Symmetry

Generative modeling of physical systems, such as molecules, requires learning distributions that are invariant under global symmetries, such as rotations in three-dimensional space. Equivariant diffusion and flow matching models can incorporate such invariances effectively, even when trained on a non-invariant empirical distribution, but they typically rely on costly multi-step sampling. Recently, drifting models have emerged as an efficient alternative, enabling single-step generation and achieving state-of-the-art performance in generative modeling tasks. However, we show that drifting models face a symmetry-specific challenge, since an equivariant generator does not generally produce the same drifting field as the one obtained from the symmetrized target distribution. Addressing this issue would require expensive symmetrization of the empirical distribution. To avoid this cost, we propose SymDrift, a framework that makes the drifting field itself symmetry-aware. We introduce two complementary strategies: (i) a symmetrized drift in coordinate space based on optimal alignment, and (ii) a $G$-invariant embedding that removes symmetry ambiguity by construction. Empirically, SymDrift outperforms existing one-shot methods on standard benchmarks for conformer and transition state generation, while remaining competitive with significantly more expensive multi-step approaches. By enabling one-shot inference, SymDrift reduces computational overhead by up to 40$\times$ compared to existing baselines, making it promising for high-throughput applications such as virtual drug screening and large-scale reaction network exploration.

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

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.

Diffusion models undergo a phase transition in a critical time window during generation dynamics, with two complementary diagnoses of criticality. The symmetry breaking picture views the critical window as when trajectories bifurcate into different semantic minima of the energy landscape, whereas the nonlocality picture views the critical window as when local denoising fails. We study whether two notions of such phase transitions are concurrent in modern diffusion transformers. By evaluating the dynamics and outcomes of the generation trajectory, we observe a near-simultaneous occurrence of the non-locality and symmetry breaking critical times. Our work is the first to unify the two notions of phase transitions in practice: it provides a concrete diagnostic for when and why diffusion models rely on conditioning and global denoising, enabling principled evaluation of model efficiency and guiding the design of architectures and sampling schemes that avoid unnecessary computation.

We consider the problem of learning data-driven replicas for stiff systems of ordinary differential equations arising in chemical kinetics that can be evaluated with high computational efficiency. We first focus on training emulators for families of reaction equations under varying reaction rates, using conditional residual networks or long-short term memory architectures. We then apply a recently proposed data-driven framework known as ``inVAErt networks'' to address the ill-posed inverse problem of inferring reaction rates, integration time, and possibly initial conditions from a target set of species concentrations - a problem that has received relatively little attention in the literature. The proposed approach is demonstrated on chemical systems with reversible and irreversible kinetics, spanning 2 to 20 differential equations, 3 to 20 chemical species, and 3 to 25 reaction rate parameters. Relative root mean squared errors produced by the proposed emulators range from $10^{-5}$ for lower-dimensional systems to $10^{-4}$ and $10^{-3}$ for an air pollution model and a hydrogen-air reaction system, respectively. Manifolds of non-identifiable reaction rates recovered by the proposed approach can be analytically verified for simple systems and are consistent with local identifiability analysis in higher dimensions.

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.

Continuous-memory recurrent networks must preserve phase, position, or orientation despite model imperfections and state noise. We develop a geometric framework that separates three questions: how many memory coordinates are neutrally transported, how deterministic perturbations alter their finite-horizon stability, and how ambient noise is decoded along them. Exact per-input equivariance transports analytical group tangents pathwise and, on a compact nondegenerate orbit stratum, yields at least $q=\dim(G/H)$ zero group-tangent Lyapunov exponents under stationary ergodic driving. For imperfect dynamics, a four-block tangent/normal decomposition gives local and finite-horizon bounds on tangent growth and subspace rotation, distinguishing first-order direct damage from second-order leakage through contracting normal directions. For noisy dynamics, a specified decoder maps ambient covariance $Q$ to coordinate covariance $ZQZ^\top$; under isotropic noise and fixed tangent energy, least-squares decoding and scaled-isometric action geometry minimize local diffusion. Local and finite-horizon evaluations include cases both within and outside the sufficient conditions. In a fresh twenty-seed $T^2$ replication, a decoder-covariance objective improves noisy horizon-256 memory in every pair while meeting a prespecified clean-error equivalence margin. Direct noise training also improves noisy memory but incurs a clean-error tradeoff. In coupled $T^4/T^8$ integrators, the advantage of anisotropic-covariance over isotropic regularization reverses when evaluation noise becomes isotropic. These results connect continuous symmetry to measurable limits and design choices for recurrent memory under explicit dynamical, decoder, and noise assumptions.

Anatomical mesh segmentation requires models that operate directly on irregular surface geometry while remaining robust to arbitrary patient pose and mesh resolution variation. Existing task-specific mesh and point-cloud methods are not equivariant, and can degrade sharply under test-time perturbation, for example dropping by 25-26 IoU points on intraoral scan segmentation at $40^\circ$ tilt. We present EAMS, an Equivariant Anatomical Mesh Segmentor built on Equivariant Mesh Neural Networks (EMNN), and evaluate it across four clinically distinct tasks spanning edge-, vertex-, and face-level supervision. We combine intrinsic mesh descriptors with anatomy-aware priors, including PCA-derived frames for dental arches and liver surfaces, and augment message passing to provide lightweight global context. Across intracranial aneurysm and intraoral segmentation, EAMS variants are competitive with specialized baselines on unperturbed inputs while remaining stable under geometric perturbations, and on liver surfaces they expose a favorable trade-off between canonical-pose accuracy and rotation robustness. These results show that a lightweight ($<2$M parameters) equivariant framework can deliver robust anatomical mesh segmentation across diverse supervision types without task-specific architectures.

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.

Fourier Neural Operators are deep learning models that learn mappings between function spaces and can be used to learn and solve partial differential equations (PDEs), in some cases significantly faster than traditional PDE solvers. Within the model are Fourier layers, which apply linear transformations directly to the Fourier modes, with parameters depending on the wave numbers. However, most physical systems are isotropic, with the results being independent of the coordinate system chosen, but the linear transformations do not necessarily respect these symmetries. We propose a modification to the linear transformations that ensures spatial symmetries are respected, called the Isotropic Fourier Neural Operator, which both improves model performance and reduces the number of parameters by up to a factor of 16 in 2D and 96 in 3D.

