equivariant Graph Neural Networks Across 3D Scientific Tasks

Equivariant graph neural networks provide a powerful modeling language for three-dimensional scientific data, but their reuse is often limited by implementations tied to specific tasks, outputs, and training regimes. We present GEqTrain, a configuration-driven framework that separates dataset semantics, model composition, and training objectives. Raw data are mapped to typed node-, edge-, and graph-level fields, while model stacks, losses, and training workflows are assembled declaratively through Hydra configurations. A shared equivariant backbone and training infrastructure can therefore be retargeted to a new task primarily through configuration. We demonstrate this flexibility on three different problems handled within one software stack: coarse-grained-to-atomistic backmapping of biomolecular systems, prediction of NMR chemical shifts in molecular solids, and equivariant generative modeling. Our aim is not to surpass individually optimized task-specific systems, but to show that a shared representation and training infrastructure can achieve competitive accuracy across qualitatively different tasks at the cost of a configuration change. We further introduce GEqDiff, a generative extension based on equivariant flow matching. GEqDiff treats user-defined equivariant fields as first-class generation targets, jointly transporting Cartesian positions and non-scalar node fields spanning representations up to l=3 within a single equivariant flow. We validate this capability on a controlled synthetic benchmark inspired by protein secondary-structure motifs, showing that fields with heterogeneous transformation properties can be reconstructed jointly and with high fidelity. By reducing the software overhead of moving between predictive and generative, scalar and tensorial settings, GEqTrain aims to make equivariant modeling more reproducible, extensible, and reusable.

Daniele Angioletti, Marco Nobile, Vittorio Limongelli

Faculty of Biomedical Sciences, Euler Institute, Universit´a della Svizzera italiana, 6900 Lugano, Switzerland

Flow MatchingGenerative ModelingEquivariant GNNsEquivariant Representation Learning
AskPDF
Jul 16, 2026

MIDI-RAE-JEPA: Hierarchical Representation Learning and Generation for Symbolic Music

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.

Scott H. Hawley

Department of Chemistry & Physics, Belmont University, Nashville, TN, USA

Representation LearningSelf-Supervised Pre-TrainingSymbolic Music GenerationHierarchical Representation LearningEquivariant Representation Learning
AskPDF
Jul 7, 2026

When Do Geometric Algebra Layers Beat Scalarization? A Controlled Study on SO(3)-Equivariant Vector Laws

Compact networks built from Clifford algebra Cl(3,0) primitives are exactly SO(3)-equivariant and learn synthetic 3D vector laws from few samples. We ask whether the geometric algebra structure itself contributes anything beyond exact equivariance. We compare against a minimal scalarization baseline: invariant dot products fed to a small MLP that outputs coefficients on the equivariant basis {v_i, v_i x v_j}, which is also exactly equivariant. On single-stage laws (rotation by axis-angle, cross product, central force), scalarization matches or beats the Cl(3,0) network at a fraction of the training cost, so the geometric algebra adds nothing there. On compositional targets whose computation graph nests group operations (apply R2 R1 to a point; map a local force through an orientation, then take a torque), the Cl(3,0) network beats scalarization by an order of magnitude in the low-data regime, reaching with 100 samples what the baseline needs 3000 for, and the gap survives strengthening the baseline with the triple-product invariant and 17x more parameters, external Vector Neurons and e3nn baselines, and a multiplicative coefficient network. Ablations show the required network depth tracks the rotation chain length, and scalarization falls below the constant predictor on chains of four rotations. The advantage is not composition per se: on a rotation-free nested cross product, which flattens into polynomial invariant coefficients, scalarization wins by 24x. No tested model, equivariant or not, extrapolates invariant magnitudes: on radius and separation shifts every model is worse than a constant predictor once errors are normalized. We conclude that geometric algebra layers are not a general shortcut for low-data 3D learning, but become useful precisely when the target composes group elements in depth.

