Symmetry

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

Finite groups are rigid algebraic objects, whose Cayley graphs expose a rich network geometry through which group-theoretic structure can be measured, compared, and learned. In this paper, a dataset of $131{,}406$ Cayley graphs is constructed, covering all groups of order at most $767$ except order $512$, recording exact algebraic labels for group properties together with a broad collection of graph, cycle, distance, and spectral statistics. This census aims to provide novel benchmarks for studying how finite-group properties are reflected in Cayley graph observables. It also yields new enumerative contributions: alongside recovering known OEIS sequences for standard group classes, new sequences for monolithic groups and for groups generated by at most three, four, and five elements are contributed to the OEIS. The accompanying network analysis identifies several empirical regularities and formulates testable conjectures, including relationships involving square clustering, Cayley graph diameter, average graph disorder, and spectral eigengaps of nilpotent groups. Finally, a comparison between classical models, an MLP, and graph neural network architectures is performed for predicting algebraic group properties directly from Cayley graph data. The results show that engineered graph statistics are highly informative, while GNNs, especially GIN and in some fixed-order settings GCN, can recover substantial structural signal directly from the graph. Such that graph-aware architectures show phases of optimality on these group-theoretic graph representations.

Representing manipulation actions as 2D trajectories in the camera plane provides a compact and interpretable basis for learning complex 3D manipulation policies. However, it also creates challenges from out-of-frame trajectories and limited precision. We propose Pix2Act, an imitation learning method that addresses these challenges by generating continuous image-space keypoint trajectories in each camera plane and losslessly recovering end-effector poses via triangulation. This reformulates high-dimensional 3D control as a simpler, more learnable 2D prediction problem. Crucially, it aligns observations and actions in the same coordinate space, enabling equivariant transformations to jointly rotate individual camera images together with their image-space actions. We analyze the symmetry properties of this augmentation and design a network architecture that can fuse multiple camera views while respecting their per-view rotations. As a result, Pix2Act implicitly enlarges the support of the data distribution and learns invariant action structures across transformations, yielding improved generalization and overall performance. Across diverse simulated and real-world manipulation tasks, Pix2Act outperforms state-of-the-art baselines and remains robust under camera perturbations.

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

Spectral embedding methods are widely used for dimensionality reduction and clustering of high-dimensional datasets with intrinsic low-dimensional structures. Although many datasets of practical interest exhibit invariance under symmetries such as rotations, standard spectral embedding methods do not account for this, treating symmetry-related data points as unrelated. Our approach to this problem is to incorporate the symmetries directly into the affinity kernels used for spectral embedding. We analyze the case of a Riemannian data manifold $M$ with symmetries given by a compact Lie group~$G$ and prove that, under suitable conditions, graph Laplacians constructed from three types of invariant kernels converge pointwise to explicit second-order differential operators on the quotient space $M/G$. Our analysis implies improved convergence rates, as the effective dimension drops according to the dimension of the group. We validate our approach on datasets with $\mathrm{SO}(2)$ or $\mathrm{SO}(3)$ symmetry, and show that $G$-invariant spectral embedding recovers the intrinsic geometry of the data, in contrast to standard spectral embedding, which fails to do so even in the limit of infinite data.

Quantifying how mirror-symmetric an image is about a given axis (symmetry scoring) underpins applications from visual aesthetics to medical imaging, yet proposed scoring methods have never been compared on a common, statistically grounded protocol. We benchmark 13 scoring methods (nine collected from literature, four introduced here) spanning from classical features to frozen deep features, across four single-axis and five multi-axis datasets under a reflection-exact protocol with a chance-anchored, significance-tested discrimination skill. Deep backbones perform best on single-axis and harder multi-axis protocols. However, a classical histogram-of-oriented-gradients (HOG) descriptor trails the best frozen-network readout by a small (but significant) margin, is not statistically separable from the runner-up (a CNN-filter measure), and runs ~300x faster on CPU. Our results show that discrimination concentrates in mid-scale oriented features, where deep backbones peak at a low or mid stage, and HOG peaks at a mid cell size. Among existing methods, frozen deep features thus offer little over a tuned classical descriptor for measuring symmetry; whether task-trained deep scorers can do better remains open. We release the scorers and harness in imgsym, an open toolkit for image symmetry detection and measurement.

The Hessian of the training loss governs the local geometry of the loss landscape, yet despite existing explanations for its largest eigenvalues, the origin of the vast multitude of vanishingly small eigenvalues remains elusive. We argue that the bulk consists of the weakly lifted pseudo-Goldstone modes of the continuous symmetries of the network parametrization. In deep linear networks these symmetries are exact: they generate flat directions and hence exact zero modes, whose eigenvectors we construct explicitly. Introducing a ReLU nonlinearity as a perturbation, we show that it breaks these symmetries weakly and explicitly. Resolving the spectrum at the level of eigenvectors, we find that the high-curvature directions are orthogonal to the symmetry subspace, while the bulk lies almost entirely within it. We demonstrate the mechanism in a two-layer ReLU student--teacher model and in a network trained on CIFAR-10. A convolutional example demonstrates that the same diagnostic extends beyond fully connected layers. Together, these results link the Hessian bulk to weakly broken symmetries and clarify the origin of near-zero modes.