Bayesian Optimization is an iterative method, tailored to optimizing expensive black box objective functions. Surrogate models like Gaussian Processes, which are the gold standard in Bayesian Optimization, can be inefficient for inputs with permutation symmetries, as the most common kernels employed are better suited for vector inputs rather than unordered sets of items. Motivated by this issue, we turn to permutation invariant Bayesian Optimization for well placement in Carbon Capture and Storage projects. The high fidelity black box simulator is instructed to operate wells under group control, giving rise to permutation symmetries within injector and producer groups that cannot be exploited with standard GP kernels. In this work, our main contribution is a novel Gaussian Process kernel (GP-Perm) that encodes permutation invariance by comparing sets through a stable divergence between their induced empirical representations, and can be combined with standard kernels for additional vector-valued inputs. As a learned invariant baseline, we also consider a Deep Kernel Learning model (DKL-DS) using the Deep Sets architecture to learn a permutation-invariant embedding. We evaluate the proposed methodology across 8 use cases, comprising seven synthetic benchmarks and one realistic CCS case study (Johansen formation)

The performance of machine learning models often relies on large labeled datasets; however, data collected from diverse sources can contain label noise. Recent work has shown that, in noisy settings, there may exist a subset of the training data on which models can achieve performance comparable to training on a noise-free dataset. A widely used method for identifying such subsets is cutstats, which employs k-nearest neighbors (k-NN) to detect low-noise samples. However, its performance on high-dimensional data remains largely unexplored. In this work, we formally establish that the performance of a classifier trained on a subset of a noisy dataset selected via cutstats is influenced by the accuracy of k-NN. We further demonstrate that, in noisy environments, exploiting data invariance and knowledge of underlying symmetries can significantly enhance the performance of k-NN, bringing it closer to the Bayes optimal classifier even in high-dimensional regimes. Finally, we show that for real-world scenarios, where information about the underlying invariance is only partially known, learnt invariant representations can still facilitate the identification of near-optimal subsets.

Model merging aims to combine multiple models into one without additional training. Naïve parameter-space averaging can be fragile under architectural symmetries, as their geometry does not take them into account. In this work we show that not only the geometry, but also the averaging procedure itself, must be symmetry-invariant to achieve symmetry-aware merges. Consequently, we propose a general solution: merging as Fréchet averaging, i.e., selecting parameters that minimize a sum of geodesic distances on an appropriate manifold. In this view, the key design choice is the overall geometry, i.e., the choice of metric, manifold, and distance approximation, that determines what it means for two models to be "close". We show that Fréchet averaging, combined with simplifying assumptions, contains Fisher merging. Building on this, we examine the particular case of low-rank adapters (LoRA), whose symmetries induce a distinct geometry: that of a quotient manifold. We outline the limitations of current LoRA merging methods, propose a practical algorithm for this setting, and show how they compare with other commonly used approaches.

First-principles atomistic simulations are essential for understanding complex material phenomena but are fundamentally limited by their computational cost. While Machine Learning Interatomic Potentials (MLIPs) have drastically improved cost for a given accuracy, their inference cost remains a bottleneck for massive systems or long timescales. To address this, we introduce a multifidelity "Mixture-of-Experts" framework based on the E(3)-equivariant Allegro architecture. Our method spatially partitions the simulation domain into a chemically complex region (e.g., reactive interfaces) and a simple region (e.g., bulk lattice), assigning models of varying capacity to each. Among the challenges in such static domain decomposition, the mechanical mismatch between models at the interface is particularly critical, as it can generate artificial stress fields and instability. We address this challenge with a co-training strategy in which the loss function includes agreement constraints -- penalties on per-atom energy and force discrepancies between models evaluated on shared bulk environments -- forcing the independent models to learn a consistent physical description of the bulk material. We validate this approach on a realistic Pt+CO catalytic system, demonstrating that the co-trained models maintain exact energy conservation, align their bulk mechanical response (e.g., equation of state and bulk modulus), and achieve predictive accuracy comparable to a full high-fidelity simulation at more than twice the computational speed.

Deep Reinforcement Learning (DRL) algorithms often require a large amount of data and struggle in sparse-reward domains with long planning horizons and multiple sub-goals. In this paper, we propose a neuro-symbolic extension of Proximal Policy Optimization (PPO) that transfers partial logical policy specifications learned in easier instances to guide learning in more challenging settings. We introduce two integrations of symbolic guidance: (i) H-PPO-Product, which biases the action distribution at sampling time, and (ii) H-PPO-SymLoss, which augments the PPO loss with a symbolic regularization term. We evaluate our methods on three benchmarks (OfficeWorld, WaterWorld, and DoorKey), showing consistently faster learning and higher return at convergence than PPO and a Reward Machine baseline, also under imperfect symbolic knowledge.

Overparameterization is central to the success of deep learning, yet the mechanisms by which it improves optimization remain incompletely understood. We analyze weight-space symmetries in neural networks and show that overparameterization introduces additional symmetries that benefit optimization in two distinct ways. First, we prove that these symmetries act as a form of diagonal preconditioning on the Hessian, enabling the existence of better-conditioned minima within each equivalence class of functionally identical solutions. Second, we show that overparameterization increases the probability mass of global minima near typical initializations, making these favourable solutions more reachable. These results offer a potential link between loss landscape geometry and simplicity bias. Empirically, we observe wider networks have lower top eigenvalues, smaller condition numbers and faster convergence, matching our analysis. Our analysis provides a unified framework for understanding overparameterization and width growth as a geometric transformation of the loss landscape.