Fabien Polly

Equivariant Neural NetworksEquivariant Representation LearningInvariant Representation Learning
AskPDF
Jul 4, 2026

Foundations of Equivariant Deep Learning: Unifying Graph and Sheaf Neural Networks

Symmetry is everywhere in nature and society. Geometric deep learning exploits symmetries in data to improve the performance and efficiency of deep learning systems. In this paper, we extend geometric deep learning to utilize richer symmetry structures. Specifically, we develop order-equivariant neural networks (OENN), which generalize standard graph message passing and sheaf neural networks via the theory of equivariant bundles over face posets (face categories). We (i) characterize all linear order-equivariant maps, (ii) build OENN layers, and (iii) prove universal approximation theorems (UATs) for continuous order-equivariant maps, which are new results even when restricted to sheaf neural networks (for which no UAT was known before). We illustrate the framework on graph and sheaf models. Our results can also be seen as extending the known UAT for graph neural networks to a more general setting that subsumes sheaf neural networks as well. In addition, we show that OENN can be extended further to CENN, Category-Equivariant Neural Network, which gives the general form of equivariant neural networks as well as of equivariant universal approximation theorems, allowing us to leverage categorical symmetry in data (e.g., non-invertible symmetries on multiple objects with compositional relations on those symmetries).

Yoshihiro Maruyama

Department of Mathematical and Information Sciences, Kyoto University, Kyoto, Japan.

Graph Neural NetworksEquivariant Neural NetworksSheaf Neural NetworksNeural Network Approximation TheoryEquivariant Representation Learning
AskPDF
Jun 26, 2026

A Unified Framework for Vision Transformers Equivariant to Discrete Subgroups of O(2)\mathrm{O}(2)

Vision transformers have become a dominant architecture for visual recognition. However, standard models do not explicitly encode the planar symmetries that arise in many vision domains. We introduce a family of vision transformers equivariant to arbitrary discrete subgroups of O(2)\mathrm{O}(2), providing a unified framework that generalizes prior flipping- and D4D_4-equivariant transformer architectures. Our construction yields equivariant analogues of the core transformer components, together with expressivity guarantees for the resulting layers. In particular, we show that whenever H≤GH \le G, the class of GG-equivariant ViTs embeds naturally into the class of HH-equivariant ViTs. We also prove that, in the single-head setting, the corresponding equivariant self-attention layer realizes every GG-equivariant self-attention map representable by ordinary self-attention. We further construct a D6D_6-equivariant model based on hexagonal patches, making the architecture compatible with six-fold rotational symmetries. We evaluate the resulting models on the PatternNet aerial image dataset in artificially data-scarce regimes across subgroups of D4D_4 and D6D_6. Our experiments compare two equivariant attention mechanisms and analyze how the choice of homogeneous-space configurations used in the nonlinearities affects performance. Preliminary results under matched parameter budgets indicate that equivariance can improve recognition accuracy, motivating further study of how discrete symmetry groups shape transformer-based visual recognition models.

Tīkun Ông, Georg Bökman

Independent Researcher · University of Amsterdam

Vision TransformerEquivariant Neural NetworksEquivariant Representation LearningImage Classification
AskPDF
Jun 4, 2026

Equivariant Neural Belief Propagation

Probabilistic inference over spatially embedded variables requires beliefs that respect SE(3)SE(3) symmetry, yet existing equivariant networks produce only scalars and vectors -- not the rank-2 precision tensors needed for anisotropic uncertainty, and single-component messages collapse multi-modal energy landscapes to physically meaningless averages. We introduce Equivariant Neural Belief Propagation (ENBP), a factor-graph framework whose messages are equivariant Gaussian mixture models with sufficient statistics that transform exactly under SE(3)SE(3). Rank-2 precision matrices are synthesised via equivariant outer products, ingested through differentiable spectral decomposition, and kept tractable by a greedy KL-based mixture reduction that provably commutes with SE(3)SE(3). On GEOM-QM9 and GEOM-Drugs, ENBP achieves 98.9% conformational coverage at 0.090 A˚\mathring{A} error with sub-second latency -- over 100×100\times faster than diffusion baselines at higher accuracy. On multi-body robotic inference, vanilla loopy BP diverges at 15+ agents while ENBP converges with near-zero collision rates and machine-precision equivariance error (∼10−7{\sim}10^{-7} vs.\ 10−110^{-1} for augmented baselines).