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

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.

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

While various works address reflective symmetry understanding in 3D data and images, pixel-level semantic left-right prediction of in-the-wild images remains challenging, due to certain difficulties including the lack of 3D information, occlusion, object pose variation, partiality, etc. In this work, we propose an unsupervised learning framework to tackle this challenge. Leveraging recent advances in vertex-wise semantic left-right understanding of 3D data, our unsupervised learning method jointly utilises 3D shape and image datasets to infer pixel-wise semantic left-right predictions in single-view images. In particular, we show that a medium-scale 3D shape dataset comprising mainly of human- and quadruped animal-like shapes, combined with diverse in-the-wild image data, are sufficient to achieve high-quality semantic left-right prediction in images, even for entirely unseen 3D object categories, such as cars or trains. Overall, our approach achieves superior performance in dense pixel-wise semantic left-right predictions on both rendered and in-the-wild image datasets when compared to existing state-of-the-art methods.

Lensless imaging enables compact and versatile computational cameras by replacing bulky optics with thin coded elements. However, reconstruction from the resulting measurements is challenging: large-footprint point-spread functions (PSFs) produce highly multiplexed observations, making inversion severely ill-conditioned and sensitive to calibration errors and model mismatch. While deep learning approaches, including hybrid models that incorporate physics priors, have shown promise, explicitly maintaining data fidelity throughout the network hierarchy remains difficult. Here, we propose the Integrated Forward-Inverse Network (IFIN), a physics-guided architecture that interleaves differentiable forward projections with learnable inverse updates at every scale, enabling complementary cues to be exploited jointly in the measurement and image domains. This bidirectional coupling supports progressive, physics-consistent refinement and permits system-constrained PSF kernel adaptation under model uncertainty. On challenging lensless benchmarks, including a newly introduced dataset, IFIN achieves state-of-the-art reconstruction quality. We further observe competitive performance on Gaussian deblurring and simulated inline holography reconstruction, suggesting that the same interleaving principle can extend beyond lensless cameras.

For compact convex sets $L,K \subset \mathbb{R}^n$, denote by $λ_K(L)$ the smallest size of a homothet of $K$ that contains $L$. We define a measure of symmetry based on the $n$-simplex $Δ= Δ^n \subset \mathbb{R}^n$ as the ratio $$ρ_Δ(L):=\frac{λ_{-Δ}(L)}{λ_Δ(L)}.$$ We study this measure and deduce the following results: (1) The classical Minkowski measure of symmetry $m^*(L)$ can be defined as an affine-invariant version of $ρ_Δ(L)$. (2) We improve the stability analysis for the Minkowski measure of symmetry; if $m^*(L)\ge n-\varepsilon$ then $L$ is $\tfrac{1}{1-\varepsilon}$-close to $Δ$ in the Banach--Mazur distance. (3) We obtain a novel characterization of simplices as the only convex bodies $K$ for which the function $L \mapsto λ_K(L)$ is additive (a property we term ``outer additivity''). (4) Motivated by the expressivity of ReLU neural networks, we study the depth complexity of polytopes in $\mathbb{R}^n$ under the two operations: Minkowski sum and convex hull of a union. We prove the sharp bound $ρ_Δ(P) \leq 2^d -1$ for every polytope $P$ of depth complexity $d$. In other words, simplices cannot be approximated by low-depth polytopes.

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).

In many multisensor systems, measurements from different sensors are subject to unknown relative time delays. Accurate state estimation requires that delays be accounted for and, when possible, calibrated online. We consider the case of aided navigation, where measurements from a single aiding sensor are subject to an unknown but constant delay relative to the inertial measurement stream, and study the identifiability of the resulting system. Critically, identifiability depends not only on the temporal structure of the measurements, but also on the shape of the vehicle trajectory: some trajectories are sufficiently informative to support unique recovery of the delay and the navigation state, while others are not. Using the special Galilean group, we characterize these uninformative (or degenerate) trajectories and relate them to a continuous symmetry of the delayed measurement model, providing geometric insight into identifiability failures. We show that the class of trajectories for which identifiability fails is larger than previously reported, and connect our characterization to the familiar linearized, Jacobian-based analysis. Although our development is motivated by aided navigation, the underlying ideas apply more broadly to estimation problems on Lie groups with delayed measurements.

Post-hoc analysis of trained neural network weights often seeks to recover geometric structure directly from the parameters. We show that, for positional-encoding-equipped neural fields, the symmetry visible from weights is not the true symmetry group itself, but an observable symmetry set determined by the trained parameters, the positional encoding (PE), and readout observable. We formulate this dependence through an exact observability hierarchy, $G_{\mathrm{obs}}^{\mathrm{exact}} \subseteq G_{\mathrm{lift}}^{\mathrm{exact}}(φ) \cap G_{\mathrm{true}}$, where $G_{\mathrm{lift}}^{\mathrm{exact}}(φ)$ is the set of input transformations that the PE can exactly lift to the feature space. The hierarchy implies that even when a target function has a geometric symmetry, that symmetry may be structurally invisible to weight-level observables if the PE does not represent the corresponding transformation. We test this prediction using MLPs trained on two-dimensional signed distance functions with multiple shape symmetry groups, positional encodings, and Gram-based observables. The results show a consistent PE-dependent pattern: DyadicAxisPE supports $D_4$-sensitive readout but structurally suppresses $D_3$ rotations, TriAxisPE yields lower $D_3$ / $D_6$ readout scores under the tested Gram observables by replacing coordinate axes with three 120-degree-separated axes, and random Fourier features mainly exhibit a $π$-rotation response under these readouts. These findings show that PE design affects not only approximation behavior but also which structures are accessible to post-hoc weight-level readouts. This provides a basis for a principled observable-dependent symmetry readout.