Recent work revisiting measurability in the fundamental theorem of statistical learning imposes Borel measurability of ghost-gap suprema. We show that, at the one-sided ghost-gap interface actually used by the standard symmetrization proof, this requirement is stronger than necessary. For any Borel-parameterized concept class on a Polish domain, the bad event "there exists a hypothesis whose ghost empirical error exceeds its training empirical error by at least ε/2" is analytic. By Choquet capacitability, it is therefore measurable in the completion of every finite Borel measure. We then construct a concept class whose bad event is null-measurable but not Borel, giving a strict separation from the Borel supremum condition. Finally, we prove closure under patching, fixed and countable interpolation, and fiber-product amalgamation, showing that the weaker regularity level is stable under natural concept-class constructors. In the realizable setting, where targets belong to the class and are measurable, these results weaken the measurability hypothesis needed by the symmetrization route from finite VC dimension to PAC learnability. The main results and the descriptive-set-theoretic infrastructure used by them are formalized in Lean 4.

The need to burn 100% H2 in high efficient gas turbines featuring low NOx combustion in premix mode require the complete redesign of the combustion system to ensure stable operation without any flashback. Since all engine frames featuring a power range from 4 MW up to 600 MW are affected, a huge design effort is expected. To reduce this effort, especially to transfer knowledge between the different engine classes, generative design methods using latest AI technology will provide promising potential. In this work, this challenge is approached utilizing the current advances in generative artificial intelligence. We train an Invertible Neural Network (INN) on an expandable database of geometrically parameterized combustor designs with simulated performance labels. Utilizing the INN in its inverse direction, multiple design proposals are generated which fulfill specified performance labels.

Metanetworks are neural architectures designed to operate directly on pretrained weights to perform downstream tasks. However, the parameter space serves only as a proxy for the underlying function class, and the parameter-function mapping is inherently non-injective: distinct parameter configurations may yield identical input-output behaviors. As a result, metanetworks that rely solely on raw parameters risk overlooking the intrinsic symmetries of the architecture. Reasoning about functional identity is therefore essential for effective metanetwork design, motivating the development of equivariant metanetworks, which incorporate equivariance principles to respect architectural symmetries. Existing approaches, however, typically enforce strict equivariance, which imposes rigid constraints and often leads to sparse and less expressive models. To address this limitation, we introduce the novel concept of quasi-equivariance, which allows metanetworks to move beyond the rigidity of strict equivariance while still preserving functional identity. We lay down a principled basis for this framework and demonstrate its broad applicability across diverse neural architectures, including feedforward, convolutional, and transformer networks. Through empirical evaluation, we show that quasi-equivariant metanetworks achieve good trade-offs between symmetry preservation and representational expressivity. These findings advance the theoretical understanding of weight-space learning and provide a principled foundation for the design of more expressive and functionally robust metanetworks.

A widely cited result by Dong et al. (2021) showed that Transformers built from self-attention alone, without skip connections or feed-forward layers, suffer from rapid rank collapse: all token representations converge to a single direction. The proposed remedy was the MLP. We show that this picture, while correct in the regime studied by Dong, is incomplete in ways that matter for architectural understanding. Three results are established. First, layer normalisation is precisely affine-rank-neutral: it preserves the affine rank of the token representation set exactly. The widespread claim that LN "plays no role" is imprecise; the correct statement is sharper. Second, residual connections generically obstruct rank collapse in real Transformers such as BERT-base, in a measure-theoretic sense, without contribution from the MLP. The MLP's irreplaceable function is different: generating feature directions outside the linear span of the original token embeddings, which no stack of attention layers can produce. Third, a phenomenon distinct from rank collapse is identified: head-channel non-identifiability. After multi-head attention sums per-head outputs through the output projection, individual contributions cannot be canonically attributed to a specific head; n(H-1)d_k degrees of freedom per layer remain ambiguous when recovering a single head from the mixed signal. The MLP cannot remedy this because it acts on the post-summation signal. A constructive partial remedy is proposed: a position-gated output projection (PG-OP) at parameter overhead below 1.6% of the standard output projection. The four collapse phenomena identified in the literature -- rank collapse in depth, in width, head-channel non-identifiability, and entropy collapse -- are unified under a symmetry-breaking framework, each corresponding to a distinct symmetry of the Transformer's forward pass.

Symmetry detection is a fundamental problem in computer vision, and symmetries serve as powerful priors for downstream tasks. However, existing learning-based methods for detecting 3D symmetries from single images have been almost exclusively trained and evaluated on object-centric or synthetic datasets, and thus fail to generalize to real-world scenes. Furthermore, due to the inherent scale ambiguity of monocular inputs, which makes localizing the 3D plane an ill-posed problem, many existing works only predict the plane's orientation. In this paper, we address these limitations by presenting the first framework for detecting 3D-grounded reflectional symmetries from single, in-the-wild RGB images, focusing on architectural landmarks. We introduce two key innovations: (1) a scalable data annotation pipeline to automatically curate a large-scale dataset of architectural symmetries, ArchSym, from SfM reconstructions by leveraging cross-view image matching; and building on the dataset, (2) a single-view symmetry detector that accurately localizes symmetries in 3D by parameterizing them as signed distance maps defined relative to predicted scene geometry. We validate our symmetry annotation pipeline against geometry-based alternatives and demonstrate that our symmetry detector significantly outperforms state-of-the-art baselines on our new benchmark.

Diffusion-based generative models have reformed generative AI, and also enabled new capabilities in the science domain, e.g., fast generation of 3D structures of molecules. In such tasks, there is often a symmetry in the system, identifying elements that can be converted by certain transformations as equivalent. Equivariant diffusion models guarantee a symmetric distribution, but miss the opportunity to make learning easier, while alignment-based simplification attempts fail to preserve the target distribution. In this work, we develop quotient-space diffusion models, a principled generative framework to fully handle and leverage symmetry. By viewing the intrinsic generation process on the quotient space, the exact construction that removes symmetry redundancy, the framework simplifies learning by allowing model output to have an arbitrary intra-equivalence-class movement, while generating the correct symmetric target distribution with guarantee. We instantiate the framework for molecular structure generation which follows $\mathrm{SE}(3)$ (rigid-body movement) symmetry. It improves the performance over equivariant diffusion models and outperforms alignment-based methods universally for small molecules and proteins, representing a new framework that surpasses previous symmetry treatments in generative models.