Zehua Cheng, Wei Dai, Jiahao Sun

Department of Computer Science, University of Oxford, Oxford, United Kingdom · FLock.io, London, United Kingdom

Equivariant Representation LearningFactor Graph OptimizationGaussian Mixture ModelsBayesian InferenceEquivariant Neural NetworksProbabilistic Graphical Models+2
AskPDF
Jun 2, 2026

Exact equivariance, kept through training, buys zero-shot generalisation across the symmetry group

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 GG acting on latents by an orthogonal representation ρ(g)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 ∼10−6\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 ×1.00\times 1.00 vs a higher-capacity non-equivariant baseline ×12.7\times 12.7 in 2D, ×17.2\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 (×1.000\times 1.000, rotation residual 1.5×10−161.5\times 10^{-16}) while a 4.5×4.5\times-larger baseline degrades ×11\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 ≈0.43\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.

Hongbo Wang

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA

World Models for RoboticsWorld ModelsEquivariant Neural NetworksLatent World ModelsEquivariant Representation LearningStructure-Preserving Neural Networks
AskPDF
May 29, 2026

SSR: Scaling Surefooted and Symmetric Humanoid Traversal to the Open World

Extending humanoid traversal to the open world is key to practical deployment in human environments, but remains challenging. The robot must use vision to ensure safe and reliable foot placement on heterogeneous terrain under highly dynamic motion, while producing coordinated, natural whole-body behaviors. We propose SSR, an efficient end-to-end framework for egocentric vision-based humanoid traversal that jointly learns these capabilities. SSR introduces imagined foothold guidance, which learns to model forthcoming swing-foot contacts and evaluates their support to guide pre-touchdown swings toward stable regions, reducing edge slips. It further employs equivariant latent-space symmetry augmentation to efficiently induce bilateral coordination under high-dimensional visual observations, and uses terrain-specific multi-discriminator motion priors to encourage human-like behavior across scenes. Extensive experiments show that SSR achieves safe, stable, and high-quality locomotion on diverse real-world terrains, including stairs with varied structures and extreme challenges such as wide gaps and high platforms, while enabling reliable long-horizon traversal in open outdoor environments.

Ruiqi Yu, Yiwen Wang, Yuan Hao +2

Zhejiang University

Humanoid Robot LocomotionTerrain-Aware Robot LocomotionRobot LocomotionEquivariant Representation LearningVision-Based Robot Control
AskPDF
May 26, 2026

On the Equivariant Learning of the QQ-tensor Order Parameter

We construct and evaluate group-equivariant neural networks for the prediction of the two-dimensional QQ-tensor order parameter of nematic liquid crystals from synthetically generated microscopic textures. Seven architectures, equivariant to cyclic groups CkC_k of order kk for k=4, 8, 16, 32, 64, 128, 256k=4,\,8,\,16,\,32,\,64,\,128,\, 256, are built using a combination of weight-sharing constraints, equivariant activations and regularization techniques. To do this, we construct rotation-like permutation matrix groups with elements ϱCk(g)\varrho_{C_k}(g) that act on row-wise vectorized images, thereby approximating a 2πk\frac{2π}{k} rotation of the circular subdomain on square images. We show that all seven equivariant models satisfy the QQ-tensor equivariance constraint to within single-precision floating point accuracy. Comparing against approximate parameter-matched non-equivariant benchmarks, with and without data augmentation, we find that the equivariant models consistently achieve lower errors and generalize more robustly to unseen defect configurations. Performance increases with group order, suggesting that the incorporation of finer rotational symmetry leads to lower errors.

Julia Navarro, Mark Wilkinson

Nottingham Trent University, UK · Berea College, USA

Equivariant Neural NetworksMaterials ScienceEquivariant Representation Learning
AskPDF
May 26, 2026

How the Optimizer Shapes Learned Solutions in Equivariant Neural Networks

Equivariant neural networks encode geometric symmetries by construction, yet they are often difficult to optimize and can underperform less constrained architectures. A growing body of work addresses this through architectural modifications such as constraint relaxation or approximate equivariance, while the role of the optimizer remains comparatively underexplored. We study this direction by comparing Muon and Adam across several equivariant and geometric architectures under pointcloud and molecular learning settings. On ModelNet40, where the comparison is clearest, Muon consistently improves over Adam across all architectures considered. We then analyze the trained ModelNet40 checkpoints through Hessian estimates, loss surface visualizations, and spectral properties of learned weights and intermediate representations. The checkpoints reached by Muon have larger Hessian curvature summaries but more regular loss surfaces, and their learned weights and representations have higher stable and effective ranks. These observations suggest that the interaction between optimizer design and geometric inductive bias deserves further attention from the community.