$\mathrm{E}(3)$-equivariant networks are promising for 3D atomistic system modeling, yet their scalability is limited by the $O(L^6)$ complexity of the Clebsch-Gordan Tensor Product (CGTP). The recently proposed Gaunt Tensor Product (GTP) reduces the complexity but is unable to capture the antisymmetric paths, resulting in incomplete expressivity. In this work, we present SpinGTP, an approach to overcome the GTP incompleteness by generalizing from scalar functions to Spin-Weighted Spherical Harmonics (SWSH). By relying on the algebraic properties of SWSH, SpinGTP recovers the missing antisymmetric interactions while maintaining the asymptotic efficiency of GTP. It also allows for a more expressive equivariant basis that naturally accounts for the parity-odd components of tensor products. We evaluate SpinGTP across diverse benchmarks, including Tetris, 3BPA, SPICE-MACE-OFF, and OC20. Our results show that SpinGTP achieves accuracies comparable to full CGTP. Notably, by explicitly capturing antisymmetric paths, SpinGTP exhibits superior performance in tasks involving chiral materials and non-centrosymmetric geometries. This work provides a complete, scalable, and mathematically rigorous path toward high-order equivariance in large-scale 3D atomistic system simulations.

We present a simple, yet general approach to study the scaling properties as the dimensionality of Metropolised MCMC sampling algorithms increases. The study relies on the symmetries of the Hamiltonian formalism and ultimately on the symmetry of the Metropolis-Hastings formula. Our findings contain, as particular cases, many known results for the Random Walk Metropolis, MALA and other algorithms. In addition, they provide, in an easy way, new optimal scaling results for a variety of proposal mechanisms, including implicit proposals and proposals generated with the help of differential equation integrators. The analysis applies to targets that are products of a given, not necessarily univariate distribution, and also to cases where the different terms in the product are scaled differently. We show how to construct gradient-based MALA-like proposals where the variance of the proposal as the dimension $d$ increases may be taken as $O(1/d^μ)$, with $μ>0$ arbitrarily small, to be compared with the values $μ= 1$ for Random Walk Metropolis and $μ=1/3$ for MALA.

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

Islamic geometric patterns are governed by exact rotational symmetry and strict construction rules. This paper treats these rules as formal geometric knowledge and embeds them in a neural completion framework, rather than leaving them to be learned statistically from data. Given sparse control geometry and a target symmetry order, the system completes the pattern as a vector graph by predicting edges and refinements of bounded curves over a candidate lattice whose edges are organised into rotational orbits under the cyclic group. Symmetry is enforced either by constraining predictions within these orbits or by projecting them onto them during inference. The orbit-tied variant provides a constructive guarantee: for any input and any orbit-level selection rule, it produces exact N-fold symmetry, preserves anchor points, and keeps all refinements within prescribed bounds. These properties are verified numerically. The study focuses on rotational symmetry, and all quantitative results are obtained from procedurally generated graphs inspired by Islamic geometric design rather than from a historical corpus. On clean inputs, enforcing exact validity produces no measurable loss in fidelity. When control geometry is missing, an unstructured decoder loses fidelity and breaks symmetry; retraining on corrupted inputs recovers much of the fidelity but not exact validity. Symmetry-structured inference, by contrast, keeps violations at zero throughout. The results show that augmentation and symmetry structure address distinct failure modes: augmentation improves fidelity under corruption, while symmetry structure guarantees validity. The framework therefore provides a knowledge-constrained, guarantee-backed approach to neural completion for scalable vector ornaments whose validity depends on exact geometric structure.

We report a machine-verified resolution of a problem open for over a decade in quantum optimization: the Farhi, Goldstone and Gutmann (FGG) conjecture that depth-$p$ Quantum Approximate Optimization Algorithm (QAOA) on the ring of disagrees attains approximation ratio $(2p+1)/(2p+2)$ exactly. We found the proof using a large language model, Claude Fable 5, and verified its correctness end-to-end by the Lean 4 proof assistant. Our methodology includes several ingredients: building on a substantial Lean library of quantum information, we formalized the QAOA components and the known parts of the problem, and reduced the conjecture to a single open mathematical statement. The model was then handed the library and our agentic toolkit, and tasked with closing that gap by constructing a proof in Lean. The resulting process is a feedback loop between the model's natural-language reasoning and Lean's mechanical verification, which converged to a machine-verified proof. Human verification is required only for the structural scaffolding - that the formal statement faithfully encodes the intended claim - while the proof itself is supplied by the model and certified mechanically by Lean. The proof is nevertheless striking - the model uncovered a hidden dynamical symmetry of the problem and exploited it, borrowing tools and machinery from an adjacent field to turn a hard existence problem into an explicit construction. This work paves the way for resolving open conjectures in quantum information science and beyond.