When approximating an intractable density via variational inference (VI) the variational family is typically chosen as a simple parametric family that very likely does not contain the target. This raises the question: Under which conditions can we recover characteristics of the target despite misspecification? In this work, we extend previous theoretical results on robust VI with location-scale families under target symmetries in two substantial ways: (1) We open them up to a wider range of divergences by providing sufficient conditions for exact recovery of the target mean and correlation matrix when using the forward Kullback-Leibler divergence and $α$-divergences. (2) By doing so, we find that we can drop the restrictive assumption of a log-concave target made in previous work, allowing us to give guarantees for a wider range of targets, including multi-modal ones. In our experiments, we show how our guarantees can serve as guidelines for the choice of the variational family and $α$-value and we illustrate on a diverse set of examples how and why optimization can fail in the absence of our sufficient conditions.

Graph neural networks face two fundamental challenges rooted in the linear structure of Euclidean vector spaces: (1) Current architectures represent geometry through vectors (directions, gradients), yet many tasks require matrix-valued representations that capture relationships between directions-such as how atomic orientations covary in a molecule. These second-order representations are naturally captured by points on the symmetric positive definite matrices (SPD) manifold; (2) Standard message passing applies shared transformations across edges. Sheaf neural networks address this via edge-specific transformations, but existing formulations remain confined to vector spaces and therefore cannot propagate matrix-valued features. We address both challenges by developing the first sheaf neural network operates natively on the SPD manifold. Our key insight is that the SPD manifold admits a Lie group structure, enabling well-posed analogs of sheaf operators without projecting to Euclidean space. Theoretically, we prove that SPD-valued sheaves are strictly more expressive than Euclidean sheaves: they admit consistent configurations (global sections) that vector-valued sheaves cannot represent, directly translating to richer learned representations. Empirically, our sheaf convolution transforms effectively rank-1 directional inputs into full-rank matrices encoding local geometric structure. Our dual-stream architecture achieves SOTA on 6/7 MoleculeNet benchmarks, with the sheaf framework providing consistent depth robustness.

Symmetric objects are common in daily life and industry, yet their inherent orientation ambiguities that impede the training of deep learning networks for pose estimation are rarely discussed in the literature. To cope with these ambiguities, existing solutions typically require the design of specific loss functions and network architectures or resort to symmetry-invariant evaluation metrics. In contrast, we focus on the numeric representation of the rotation itself, modifying trigonometric identities with the degrees of symmetry derived from the objects' shapes. We use our representation, SARR, to obtain canonic (symmetry-resolved) poses for the symmetric objects in two popular 6D pose estimation datasets, T-LESS and ITODD, where SARR is unique and continuous w.r.t. the visual appearance. This allows us to use a standard CNN for 3D orientation estimation whose performance is evaluated with the symmetry-sensitive cosine distance $\text{AR}_{\text{C}}$. Our networks outperform the state of the art using $\text{AR}_{\text{C}}$ and achieve satisfactory performance when using conventional symmetry-invariant measures. Our method does not require any 3D models but only depth, or, as part of an additional experiment, texture-less RGB/grayscale images as input. We also show that networks trained on SARR outperform the same networks trained on rotation matrices, Euler angles, quaternions, standard trigonometrics or the recently popular 6d representation -- even in inference scenarios where no prior knowledge of the objects' symmetry properties is available. Code and a visualization toolkit are available at https://github.com/akriegler/SARR .

This paper investigates communication-efficient neural network transmission by exploiting structured symmetry constraints in convolutional kernels. Instead of transmitting all model parameters, we propose a degrees-of-freedom (DoF) based codec that sends only the unique coefficients implied by a chosen symmetry group, enabling deterministic reconstruction of the full weight tensor at the receiver. The proposed framework is evaluated under quantization and noisy channel conditions across multiple symmetry patterns, signal-to-noise ratios, and bit-widths. To improve robustness against transmission impairments, a projection step is further applied at the receiver to enforce consistency with the symmetry-invariant subspace, effectively denoising corrupted parameters. Experimental results on MNIST and CIFAR-10 using a DeepCNN architecture demonstrate that DoF-based transmission achieves substantial bandwidth reduction while preserving significantly higher accuracy than pruning-based baselines, which often suffer catastrophic degradation. Among the tested symmetries, \textit{central-skew symmetry} consistently provides the best accuracy-compression tradeoff, confirming that structured redundancy can be leveraged for reliable and efficient neural model delivery over constrained links.

Recent work has shown that parameterizing and optimizing coordinate transformations using normalizing flows, i.e., invertible neural networks, can significantly accelerate the convergence of spectral approximations. We present the first error estimates for approximating functions using Hermite expansions composed with adaptive coordinate transformations. Our analysis establishes an equivalence principle: approximating a function $f$ in the span of the transformed basis is equivalent to approximating the pullback of $f$ in the span of Hermite functions. This allows us to leverage the classical approximation theory of Hermite expansions to derive error estimates in transformed coordinates in terms of the regularity of the pullback. We present an example demonstrating how a nonlinear coordinate transformation can enhance the convergence of Hermite expansions. Focusing on smooth functions decaying along the real axis, we construct a monotone transport map that aligns the decay of the target function with the Hermite basis. This guarantees spectral convergence rates for the corresponding Hermite expansion. Our analysis provides theoretical insight into the convergence behavior of adaptive Hermite approximations based on normalizing flows, as recently explored in the computational quantum physics literature.

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

Parameter space is not function space for neural network architectures. This fact, investigated as early as the 1990s under terms such as reverse engineering," or parameter identifiability", has led to the natural question of parameter space symmetries\textemdash the study of distinct parameters in neural architectures which realize the same function. Indeed, the quotient space obtained by identifying parameters giving rise to the same function, called the \textit{neuromanifold}, has been shown in some cases to have rich geometric properties, impacting optimization dynamics. Thus far, techniques towards complete classifications have required the analyticity of the activation function, notably excising the important case of ReLU. Here, in contrast, we exploit the non-differentiability of the ReLU activation to provide a complete classification of the symmetries in the shallow case.