Teodor-Mihai Stupariu, Andrei Manolache

1Bitdefender, Romania · 2Tudor Vianu High School of Com · University of Stuttgart, Germany +1

Neural Representation GeometryNeural Network OptimizationEquivariant Neural NetworksMuon OptimizerEquivariant Representation LearningNeural Network Training Dynamics
AskPDF
May 19, 2026

Exact Symmetry as Algebra: Group-Structured Tensors that Enforce Physical Selection Rules

Symmetry determines which physical responses of molecules and crystals are allowed and underlies many models used throughout the physical sciences. Yet several widely used machine-learned models learn symmetry only approximately from data, raising the question of whether approximate symmetry is sufficient for scientific prediction. Using the Orb-v3 force field, we show that it is often not: the relaxed space group of a crystal depends on the orientation of the input for 47%47\% of a 150150-crystal subsample, and relaxation lowers the space group for half of all 399399 crystals studied. To characterize, quantify, and correct such failures, we introduce a tensor algebra, ⋆G\star_G, defined by convolution over a finite symmetry group and applicable to tensors with noncommuting entries. We prove that the symmetry group determines the algebra's block-diagonal transform and use this structure to analyze model errors on symmetric targets. The resulting projections decompose the average squared error into orthogonal components arising from selection-rule violations, inconsistencies across symmetry-related inputs, and residual error. The removable components reveal how much error arises from broken symmetry, provide reference-free lower bounds on prediction error, and can be eliminated without sacrificing accuracy. For operators, projection followed by blockwise truncation yields the optimal symmetric low-rank approximation. Applying these projections to force predictions reduces both rates of space-group change to at most 2%2\%. Exact selection rules, reference-free error bounds, and symmetry-preserving corrections are therefore available for any trained model without retraining.

Paulina Hoyos, Shashanka Ubaru, Dongsung Huh +5

UT Austin · IBM Research · Independent +2

Scientific DiscoverySymmetry DiscoveryEquivariant Neural NetworksTensor DecompositionEquivariant Representation LearningSingular Value Decomposition
AskPDF
May 5, 2026

PHALAR: Phasors for Learned Musical Audio Representations

Stem retrieval, the task of matching missing stems to a given audio submix, is a key challenge currently limited by models that discard temporal information. We introduce PHALAR, a contrastive framework achieving a relative accuracy increase of up to ≈70%\approx 70\% over the state-of-the-art while requiring <50%<50\% of the parameters and a 7×\times training speedup. By utilizing a Learned Spectral Pooling layer and a complex-valued head, PHALAR enforces pitch-equivariant and phase-equivariant biases. PHALAR establishes new retrieval state-of-the-art across MoisesDB, Slakh, and ChocoChorales, correlating significantly higher with human coherence judgment than semantic baselines. Finally, zero-shot beat tracking and linear chord probing confirm that PHALAR captures robust musical structures beyond the retrieval task.

Davide Marincione, Michele Mancusi, Giorgio Strano +4

Department of Computer Science, Sapienza University of Rome, Italy · Moises Systems, Inc. · Paradigma, Inc.

Audio Representation LearningRepresentation LearningAudio UnderstandingMusic Information RetrievalEquivariant Representation LearningMusic Understanding
AskPDF
Apr 14, 2026

Image-to-Image Translation Framework Embedded with Rotation Symmetry Priors

Image-to-image translation (I2I) is a fundamental task in computer vision, focused on mapping an input image from a source domain to a corresponding image in a target domain while preserving domain-invariant features and adapting domain-specific attributes. Despite the remarkable success of deep learning-based I2I approaches, the lack of paired data and unsupervised learning framework still hinder their effectiveness. In this work, we address the challenge by incorporating transformation symmetry priors into image-to-image translation networks. Specifically, we introduce rotation group equivariant convolutions to achieve rotation equivariant I2I framework, a novel contribution, to the best of our knowledge, along this research direction. This design ensures the preservation of rotation symmetry, one of the most intrinsic and domain-invariant properties of natural and scientific images, throughout the network. Furthermore, we conduct a systematic study on image symmetry priors on real dataset and propose a novel transformation learnable equivariant convolutions (TL-Conv) that adaptively learns transformation groups, enhancing symmetry preservation across diverse datasets. We also provide a theoretical analysis of the equivariance error of TL-Conv, proving that it maintains exact equivariance in continuous domains and provide a bound for the error in discrete cases. Through extensive experiments across a range of I2I tasks, we validate the effectiveness and superior performance of our approach, highlighting the potential of equivariant networks in enhancing generation quality and its broad applicability. Our code is available at https://github.com/tanfy929/Equivariant-I2I