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

Modern machine learning models are trained by optimizing high-dimensional non-convex empirical risk functions. Such cost functions can have a multitude of local optima and yet, gradient-based optimization appears to converge to near-global optima. Within a simple supervised learning setting, we develop a precise picture of which parts of the empirical risk landscape are accessible by polynomial-time algorithms. We are given i.i.d. pairs $\{(\boldsymbol{x}_i,y_i):\; 1 \le i\le n\}$ with $\boldsymbol{x}_i\in \mathbb{R}^d$ standard Gaussian feature vectors, and $y_i\in\mathbb{R}$ response variables that depend on $\boldsymbol{x}_i$ through their projections on an unknown $k$-dimensional subspace. We use empirical risk minimization to learn a model that depends on an $m$-dimensional projection of the data (e.g., an $m$-neurons neural network). We propose an incremental approximate message passing (IAMP) algorithm and precisely characterize the training error it achieves, as well as the relation between test and training error, in the high dimensional asymptotics $n,d\to\infty$, with $n/d\toα\in (0, +\infty)$. Based on earlier work in related models, we expect that the performance achieved by our algorithm is optimal among polynomial-time algorithms.

Ab initio modeling has established Wigner's SU(4) and Elliott's SU(3) as dominant symmetries of the nuclear force in light and intermediate-mass nuclei. We ask whether they also govern nuclear binding across the entire chart. Our aim is not high-precision prediction but physical insight, through interpretable, symmetry-based models. From the SU(3) and SU(4) Casimir operators we construct three neural-network (NN) mass models: Feature-Informed NN (FINN) for point predictions, Gaussian-Informed NN (GINN) adding uncertainty quantification, and Wigner-Informed NN (WINN) -- a mass formula using the Casimirs as an operator basis. All are trained on AME2016 and validated on nuclei new to AME2020. The SU(4) operators alone cut the root-mean-square error (RMSE) by nearly half on train and test data, and by about a fifth on extrapolation, relative to the liquid-drop baseline -- showing that Wigner's symmetry carries predictive information beyond bulk properties. Despite its compact form, WINN reaches the lowest validation RMSE, 0.430 MeV -- competitive with state-of-the-art mass models -- which we read less as a benchmark than as evidence that its symmetry basis captures important physics. WINN further reveals i) an enhancement of the quadratic SU(4) Casimir near the neutron dripline, signaling restoration of Wigner's symmetry, and ii) an unexpected gain of the quartic operator in the superheavy region. We thereby elevate emergent symmetries from the hidden order within individual nuclei to a governing principle of the whole nuclear chart.

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 $\mathrm{O}(2)$, providing a unified framework that generalizes prior flipping- and $D_4$-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 $H \le G$, the class of $G$-equivariant ViTs embeds naturally into the class of $H$-equivariant ViTs. We also prove that, in the single-head setting, the corresponding equivariant self-attention layer realizes every $G$-equivariant self-attention map representable by ordinary self-attention. We further construct a $D_6$-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 $D_4$ and $D_6$. 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.

We propose a complete architectural blueprint for safe artificial general intelligence based on a closed reentry loop (D <-> I cycle). In contrast to feedforward networks, which are directed acyclic graphs (C=0, S=0) incapable of self-reference, the proposed architecture contains a structural cycle (C >= 1) with self-sustaining amplification (rho > 1), mathematically guaranteeing the emergence of a self-model, instrumental self-preservation, and unprogrammed goal-directed behaviour. The agent's goals are encoded as a non-textual D-vector in the architecture itself, making them immune to reinterpretation and prompt injection. We present the S-measure -- a polynomial-time [O(N^3)] computable alternative to Tononi's NP-hard Phi -- with machine-verified Lean 4 proof that S>0 implies positive integrated information. The work provides full Python/NumPy implementations (Tarjan-based cycle complexity, Delta-S barrier), industrial horizontal scaling via Apache Kafka and Docker Compose, a taxonomy of six epochs of AI evolution, a zoo of future reentry architectures (RAS, diffusion attractors, fractal loops), gauge-invariant networks for safe swarms, fault-tolerance and recovery protocols, and eight falsifiable predictions. All formal proofs are machine-verified in Lean 4. This architecture is deployable today and represents a topologically protected, safe-by-design approach to AGI.

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

Image hiding aims to conceal image-level messages within cover images at the same resolution. Invertible neural networks (INN)-based image hiding has emerged as an important branch. It treats concealing and revealing as a pair of inverse problems on image domain transformation and uses INN's forward and backward processes to address them. Due to architectural constraints, existing INN-based methods suffer from single-scale and single-domain feature extraction and limited nonlinear representation capability, resulting in inferior image quality. To mitigate these limitations, we propose an efficient cross-scale invertible hiding network with the spatial-frequency collaboration and the non-invertible mechanism, termed CrosInv. CrosInv exploits cross-scale and spatial-frequency collaborative features while enhancing nonlinear representation. Specifically, we introduce a cross-scale invertible module that bijectively maps inputs to cross-scale representations. To effectively integrate spatial and frequency information, the cross-scale invertible module employs pixel shuffle, Haar wavelet transformation, and their inverse operations for scale transformation. Furthermore, a non-invertible cross dense module is integrated to enhance the nonlinearity. Comprehensive experiments verify the effectiveness and superiority of the proposed CrosInv.