Grokking occurs when a model achieves high training accuracy but generalization to unseen test points happens long after that. This phenomenon was initially observed on a class of algebraic problems, such as learning modular arithmetic (Power et al., 2022). We study grokking on algebraic tasks in a class of feature learning kernels via the Recursive Feature Machine (RFM) algorithm (Radhakrishnan et al., 2024), which iteratively updates feature matrices through the Average Gradient Outer Product (AGOP) of an estimator in order to learn task-relevant features. Our main experimental finding is that generalization occurs only when a certain symmetry in the training set is broken. Furthermore, we empirically show that RFM generalizes by recovering the underlying invariance group action inherent in the data. We find that the learned feature matrices encode specific elements of the invariance group, explaining the dependence of generalization on symmetry.

This work uncovers variational principles behind symmetrizing the Bregman divergences induced by generic mirror maps over the cone of positive definite matrices. We show that computing the canonical means for this symmetrization can be posed as minimizing the desired symmetrized divergences over a set of mean functionals defined axiomatically to satisfy certain properties. For the forward symmetrization, we prove that the arithmetic mean over the primal space is canonical for any mirror map over the positive definite cone. For the reverse symmetrization, we show that the canonical mean is the arithmetic mean over the dual space, pulled back to the primal space. Applying this result to three common mirror maps used in practice, we show that the canonical means for reverse symmetrization, in those cases, turn out to be the arithmetic, log-Euclidean and harmonic means. Our results improve understanding of existing symmetrization practices in the literature, and can be seen as a navigational chart to help decide which mean to use when.

Discrete choice models are fundamental tools in management science, economics, and marketing for understanding and predicting decision-making. Logit-based models are dominant in applied work, largely due to their convenient closed-form expressions for choice probabilities. However, they impose restrictive assumptions on the stochastic utility component, constraining our ability to capture realistic substitution patterns. We propose an amortized inference approach that relies on a neural network emulator to approximate choice probabilities for general error distributions, including those with correlated errors. We develop a specialized neural network architecture designed to respect the invariance properties of discrete choice models. We provide group-theoretic foundations for the architecture, including a proof of universal approximation given a minimal set of invariant features. Once trained, the emulator enables rapid likelihood evaluation and gradient computation. We use Sobolev training, augmenting the likelihood loss with a gradient-matching penalty, so that the emulator learns both choice probabilities and their derivatives. We show that emulator-based maximum likelihood estimators are consistent and asymptotically normal under mild approximation conditions, and we provide sandwich standard errors that remain valid for a psuedo-true parameter even with imperfect likelihood approximation. Simulations show significant gains over the GHK simulator in accuracy and speed.

Analog in-memory computing (AIMC) performs computation directly within resistive crossbar arrays, offering an energy-efficient platform to scale large vision and language models. However, non-ideal analog device properties make the training on AIMC devices challenging. In particular, its update asymmetry can induce a systematic drift of weight updates towards a device-specific symmetric point (SP), which typically does not align with the optimum of the training objective. To mitigate this bias, most existing works assume the SP is known and pre-calibrate it to zero before training by setting the reference point as the SP. Nevertheless, calibrating AIMC devices requires costly pulse updates, and residual calibration error can directly degrade training performance. In this work, we present the first theoretical characterization of the pulse complexity of SP calibration and the resulting estimation error. We further propose a dynamic SP estimation method that tracks the SP during model training, and establishes its convergence guarantees. In addition, we develop an enhanced variant based on chopping and filtering techniques from digital signal processing. Numerical experiments demonstrate both the efficiency and effectiveness of the proposed method.

Crystal structure prediction (CSP), which aims to predict the 3D atomic arrangement of a crystal from its composition, is central to materials discovery and mechanistic understanding. Crystal symmetry plays a crucial role in CSP, but given the composition in a unit cell, existing methods either struggle with the NP-hard combinatorial challenge of enforcing symmetry rigorously or rely on retrieving known templates, inherently limiting both physical fidelity and the discovery of genuinely new materials. To address this challenge, we introduce NextCrystal, a symmetry-driven generative framework that employs large language models to encode chemical semantics and directly generate fine-grained Wyckoff site patterns from atomic stoichiometry, eliminating reliance on database lookups. To overcome the combinatorial complexity of site assignments, we incorporate domain knowledge via an efficient, linear-complexity heuristic beam search, rigorously enforcing algebraic consistency between site multiplicities and atomic stoichiometry. By integrating this symmetry-consistent template into a diffusion backbone, the framework constrains the stochastic generative trajectory to a physically plausible geometric manifold. NextCrystal achieves state-of-the-art performance on stability, uniqueness, and novelty (SUN) benchmarks, as well as superior structural matching, establishing a rigorous paradigm for exploring previously unexplored crystallographic space without relying on prior structural templates. As a representative application, first-principles screening of HfO2 candidates generated by NextCrystal identifies a previously unreported dynamically stable Pnma phase, 0.056~eV/atom lower in energy than the conventional high-pressure Pnma phase.

The internal representations learned by language models consistently exhibit striking geometric structure: calendar months organize into a circle, historical years form a smooth one-dimensional manifold, and cities' latitudes and longitudes can be decoded using a linear probe. To explain this neural code, we first show that language statistics exhibit translation symmetry (for example, the frequency with which any two months co-occur in text depends only on the time interval between them). We prove that this symmetry governs these geometric structures in high-dimensional word embedding models, and we analytically derive the manifold geometry of word representations. These predictions empirically match large text embedding models and large language models. Moreover, the representational geometry persists at moderate embedding dimension even when the relevant statistics are perturbed (e.g., by removing all sentences in which two months co-occur). We prove that this robustness emerges naturally when the co-occurrence statistics are controlled by an underlying latent variable. Our results indicate that these representational manifolds originate in the statistical symmetries of natural language.

We introduce Clifford Kolmogorov-Arnold Network (ClKAN), a flexible and efficient architecture for function approximation in arbitrary Clifford Algebra spaces. We propose the use of Randomized Quasi-Monte Carlo grid generation as a solution to the exponential scaling associated with higher-dimensional algebras. Our ClKAN also introduces new batch normalization strategies to deal with variable domain input. ClKAN finds application in scientific discovery and engineering, and is validated in synthetic and physics-inspired tasks.