Feiyu Tan, Heran Yang, Qihong Duan +3

School of Mathematics and Statistics, Xi’an Jiaotong University, Shaanxi, P.R.China · MOE Key Lab for Intelligent Networks and Networks Security, Faculty of Electronic and Information Engineering, Xi’an Jiaotong University, Shaanxi, P.R.China

Image-to-Image TranslationUnpaired Image-to-Image TranslationEquivariant Neural NetworksEquivariant Representation Learning
AskPDF
Apr 9, 2026

Rotation Equivariant Convolutions in Deformable Registration of Brain MRI

Image registration is a fundamental task that aligns anatomical structures between images. While CNNs perform well, they lack rotation equivariance - a rotated input does not produce a correspondingly rotated output. This hinders performance by failing to exploit the rotational symmetries inherent in anatomical structures, particularly in brain MRI. In this work, we integrate rotation-equivariant convolutions into deformable brain MRI registration networks. We evaluate this approach by replacing standard encoders with equivariant ones in three baseline architectures, testing on multiple public brain MRI datasets. Our experiments demonstrate that equivariant encoders have three key advantages: 1) They achieve higher registration accuracy while reducing network parameters, confirming the benefit of this anatomical inductive bias. 2) They outperform baselines on rotated input pairs, demonstrating robustness to orientation variations common in clinical practice. 3) They show improved performance with less training data, indicating greater sample efficiency. Our results demonstrate that incorporating geometric priors is a critical step toward building more robust, accurate, and efficient registration models.

Arghavan Rezvani, Kun Han, Anthony T. Wu +2

University of California, Irvine

Medical Image RegistrationDeformable Image RegistrationEquivariant Neural NetworksMagnetic Resonance ImagingEquivariant Representation Learning
AskPDF
Mar 25, 2026

How unconstrained machine-learning models learn physical symmetries

The requirement of generating predictions that exactly fulfill the fundamental symmetry of the corresponding physical quantities has profoundly shaped the development of machine-learning models for physical simulations. In many cases, models are built using constrained mathematical forms that ensure that symmetries are enforced exactly. However, unconstrained models that do not obey rotational symmetries are often found to have competitive performance, and to be able to learn to a high level of accuracy an approximate equivariant behavior with a simple data augmentation strategy. In this paper, we introduce rigorous metrics to measure the symmetry content of the learned representations in such models, and assess the accuracy by which the outputs fulfill the equivariant condition. We apply these metrics to two unconstrained, transformer-based models operating on decorated point clouds (a graph neural network for atomistic simulations and a PointNet-style architecture for particle physics) to investigate how symmetry information is processed across architectural layers and is learned during training. Based on these insights, we establish a rigorous framework for diagnosing spectral failure modes in ML models. Enabled by this analysis, we demonstrate that one can achieve superior stability and accuracy by strategically injecting the minimum required inductive biases, preserving the high expressivity and scalability of unconstrained architectures while guaranteeing physical fidelity.

Michelangelo Domina, Joseph William Abbott, Paolo Pegolo +2

Laboratory of Computational Science and Modeling, Institut des Matériaux, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland

Equivariant Representation LearningPhysics-Based Simulation
AskPDF
Mar 23, 2026

Generalized multi-object classification and tracking with sparse feature resonator networks

In visual scene understanding tasks, it is essential to capture both invariant and equivariant structure. While neural networks are frequently trained to achieve invariance to transformations such as translation, this often comes at the cost of losing access to equivariant information - e.g., the precise location of an object. Moreover, invariance is not naturally guaranteed through supervised learning alone, and many architectures generalize poorly to input transformations not encountered during training. Here, we take an approach based on analysis-by-synthesis and factoring using resonator networks. A generative model describes the construction of simple scenes containing MNIST digits and their transformations, like color and position. The resonator network inverts the generative model, and provides both invariant and equivariant information about particular objects. Sparse features learned from training data act as a basis set to provide flexibility in representing variable shapes of objects, allowing the resonator network to handle previously unseen digit shapes from the test set. The modular structure provides a shape module which contains information about the object shape with translation factored out, allowing a simple classifier to operate on centered digits. The classification layer is trained solely on centered data, requiring much less training data, and the network as a whole can identify objects with arbitrary translations without data augmentation. The natural attention-like mechanism of the resonator network also allows for analysis of scenes with multiple objects, where the network dynamics selects and centers only one object at a time. Further, the specific position information of a particular object can be extracted from the translation module, and we show that the resonator can be designed to track multiple moving objects with precision of a few pixels.