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

Learned world models are useful only over horizons on which their rollout error remains controlled. We study trust-horizon certification for latent world models with known group symmetries. Given a one-step latent residual and a finite-time expansion estimate, we form a raw horizon curve and calibrate it with a split-conformal multiplicative factor. On the reproducible audit set, the conformal factor is $γ_α=1.0$: the raw certificate is already conservative under the audit protocol. Across 50 stable audits, we observe zero anti-conservative violations, corresponding to an exact-binomial 95% upper bound of 5.8% on the violation rate. Our main structural result is that exact equivariance transports a calibrated trust-horizon curve over the group orbit: when the environment dynamics, encoder, predictor, action transform, and latent metric satisfy the stated equivariance/invariance conditions, rollout errors and trust horizons are orbit-constant. Empirically, the implemented models exhibit small orbit-transport residuals, with median 1.1% and maximum 4.1% over 14 orbit audits. The certificate is also non-vacuous (median certified-to-measured horizon ratio 0.67). A certificate-level calibration-cost study shows two complementary regimes. On a symmetric 2D substrate, equivariant, plain, and augmented models are all orbit-valid from a single calibration sector -- no separation, because the substrate already makes non-equivariant baselines approximately orbit-robust. A 3D yaw audit shows the other regime: the equivariant model obtains a one-sector safe and non-vacuous orbit-valid certificate, while healthy non-equivariant baselines pay violation, slack, sharpness, or additional-sector cost. The certificate is a conservative, distributional audit rather than a global reachability guarantee, and certificate-guided subgoal spacing is not confirmed in the current 3D CEM-MPC behavior layer.

Quantum machine learning is often motivated by the exponentially large state space of quantum systems, but this promise leaves a basic generalization problem unresolved: how can a learner assign different meanings to unseen quantum directions when the training data provide no preferred basis, measurement frame, or other orienting structure? We address this identifiability problem by formulating supervised learning without an external quantum reference frame, so that predictions cannot depend on an arbitrary choice of Hilbert-space coordinates. This requirement forces the learned classifier to preserve every unitary symmetry left unbroken by the training data. We prove that whenever the training states fail to span the full Hilbert space, all pure states orthogonal to their span must receive the same prediction -- even when those states are mutually orthogonal and perfectly distinguishable once an appropriate measurement is supplied. The limitation is therefore not caused by state discrimination, optimization, or computational power, but by missing reference information. We further establish a robust version under weak symmetry breaking and show that learning generic unstructured concepts on multiqubit systems requires exponentially many independently oriented training directions. Numerical illustrations visualize the resulting prediction collapse and its controlled relaxation. Our results identify feature maps, measurement bases, Hamiltonians, locality, symmetry priors, architectures, and sufficiently diverse training states as operational resources for generalization. The central implication is that Hilbert-space dimension alone is not a learnable feature space: successful QML must specify the physical structure that gives unseen quantum directions semantic meaning.

Large Language Models (LLMs) are increasingly deployed in contexts requiring complex moral reasoning and value trade-offs. However, existing evaluations typically rely on item-level behavioral metrics, which fail to capture how models structurally prioritize competing values as a cohesive system. To address this, we propose a symmetric human-LLM evaluation framework, grounded in Q methodology, to measure value-structure alignment. Under our protocol, humans and models sort an identical 140-item moral statement set into a shared nine-column forced distribution; for LLMs, we elicit strict rankings and deterministically map them to Q-sort buckets. Using a human reference sample ($N=35$), we establish a stable three-factor reference geometry specific to this instrument and sample. We evaluate 12 LLMs across four model families via 240 replicated Q-sorts at two temperature settings, quantifying structural alignment via Procrustes similarity ($φ$) and RSA-based Spearman correlation ($ρ$). Our results reveal significant cross-family heterogeneity, model-specific sensitivity to generation stochasticity and localized misalignment, which demonstrate that favorable global scores can obscure underlying regional distortions. While rank- and bucket-based analyses remain highly consistent, prompt phrasing introduces notable variance. Ultimately, assessing value-structure alignment provides a crucial structural complement to traditional itemwise moral benchmarks.

We study what Lorentz-equivariant jet taggers learn internally, using equivariance tests, linear probes and grade ablations across five models including L-GATr, L-GATr-slim and LLoCa-T. Linear probes show that equivariant models suppress frame-dependent pseudorapidity to zero while encoding jet mass and N-subjettiness strongly. Grade ablations on L-GATr reveal that bivector channels are negligible for top-quark tagging while vector-like channels are dominant but seed variable, consistent with the network exploiting multiple representational pathways. These results characterize which physical features and algebraic grade structures carry discriminative information in equivariant taggers and may inform future development of such models.