A growing number of neural architectures have been proposed to enforce geometric constraints, including projection-based networks, exponential-map updates, constrained output layers, and manifold neural ODEs. We provide a unified framework for these geometry-preserving architectures by organizing them according to where and how constraints are enforced, either throughout the intermediate layers or only at the final output. This perspective reveals several gaps in the existing theory. To address these gaps, we prove high-level approximation theorems for projected neural ODEs, intermediate augmented architectures, and final augmented architectures on prox-regular constraint sets, including smooth manifolds with boundary. Numerical experiments on synthetic dynamics over S^2, the disk, SO(3), together with real-world protein backbone data on SE(3), demonstrate exact feasibility for analytic updates and show that the final augmentation have simpler architecture and outperform in most tasks considered. When the constraint set is unknown, we learn projections via small-time heat-kernel limits, showing diffusion/flow-matching can be used as data-based projections. Moreover, we also the demonstrate the usefulness of the architectures that enforce non-convex constraints for path planning on manifolds with boundary.

Progressive Learning (PL) reduces pre-training computational overhead by gradually increasing model scale. While prior work has extensively explored depth expansion, width expansion remains significantly understudied, with the few existing methods limited to the early stages of training. However, expanding width during the mid-stage is essential for maximizing computational savings, yet it remains a formidable challenge due to severe training instabilities. Empirically, we show that naive initialization at this stage disrupts activation statistics, triggering loss spikes, while copy-based initialization introduces gradient symmetry that hinders feature diversity. To address these issues, we propose SPARKLING (balancing {S}ignal {P}reservation {A}nd symmet{R}y brea{K}ing for width-progressive {L}earn{ING}), a novel framework for mid-stage width expansion. Our method achieves signal preservation via RMS-scale consistency, stabilizing activation statistics during expansion. Symmetry breaking is ensured through asymmetric optimizer state reset and asymmetric learning rate re-warmup. Extensive experiments on dense and Mixture-of-Experts (MoE) models demonstrate that, across multiple width axes and optimizer families, SPARKLING consistently outperforms training from scratch and reduces training cost by up to 35% under $2\times$ width expansion.

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.

Low-resolution image representation can be regarded as a special form of sparse representation that retains only low-frequency information while discarding high-frequency components. This property reduces storage and transmission costs and benefits various image processing tasks. However, a key challenge is to preserve essential visual content while maintaining the ability to accurately reconstruct the original images. This work proposes LR2Flow, a nonlinear framework that learns low-resolution image representations by integrating wavelet tight frame blocks with normalizing flows. We conduct a reconstruction error analysis of the proposed network, which demonstrates the necessity of designing invertible neural networks in the wavelet tight frame domain. Experimental results on various tasks, including image rescaling, compression, and denoising, demonstrate the effectiveness of the learned representations and the robustness of the proposed framework. Code is available at https://github.com/Evanescentlove/LR2Flow-main.

Many learning problems are organized by group symmetries. While invariance is often imposed through architectures or group averaging, we ask when it can emerge from training on a finite random subset of an orbit. We study this question in classical Hopfield networks, where strict memorization can be expressed as a linear margin problem. Reparameterizing minimization of energy flow (MEF) as an exponential loss connects gradient descent to the corresponding minimum-norm hard-margin memorizer. Our main result shows that, for independent uniform samples from any finite permutation orbit, the exact sample hard-margin support vector machine (HSVM) concentrates exponentially around the invariant full-orbit HSVM. Consequently, an orbit-size-independent polynomial number of samples suffices both for approximate parameter invariance and for simultaneous memorization of every orbit element; directional convergence transfers this conclusion asymptotically to MEF gradient descent. For graph-isomorphism orbits, we characterize the invariant parameters as a three-dimensional subspace and show that every such orbit is memorizable. For cliques of fixed linear density, additional symmetry sharpens the uniform memorization bound to $O(v^4\log(1/δ))$, exponentially smaller than the orbit size. Together with experiments across several learning rules, these results give a finite-sample account of how optimization bias can recover symmetry from partial group-structured data.

Learned image reconstruction has become a pillar in computational imaging and inverse problems. Among the most successful approaches are learned iterative networks, which are formulated by unrolling classical iterative optimisation algorithms for solving variational problems. While the underlying algorithm is usually formulated in the functional analytic setting, learned approaches are often viewed as purely discrete. In this survey we present a unified operator view for learned iterative networks. Specifically, we formulate a learned reconstruction operator, defining how to compute, and separately the learning problem, which defines what to compute. In this setting we present common approaches and show that many approaches are closely related in their core. We review linear as well as non-linear inverse problems in this framework and present a short numerical study to conclude.

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.

Deep neural networks have been widely used as universal approximators for functions with inherent physical structures, including permutation symmetry. In this paper, we construct symmetric deep neural networks to approximate symmetric Korobov functions and prove that both the convergence rate and the constant prefactor scale at most polynomially with respect to the ambient dimension. This represents a substantial improvement over prior approximation guarantees that suffer from the curse of dimensionality. Building on these approximation bounds, we further derive a generalization-error rate for learning symmetric Korobov functions whose leading factors likewise avoid the curse of dimensionality.

Beside the standard stochastic gradient descent (SGD) method, the Adam optimizer due to Kingma & Ba (2014) is currently probably the best-known optimization method for the training of deep neural networks in artificial intelligence (AI) systems. Despite the popularity and the success of Adam it remains an \emph{open research problem} to provide a rigorous convergence analysis for Adam even for the class of strongly convex SOPs. In one of the main results of this work we establish convergence rates for Adam in terms of the number of gradient steps (convergence rate \nicefrac{1}{2} w.r.t. the size of the learning rate), the size of the mini-batches (convergence rate 1 w.r.t. the size of the mini-batches), and the size of the second moment parameter of Adam (convergence rate 1 w.r.t. the distance of the second moment parameter to 1) for the class of strongly convex SOPs. In a further main result of this work, which we refer to as \emph{Adam symmetry theorem}, we illustrate the optimality of the established convergence rates by proving for a special class of simple quadratic strongly convex SOPs that Adam converges as the number of gradient steps increases to infinity to the solution of the SOP (the unique minimizer of the strongly convex objective function) if and \emph{only} if the random variables in the SOP (the data in the SOP) are \emph{symmetrically distributed}. In particular, in the standard case where the random variables in the SOP are not symmetrically distributed we \emph{disprove} that Adam converges to the minimizer of the SOP as the number of Adam steps increases to infinity. We also complement the conclusions of our convergence analysis and the Adam symmetry theorem by several numerical simulations that indicate the sharpness of the established convergence rates and that illustrate the practical appearance of the phenomena revealed in the \emph{Adam symmetry theorem}.