Lazar Supic, Alec Mullen, E. Paxon Frady

Redwood Center for Theoretical Neuroscience, UC Berkeley · Neuromorphic Computing Lab, Intel

Object-Centric Representation LearningMulti-Object TrackingInvariant Representation LearningEquivariant Representation Learning
AskPDF
Mar 12, 2026

Disentangled Representation Learning through Unsupervised Symmetry Group Discovery

Symmetry-based disentangled representation learning leverages the group structure of environment transformations to uncover the latent factors of variation. Prior approaches to symmetry-based disentanglement have required strong prior knowledge of the symmetry group's structure, or restrictive assumptions about the subgroup properties. In this work, we remove these constraints by proposing a method whereby an embodied agent autonomously discovers the group structure of its action space through unsupervised interaction with the environment. We prove the identifiability of the true symmetry group decomposition under minimal assumptions, and derive two algorithms: one for discovering the group decomposition from interaction data, and another for learning Linear Symmetry-Based Disentangled (LSBD) representations without assuming specific subgroup properties. Our method is validated on three environments exhibiting different group decompositions, where it outperforms existing LSBD approaches.

Barthélémy Dang-Nhu, Louis Annabi, Sylvain Argentieri

Sorbonne Universit´e, CNRS, Institut des Syst`emes Intelligents et de Robotique, ISIR, F-75005 Paris, France

Disentangled Representation LearningUnsupervised Representation LearningSymmetry DiscoveryEquivariant Representation Learning
AskPDF
Mar 7, 2026

LEPA: Learning Geometric Equivariance in Satellite Remote Sensing Data with a Predictive Architecture

Geospatial foundation models provide precomputed embeddings that serve as compact feature vectors for large-scale satellite remote sensing data. While these embeddings can reduce data-transfer bottlenecks and computational costs, Earth observation (EO) applications can still face geometric mismatches between user-defined areas of interest and the fixed precomputed embedding grid. Standard latent-space interpolation is unreliable in this setting because the embedding manifold is highly non-convex, yielding representations that do not correspond to realistic inputs. We verify this using Prithvi-EO-2.0 to understand the shortcomings of interpolation applied to patch embeddings. As a substitute, we propose a Learned Equivariance-Predicting Architecture (LEPA). Instead of averaging vectors, LEPA conditions a predictor on geometric augmentations to directly predict the transformed embedding. We evaluate LEPA on NASA/USGS Harmonized Landsat-Sentinel (HLS) imagery and ImageNet-1k. Experiments show that standard interpolation achieves a mean reciprocal rank (MRR) below 0.2, whereas LEPA increases MRR to over 0.8, enabling accurate geometric adjustment without re-encoding.

Erik Scheurer, Rocco Sedona, Stefan Kesselheim +1

Jülich Supercomputing Centre (JSC), Forschungszentrum Jülich, 52428 Jülich, Germany · Institute for Visualization and Interactive Systems (VIS), University of Stuttgart, 70569 Stuttgart · School of Engineering and Natural Sciences (SENS), University of Iceland, 102 Reykjavík, Iceland

Equivariant Representation LearningGeometric Representation LearningRemote SensingGeospatial Foundation ModelsPredictive Representation Learning
AskPDF
Mar 4, 2026