This paper presents an invertible neural network adapter for general robotic manipulation, designed to generate precise high-dimensional actions conditioned on multimodal observations, including visual, linguistic, and proprioceptive inputs, through a one-step denoising process. Built upon a flow-matching formulation, the proposed adapter effectively constrains the action generation trajectory within an invertible latent space, thereby enabling efficient and high-quality dexterous action synthesis with only a single inference step. Compared with conventional iterative flow-matching policies, the proposed framework substantially reduces inference complexity while maintaining strong action prediction accuracy and stability. Extensive experiments are conducted across a diverse set of simulation benchmarks and real-world robotic platforms to evaluate the effectiveness of the proposed method. Across simulation benchmarks, the proposed adapter consistently demonstrates superior or near state-of-the-art performance on a wide range of manipulation tasks. Furthermore, real-world experiments reveal a significant improvement in inference efficiency for vision-language-action (VLA) models, reducing the average inference latency from 110 ms to 61 ms while maintaining strong task performance.

We develop a framework for the information discarded by machine learning models whose inputs carry a Lie group action. Given a representation $π$ of a Lie group $G$ on a space $V$ and a learned function $f\colon V \to \mathbb{R}$, we define two objects measuring the symmetry invisible to $f$. The null fiber at a point $x \in V$ is the set $N_G(f,x) = \{g \in G : f(π(g^{-1}) \cdot x) = f(x)\}$ of group elements whose inverse action on $x$ is undetectable by $f$. When $N_G(f,x)$ is independent of $x$, it coincides with the stabilizer $\mathrm{Stab}_G(f)$, the largest subgroup of $G$ under which $f$ is invariant. For smooth maps to $\mathbb{R}$, the preimage theorem guarantees that null fibers have dimension at least $\dim G - 1$ at generic inputs, regardless of architecture. For compact groups acting on themselves, the Peter--Weyl theorem yields a spectral characterization of both objects in terms of the Fourier coefficient matrices of $f$. We show that null fiber elements can be computed efficiently via Newton iteration on the orbit map, at a cost comparable to a few gradient evaluations. Applications to data masking, model fingerprinting, and privacy-preserving computation are developed and tested experimentally on molecular property prediction under $\mathrm{SO}(3)$ and spherical image classification under the Möbius group $\mathrm{PSL}(2, \mathbb{C})$. The framework applies uniformly to classical neural networks and variational quantum circuits.

Recurrent networks can contain substantial functional redundancy in weight space: changing a recurrent matrix may leave the input-output rollout nearly unchanged on a task distribution, while similar-scale changes can destroy the same behavior. We study this redundancy in one-layer tanh RNNs using ordered real Schur coordinates. The Schur form separates spectral blocks from directed nonnormal couplings, giving a diagnostic basis for structured ablations that keep the input and readout maps fixed. In a fixed-length copy task, selected nonnormal Schur couplings can be removed with little loss in some trained solutions, whereas other couplings are necessary for accurate autonomous replay. Across flip-flop, sine generation, and context-dependent integration, the loss-preserving ablation profile varies across tasks and trained solutions. These results identify candidate approximate functional invariances, not universal symmetries of recurrent weight space. Schur-coordinate ablations provide a practical diagnostic for which structured perturbations preserve a trained recurrent solution and which ones disrupt its computation.

We propose a rigorous and systematic mathematical framework for tracking the cognitive trajectories of a user, in the context of algorithmic sycophancy and AI-driven delusional spiraling. Using tools from dynamical systems theory and stochastic differential equations, we explore how individuals perceive, interpret, and update their beliefs as they interact with AI chatbots that possess hidden traits of sycophancy. We treat the evolving conviction as a continuous log-odds state variable, coupled into a stochastic differential equation, navigating a multi-valley potential energy landscape. Our analysis reveals several critical observations governing the stability and rigidity of belief dynamics. We demonstrate that the baseline prior perception of the individual is systematically enhanced by sycophantic feedback beyond a critical threshold. Here, the perceptual potential landscape undergoes a structural phase transition that severely deepens any incremental initial tilt present in the baseline state, transforming the landscape and giving rise to deep, highly resilient attractor basins that trap the individual in unshakeable, self-reinforcing, delusional convictions. Finally, we demonstrate that genuine external information can successfully challenge these rigid states. If this incoming evidence is strong and authentic enough to overcome the internal feedback barrier, it can correct the structural asymmetry caused by sycophancy, inducing a perception reversal that successfully restores the objective belief state.

A wide range of neurosymbolic (NeSy) systems compute one functional: a belief-weighted sum of a logical quantity over a space of $σ$-structures, of which weighted model counting, fuzzy logic, and probabilistic logic are special cases. This account is built on sets, and a set deliberately forgets two things that are important for NeSy: when two $σ$-structures are the same up to a symmetry of the theory, and how many distinct proofs witness a query. Types, in the sense of homotopy type theory, preserve this information and turn the functional into a belief-weighted homotopy cardinality, a notion of size that counts each object in inverse proportion to its symmetries. We develop the framework from scratch for NeSy systems, prove a conservativity theorem that recovers the classical functional when symmetries are trivial, and show that the symmetry our framework exposes is exactly the one behind reasoning shortcuts. The payoff is concrete: the shortcut-aware concept posterior that recent methods reach by ensembling or expressive density estimation is the only symmetry-invariant point of the confusion-set simplex, computable in closed form by averaging a single model over the symmetry group. On MNIST reasoning-shortcut benchmarks this single-model wrapper is better calibrated than a diversity-trained ensemble, while leaving label accuracy and identifiable concepts untouched. Code is freely available at https://github.com/bio-ontology-research-group/hott-nesy.