Initial-boundary value problems (IBVPs) provide the essential framework for modelling a wide range of phenomena in physics and engineering. We introduce a novel method for efficiently solving IBVPs using Lie symmetries to enforce the associated partial differential equation (PDE) exactly by construction. By leveraging symmetry transformations, our model embeds the underlying physical laws and learns the solution solely from initial and boundary data. Consequently, the boundary loss directly quantifies domain-wide error, enabling rigorous error estimation for well-posed IBVPs. We implement LieSolver and demonstrate its application to linear homogeneous PDEs, showing that it outperforms physics-informed neural networks (PINNs) in both speed and accuracy while yielding compact models. Overall, our approach significantly enhances the efficiency and reliability of predictions for PDE-constrained problems.

Normalizing flows are deep generative models that enable efficient likelihood estimation and sampling through invertible transformations. A key challenge is to design linear layers that enhance expressiveness while maintaining efficient computation of the Jacobian determinant and inverse. We introduce a novel invertible linear layer based on the product of circulant and diagonal matrices. This decomposition reduces parameter complexity from $\mathcal{O}(n^2)$ to $\mathcal{O}(mn)$ using $m$ diagonal matrices and $m-1$ circulant matrices while still approximating general linear transformations. By leveraging the Fast Fourier Transform, our approach reduces the time complexity of matrix inversion from $\mathcal{O}(n^3)$ to $\mathcal{O}(mn\log n)$ and that of computing the log-determinant from $\mathcal{O}(n^3)$ to $\mathcal{O}(mn)$, where $n$ is the input dimension. We build upon this layer to develop Circulant-Diagonal Flow (CDFlow), which achieves strong density estimation on natural image datasets and effectively models data with inherent periodic structure. Furthermore, CDFlow significantly accelerates key operations in normalizing flows, providing practical benefits for scalable generative modeling.

We introduce Conditional Clifford-Steerable CNNs (C-CSCNNs), a unified framework that incorporates equivariance to arbitrary pseudo-Euclidean groups and significantly improves the expressivity of standard CSCNNs. We show that the kernel basis of the standard formulation is incomplete, limiting model capacity. To address this, we augment the kernels with equivariant representations of the input feature field. We derive the equivariance constraint for these input-dependent kernels and show how it can be solved efficiently via implicit parameterization. We empirically validate on multiple PDE forecasting tasks, including fluid dynamics and relativistic electrodynamics, where our method consistently outperforms standard CSCNNs and performs on par with state-of-the-art baselines.

Flow matching models generate high-fidelity molecular geometries but incur significant computational costs during inference, requiring hundreds of network evaluations. This inference overhead becomes the primary bottleneck when such models are employed in practice to sample large numbers of molecular candidates. This work discusses a training-free caching strategy that accelerates molecular geometry generation by predicting intermediate hidden states across solver steps. The proposed method operates directly on the SE(3)-equivariant backbone, is compatible with pretrained models, and is orthogonal to existing training-based accelerations and system-level optimizations. Experiments on the GEOM-Drugs dataset demonstrate that caching achieves a twofold reduction in wall-clock inference time at matched sample quality and a speedup of up to 3x compared to the base model with minimal sample quality degradation. Because these gains compound with other optimizations, applying caching alongside other general, lossless optimizations yield as much as a 7x speedup.

Nanoporous materials hold promise for diverse sustainable applications, yet their vast chemical space poses challenges for efficient design. Machine learning offers a compelling pathway to accelerate the exploration, but existing models lack either interpretability or fidelity for elucidating the correlation between crystal geometry and property. Here, we report a three-dimensional periodic space sampling method that decomposes large nanoporous structures into local geometrical sites for combined property prediction and site-wise contribution quantification. Trained with a constructed database and retrieved datasets, our model achieves state-of-the-art accuracy and data efficiency for property prediction on gas storage, separation, and electrical conduction. Meanwhile, this approach enables the interpretation of the prediction and allows for accurate identification of significant local sites for targeted properties. Through identifying transferable high-performance sites across diverse nanoporous frameworks, our model paves the way for interpretable, symmetry-aware nanoporous materials design, which is extensible to other materials, like molecular crystals and beyond.

While transformers excel in many settings, their application in the field of automated planning is limited. Prior work like PlanGPT, a state-of-the-art decoder-only transformer, struggles with extrapolation from easy to hard planning problems. This in turn stems from problem symmetries: planning tasks can be represented with arbitrary variable names that carry no meaning beyond being identifiers. This causes a combinatorial explosion of equivalent representations that pure transformers cannot efficiently learn from. We propose a novel contrastive learning objective to make transformers symmetry-aware and thereby compensate for their lack of inductive bias. Combining this with architectural improvements, we show that transformers can be efficiently trained for either plan-generation or heuristic-prediction. Our results across multiple planning domains demonstrate that our symmetry-aware training effectively and efficiently addresses the limitations of PlanGPT.

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.

Natural images exhibit strong geometric regularities: local structures, such as edges, corners, and textures, appear in many orientations and mirror configurations. Since Vision Transformers (ViTs) operate on square image patches, these transformations naturally correspond to the dihedral symmetry group $\mathrm{D}_8$, also known as the octic group. Recent work has shown that ViTs can be made reflection equivariant and more efficient than standard ViTs simultaneously by implementing the linear layers in the Fourier domain of the reflection group. In this work, we extend the equivariance to reflections and rotations and analyze the scalability of the resulting networks. Our Quick ViTs, based on octic equivariant linear layers, achieve 5.33x reductions in FLOPs and up to 8x reductions in memory compared to ordinary linear layers. By analyzing the arithmetic intensity of these layers, we identify theoretical limits on how much the FLOP savings translate into throughput improvements on modern GPUs. However, these limitations disappear as the embedding dimensions increase. Enabled by their computational efficiency, we conduct a broader empirical evaluation of equivariant ViTs than in previous work. Upon training supervised (DeiT-III) and self-supervised (DINOv2) on ImageNet-1K, we find that our Quick ViTs match or exceed baseline accuracy while at the same time providing substantial efficiency gains.