Soft Equivariance Regularization for Invariant Self-Supervised Learning

Self-supervised learning (SSL) typically learns representations invariant to semantic-preserving augmentations. While effective for recognition, enforcing strong invariance can suppress transformation-dependent structure that is useful for robustness to geometric perturbations and spatially sensitive transfer. A growing body of work, therefore, augments invariance-based SSL with equivariance objectives, but these objectives are often imposed on the same final representation. We empirically observe a trade-off in this coupled setting: pushing equivariance regularization toward deeper layers improves equivariance scores but degrades ImageNet-1k linear evaluation, motivating a layer-decoupled design. Motivated by this trade-off, we propose Soft Equivariance Regularization (SER), a plug-in regularizer that decouples where invariance and equivariance are enforced: we keep the base SSL objective unchanged on the final embedding, while softly encouraging equivariance on an intermediate spatial token map via analytically specified group actions ρgρ_g applied directly in feature space. SER learns/predicts no per-sample transformation codes/labels, requires no auxiliary transformation-prediction head, and adds only 1.008x training FLOPs. On ImageNet-1k ViT-S/16 pretraining, SER improves MoCo-v3 by +0.84 Top-1 in linear evaluation under a strictly matched 2-view setting and consistently improves DINO and Barlow Twins; under matched view counts, SER achieves the best ImageNet-1k linear-eval Top-1 among the compared invariance+equivariance add-ons. SER further improves ImageNet-C/P by +1.11/+1.22 Top-1 and frozen-backbone COCO detection by +1.7 mAP. Finally, applying the same layer-decoupling recipe to existing invariance+equivariance baselinesimproves their accuracy, suggesting layer decoupling as a general design principle for combining invariance and equivariance.

Joohyung Lee, Changhun Kim, Hyunsu Kim +2

AITRICS · Voronoi Inc. · Korea Advanced Institute of Science and Technology (KAIST)

Self-Supervised Visual Representation LearningEquivariant Representation LearningInvariant Representation LearningEquivariant Neural Networks
AskPDF
Mar 3, 2026

Coalgebras for categorical deep learning: Representability and universal approximation

Categorical deep learning (CDL) has recently emerged as a framework that leverages category theory to unify diverse neural architectures. While geometric deep learning (GDL) is grounded in the specific context of invariants of group actions, CDL aims to provide domain-independent abstractions for reasoning about models and their properties. In this paper, we contribute to this program by developing a coalgebraic foundation for equivariant representation in deep learning, as classical notions of group actions and equivariant maps are naturally generalized by the coalgebraic formalism. Our first main result demonstrates that, given an embedding of data sets formalized as a functor from SET to VECT, and given a notion of invariant behavior on data sets modeled by an endofunctor on SET, there is a corresponding endofunctor on VECT that is compatible with the embedding in the sense that this lifted functor recovers the analogous notion of invariant behavior on the embedded data. Building on this foundation, we then establish a universal approximation theorem for equivariant maps in this generalized setting. We show that continuous equivariant functions can be approximated within our coalgebraic framework for a broad class of symmetries. This work thus provides a categorical bridge between the abstract specification of invariant behavior and its concrete realization in neural architectures.

Dragan Mašulović

University of Novi Sad, Faculty of Sciences Department of Mathematics and Informatics Trg Dositeja Obradovića 3, 21000 Novi Sad, Serbia

Equivariant Representation LearningInvariant Representation LearningNeural Network Approximation TheoryCategory Theory
AskPDF
Feb 20, 2026

Balancing Symmetry and Efficiency in Graph Flow Matching

Equivariance is central to graph generative models, as it ensures the model respects the permutation symmetry of graphs. However, strict equivariance can increase computational cost due to added architectural constraints, and can slow down convergence because the model must be consistent across a large space of possible node permutations. We study this trade-off for graph generative models. Specifically, we start from an equivariant discrete flow-matching model, and relax its equivariance during training via a controllable symmetry modulation scheme based on sinusoidal positional encodings and node permutations. Experiments first show that symmetry-breaking can accelerate early training by providing an easier learning signal, but at the expense of encouraging shortcut solutions that can cause overfitting, where the model repeatedly generates graphs that are duplicates of the training set. On the contrary, properly modulating the symmetry signal can delay overfitting while accelerating convergence, allowing the model to reach stronger performance with 19%19\% of the baseline training epochs.