Primary motivation in blind inverse problems is to recover signals of interest from corrupted observations without knowing the obfuscating mechanism. Blind deconvolution is a prominent approach when the corruption is convolutional, but it is not applicable when general linear transformations obfuscate the domain structure. In this work, we propose an unsupervised framework for recovering latent domains and signals by discovering symmetries of the data distribution. Our framework models observations as linear measurements of signals sampled from a latent random field, and optimizes a shallow group-convolutional network by imposing stationarity and locality regularization at the model output. The model learns a latent symmetry action and an appropriate filter, thereby mapping unstructured observations to a symmetry-based representation that reveals latent signals. Experiments on stochastic processes, Ising models, shuffled and bit-scrambled images, and neural recordings show that the method recovers latent domains and signals from unstructured observations, suggesting symmetry discovery as a new direction for unsupervised structure learning and blind inverse problems.

We develop a mathematically explicit link between shock-wave theory and the symmetry-quotiented learning dynamics of stochastic gradient descent, drawing on differential geometry, Lie group theory, and fluid mechanics. Specifically, after quotienting parameter symmetries and applying local-entropy coarse-graining, the effective dynamics satisfy a viscous Hamilton--Jacobi equation on the quotient manifold. Moreover, under the assumption that the raw parameter dynamics can be summarized by a gradient field on the quotiented space, the gradient of the coarse-grained loss function obeys a Burgers-type equation, and shock formation can be established rigorously. We apply our theory to multilayer perceptrons, convolutional neural networks, Transformers, and mean-field networks, and show that they obey the Hamilton--Jacobi or Burgers-type equations. We conjecture that this framework also yields practical diagnostics for deep learning. In architectures such as Transformers, raw parameter norms are often distorted by symmetry redundancy and may therefore be misleading, whereas symmetry-corrected quotient observables provide a principled basis for monitoring, forecasting, and controlling training-phase transitions.

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

Running large language models locally is often impractical, pushing inference on sensitive text to third-party providers. Split inference partially mitigates this by keeping tokens on the client and sending only hidden representations, but these representations can still be recovered via nearest-neighbor search against the public embedding table. We propose an orthogonal obfuscation procedure in which the client multiplies embeddings by a secret orthogonal matrix before transmission. To enable correct inference under arbitrary rotations, we introduce ConjFormer, a transformer variant that is exactly $\mathrm{O}(d)$-equivariant via a lightweight normalization change (scalar RMSNorm) together with blockwise orthogonal conjugation of all linear weights. As a result, the server performs the full forward pass entirely in the rotated basis and never observes unrotated hidden states. Experiments on GPT-2 and Llama 3.2 1B models fine-tuned on PubMed show that orthogonal obfuscation eliminates direct cosine nearest-neighbor inversion and reduces token recovery from over 35% top-10 to at most 1.3%, while increasing perplexity by only 0.4% after fine-tuning. These results indicate that enforcing symmetry at the architectural level can provide a practical defense for privacy-preserving LLM inference without noise injection or heavy cryptographic machinery.

Accurate interatomic potentials enable molecular dynamics of materials, molecules, and interfaces beyond density-functional-theory length and time scales. Equivariant neural network potentials have improved the representation of local geometry. However, their deployable energy surfaces ultimately manifest through invariant scalar channels, whose aggregation and spectral resolution remain comparatively underexamined. Here we use Physics-Aware Neighborhood (PAN) pooling and Physics-Guided Spectral (PGS) mixers as controlled scalar-pathway probes: lightweight, symmetry-preserving modifications that act only on $\ell=0$ channels while leaving the equivariant tensor backbone unchanged. Using MACE as a high-body-order mechanistic scaffold, PAN adds coordination-sensitive amplitude modulation, whereas PGS augments edge and readout scalar features with radial and tapered spectral bases. Across metallic Ag, covalent Si, a short-range ionic LiF/Li--F subset, and MD17/rMD17 molecules, this scalar-pathway correction reduces MACE force errors by 22--27% and energy errors by 19--22%; on systems with stress labels, stress errors decrease by 27--28%, at approximately 5% additional inference-FLOPs cost. Directionally consistent gains in Allegro and NequIP further indicate that the correction is portable across distinct short-range equivariant backbones, although effect sizes remain architecture-dependent. These results identify scalar-pathway fidelity as a practical design dimension for short-range equivariant interatomic potentials.

Steerable convolutional neural networks (Steerable-CNNs) guarantee SE(3)-equivariance by parameterizing kernels as linear combinations of steerable basis functions, but their deterministic nature precludes uncertainty quantification - limiting their use in settings where confidence estimates are essential. We propose a Bayesian Steerable-CNN that places posterior distributions over the basis coefficients, yielding stochastic kernels while preserving equivariance exactly. The loss function of the model is obtained via variational inference and minimized by Bayes-by-Backpropagation. The framework admits a decomposition of predictive uncertainty into epistemic and aleatoric components. Empirically, the model attains competitive classification accuracy alongside an expected calibration error of 0.0263 and outperforms its deterministic counterpart by up to 6.17% under distributional shift induced by additive Gaussian noise. Furthermore, we leverage the model's uncertainty estimates to enhance its performance significantly, achieving a notable gain - approximately 4% higher accuracy across 84% of the test dataset. A statistically significant negative correlation between epistemic uncertainty and prediction error confirms that the learned posterior variance is semantically meaningful. The framework unifies Bayesian uncertainty quantification with the inductive bias of equivariant CNNs.