Equivariant network architectures are a well-established tool for predicting invariant or equivariant quantities. However, almost all learning problems considered in this context feature a global symmetry, i.e. each point of the underlying space is transformed with the same group element, as opposed to a local ``gauge'' symmetry, where each point is transformed with a different group element, exponentially enlarging the size of the symmetry group. Gauge equivariant networks have so far mainly been applied to problems in quantum chromodynamics. Here, we introduce a novel application domain for gauge-equivariant networks in the theory of topological condensed matter physics. We use gauge equivariant networks to predict topological invariants (Chern numbers) of multiband topological insulators. The gauge symmetry of the network guarantees that the predicted quantity is a topological invariant. We introduce a novel gauge equivariant normalization layer to stabilize the training and prove a universal approximation theorem for our setup. We train on samples with trivial Chern number only but show that our models generalize to samples with non-trivial Chern number. We provide various ablations of our setup. Our code is available at https://github.com/sitronsea/GENet/tree/main.

We investigate the structure of Kazhdan-Lusztig polynomials of the symmetric group by leveraging computational approaches from big data, including exploratory and topological data analysis, applied to the polynomials for symmetric groups of up to 11 strands.

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.

Detecting partial extrinsic symmetry in 3D geometry is a fundamental yet persistent challenge in computer vision and graphics, critical for tasks ranging from shape completion to procedural generation. Classical transformation-space voting methods rely on pairwise matching, scaling as O(n^2) and struggling to resolve coherent multi-instance groups. Recent learning approaches advance global symmetry detection but restrict the solution space to reflection planes, failing to capture rotational or translational repetitions such as the legs of a chair or the steps of a staircase. We propose SymCL, a self-supervised contrastive learning framework that detects partial symmetries across rotation, translation, and reflection (with scale-invariant features) and requires no ground truth annotations. By mapping local geodesic patches to a latent space invariant to the Euclidean group, we reformulate symmetry detection as a density-based clustering problem, enabling the simultaneous discovery of multi-instance symmetric relationships in a single forward pass. We evaluate quantitatively on SymPartNet, a new benchmark annotating all PartNet categories with partial symmetry relations, and demonstrate class-agnostic generalization qualitatively on everyday objects outside the training distribution.

One of the primary reasons behind the success of neural networks has been the emergence of an array of new, highly-successful optimizers, perhaps most importantly the Adam optimizer. It is widely used for training neural networks, yet notoriously hard to interpret. Lacking a clear physical intuition, Adam is difficult to generalize to manifolds. Some attempts have been made to directly apply parts of the Adam algorithm to manifolds or to find an underlying structure, but a full generalization has remained elusive. In this work a new approach is presented that leverages the special structure of the manifolds which are relevant for optimization of neural networks, such as the Stiefel manifold, the symplectic Stiefel manifold and the Grassmann manifold: all of these are homogeneous spaces and as such admit a global tangent space representation. This is a common vector space, often called the Lie subspace, that makes the generalization of all steps in the Adam optimizer (as well as other optimizers) possible. It is thus possible to extend the Adam optimizer to manifolds without a projection step, something that was not possible before. The resulting algorithm is then applied to train transformers and a symplectic autoencoder for which orthogonality constraints are enforced up to machine precision and we conclusively demonstrate the advantage of the proposed optimizer over existing methods.

Training convolutional neural networks at scale demands substantial memory, largely because intermediate activations must be stored for backpropagation. Existing remedies (checkpointing, invertible architectures, or gradient-approximation methods such as randomized automatic differentiation) either add significant computation, impose architectural constraints, or require non-trivial code changes. We propose XConv, a near-drop-in replacement for standard 2D and 3D convolutional layers that addresses all three: it preserves standard backpropagation, imposes no architectural constraints, and integrates into existing codebases with minimal changes. XConv exploits the algebraic structure of convolutional weight gradients, storing highly compressed projections of the activations rather than the full tensors and approximating the gradients via multi-channel randomized trace estimation. The number of probing vectors sets a memory-accuracy tradeoff and recovers the exact gradient in the limit. We establish convergence guarantees and error bounds for the estimator, showing that its gradient-error variance is comparable to that of stochastic gradient descent. Empirically, XConv matches exact-gradient methods across classification, generative modeling, super-resolution, inpainting, and segmentation, with gaps that narrow as the number of probing vectors grows, while reducing activation memory by a factor of two or more when convolutional activations dominate, and remaining computationally competitive with optimized convolution kernels at larger batch sizes. At half precision the gradient-approximation error falls to the rounding floor, so XConv adds essentially no error beyond that of low-precision arithmetic. The savings matter most where activation memory rather than compute is the binding constraint, such as high-resolution and volumetric training and on-device finetuning.

Many sequential construction tasks have exact terminal symmetries even though execution is directed and depends on history. Process evidence supplies order; terminal correspondence transports it between equivalent outcomes; the realized state updates relevance. These roles define a carrier framework: transport what the outcome preserves; refine what history changes. SymBuild combines transported process and state residual ranks by ordinal rank meet; its top-$k$ prefix exactly equals their top-$k$ union, yielding a tight worst-case verifier query bound under prefix information. We evaluate SymBuild in three construction domains: computer-aided design (CAD) assembly, Mini-Programs, and exact-fill packing, and test additional framework instantiations in all four domains. SymBuild improves the area under the anytime verified success curve by up to 6.77, 21.75, and 8.68 points over Static in the three construction domains. Refresh gains recur beyond SymBuild under alternative aggregation, planning, and learned scoring methods; on Geometric Reasoning Network (GRN) target removal, direct Combined refresh has the lowest mean verifier query score at all three scales and reduces learned state evaluations by factors of 6.48-12.20 relative to refreshed population-based search. Together, these results support the carrier framework and demonstrate that SymBuild is an effective, analyzable method for anytime verified construction.