Benjamin Honoré, Alba Carballo-Castro, Yiming Qin +1

LTS4, EPFL, Lausanne, Switzerland

Generative ModelingEquivariant GNNsFlow MatchingEquivariant Representation LearningDiscrete Flow Matching
AskPDF
Jul 31, 2025

L-GTA: Latent Generative Modeling for Time Series Augmentation

Data augmentation is becoming increasingly important across various areas of time series analysis, including forecasting, classification, and anomaly detection. We introduce the Latent Generative Temporal Augmentation (L-GTA) model, a generative approach based on a Variational Autoencoder with a Bi-LSTM backbone and temporal self-attention. The model learns a latent representation for each timestep and applies controlled perturbations such as jittering, magnitude warping, or drift. We define an equivariance objective to further encourage consistency between latent space and data space transformations. As a result, the augmented samples show predictable and interpretable transformation signatures. We evaluate L-GTA on several real-world datasets against SOTA generative methods, including TimeGAN, TimeVAE, and Diffusion-TS, as well as direct transformation approaches. Across experiments on downstream forecasting, distribution fidelity, and controllability of transformation intensity, L-GTA consistently outperforms competing approaches. In downstream forecasting, it reduces prediction error by up to 26% compared to the strongest generative method and 27% relative to using the original data without augmentation.

Luis Roque, Vitor Cerqueira, Carlos Soares +1

Faculdade de Engenharia da Universidade do Porto, Porto, Portugal · University of Coimbra, Coimbra, Portugal · Fraunhofer Portugal AICOS, Portugal +1

Variational AutoencodersSynthetic Data AugmentationRepresentation LearningTime Series Data AugmentationEquivariant Representation Learning
AskPDF
May 26, 2025

Representation Learning for Equivariant Inference with Guarantees

In many real-world applications of regression, conditional probability estimation, and uncertainty quantification, exploiting symmetries rooted in physics or geometry can dramatically improve generalization and sample efficiency. While geometric deep learning has made empirical advances by incorporating symmetry and geometry priors, less attention has been given to statistical learning guarantees. In this paper, we introduce an equivariant representation learning framework that simultaneously addresses regression, conditional probability estimation, and uncertainty quantification while providing first-of-its-kind non-asymptotic statistical learning guarantees. Grounded in operator and group representation theory, our framework approximates the spectral decomposition of the conditional expectation operator, building representations that are both equivariant and disentangled along independent symmetry quotient groups. Empirical evaluations on synthetic datasets and real-world robotics applications confirm the potential of our approach, matching or outperforming existing equivariant baselines in regression while providing well-calibrated uncertainty estimates.

Daniel Ordoñez-Apraez, Vladimir Kostić, Alek Fröhlich +3

CSML, Italian Institute of Technology, Genova, Italy · Universit`a di Genova, Genova, Italy · University of Novi Sad, Novi Sad, Serbia +2

Disentangled Representation LearningRepresentation LearningUncertainty QuantificationConditional Distribution EstimationEquivariant Representation LearningStatistical Learning Theory
AskPDF
Mar 13, 2023

NeuSE: Neural SE(3)-Equivariant Embedding for Long-Term Object-Based Simultaneous Localization and Mapping

We present NeuSE, a novel Neural SE(3)-Equivariant Embedding for objects, and illustrate how it supports object-based Simultaneous Localization and Mapping (SLAM) for consistent spatial understanding with long-term scene changes. NeuSE is a set of latent object embeddings created from partial object observations. It serves as a compact point cloud surrogate for complete object models, encoding the full shape, scale, and transform information about an object. In addition, the inferred latent code is both SE(3) and scale equivariant, enabling strong generalization to objects of both unseen sizes and different SE(3) poses. This makes NeuSE particularly effective in real-world scenarios where objects may vary in size or spatial configuration. With NeuSE, relative frame transforms can be directly derived from inferred latent codes. Our proposed SLAM paradigm, using NeuSE for object shape, size, and pose characterization, can operate independently or in conjunction with typical SLAM systems. It directly infers SE(3) camera pose constraints that are compatible with general SLAM pose graph optimization, while maintaining a lightweight, object-centric map that adapts to real-world changes. Our evaluation is conducted on synthetic and real-world sequences with changes in both controlled and uncontrolled settings, featuring multi-category objects of various shapes and sizes. Our approach demonstrates improved localization capability and change-aware mapping consistency when working either independently or as a complement to common SLAM pipelines. Additional results are presented in the supplementary video: https://youtu.be/fZ4_2P1fQBQ

Jiahui Fu, Yilun Du, Kurran Singh +2

MIT CSAIL

Simultaneous Localization and MappingVisual SLAMObject-Centric Representation LearningEquivariant Representation Learning
AskPDF