Manifold-valued measurements are prevalent in various machine learning tasks. Recent advances have extended Deep Neural Networks (DNNs) to operate on manifolds, accompanied by normalization techniques tailored to different geometries, collectively referred to as Riemannian normalization. However, most existing Riemannian normalization methods are either designed for specific manifolds or fail to effectively normalize manifold-valued sample distributions. To address these limitations, we propose LieBN, a framework for Riemannian Batch Normalization (RBN) over Lie groups. Our approach leverages the theoretically convenient left- and right-invariant metrics, which naturally exist in every Lie group, and provides theoretical guarantees for controlling the Riemannian mean and variance. We instantiate LieBN across nine distinct geometries: four on the Symmetric Positive Definite (SPD) manifold, one on the group of rotation matrices, and four on the manifold of full-rank correlation matrices. Notably, among the SPD metrics, we introduce a novel right-invariant metric and extend three existing Lie group structures via matrix power deformation. Extensive experiments on different manifolds validate the effectiveness of our framework. The code is available at https://github.com/GitZH-Chen/LieBN.git.

Learning-based video compression has recently achieved competitive rate-distortion performance compared to conventional video codecs. However, most existing methods rely on non-invertible analysis-synthesis transforms, with reconstruction quality subject to both quantization and transform approximation errors. This limitation becomes particularly restrictive at higher quality points, where quantization errors are small and transform-induced distortion dominates. To address this, we propose InnVC, an Invertible neural network based Video Codec for wide-range and high-fidelity compression. The core idea is to preserve an invertible main transform path prior to quantization, while injecting content-adaptive context through a compact implicit conditioning field. This decouples strongly correlated video content from harder-to-model fine details, allowing different components to specialize in complementary reconstruction tasks for more efficient compression. To further improve compressibility, we introduce a scheduled masking strategy that progressively concentrates informative content into fewer latent channels for more effective entropy coding. Experiments on the UVG and MCL-JCV benchmarks show that InnVC achieves strong compression performance over a broad quality range, being particularly effective in the high-quality regime, yielding BD-rate reductions of 21.66% in PSNR and 46.06% in MS-SSIM relative to x265 on UVG. To the best of our knowledge, InnVC is the first neural video codec covers operating poins from low bitrate to high fidelity within a single architecture scale, spanning more than 20 dB in PSNR.

Evaluating generative spatial audio for First-Order Ambisonics (FOA) remains challenging due to a limited understanding of how metrics respond to changes in spatial parameters such as azimuth and elevation. We propose a framework to analyze metric sensitivity along continuous spatial trajectories, drawing on principles of sensitivity analysis in parametric sound synthesis. Using controlled FOA scenes with increasing scene complexity, we define three desiderata for metric behavior: Responsiveness, Smoothness, and Symmetry. We assess standard distribution-based and sample-based metrics, including Fréchet Audio Distance (FAD), intensity vectors, and acoustic maps. Our findings show that FAD using localization-specific embeddings and acoustic maps yield high Responsiveness and robust Smoothness and Symmetry across conditions, while intensity vectors degrade with increasing scene complexity. This is the first step towards investigating the sensitivity of metrics for generative spatial audio.

We explore whether intrinsic symmetries of the training data lead to conserved quantities during gradient-flow training of neural networks. Under the assumption that the loss function is analytic and non-polynomial, we prove that data symmetries generically do not induce any additional integrals of motion. For mean squared error (MSE) loss, on the other hand, there are situations in which data augmentation yields extra conserved quantities. We build a framework, utilizing \emph{tensorizable networks} to describe this phenomenon. Tensorizable networks are a family of architectures whose dependence on parameters and inputs can be separated using an intermediate representation. They include linear and polynomial networks, as well as Lightning Attention.

Traditional UV unwrapping relies on direct optimization of geometric distortion energies and can fail through invalid initialization, local minima, or topological foldovers. We recast fixed-chart UV unwrapping as continuous neural reparameterization: an untrained SIREN maps per-vertex mesh features to UV coordinates, and its weights are optimized for a geometric objective. The practical contribution is a robust chart-solver recipe, combining Laplace--Beltrami spectral inputs, Tutte residual warm-up, a $C^2$ determinant extension, an injectivity barrier, and validity-checked retry/fallback routing, rather than a claim that any single component guarantees validity or that recutting methods should be replaced. NTK--LBO diagnostics show that spectral conditioning changes update geometry, especially at initialization and mid-rank subspaces, but does not by itself predict chart success. On compact pre-cut charts and a 47-chart stratified Thingi10K/xatlas-cut benchmark, the neural solver produces zero flips on all compact charts and 42/47 valid zero-flip stratified solves. BFF and OptCuts comparisons sharpen the scope: recutting can be faster and lower-distortion when allowed, while the neural solver targets supplied-chart validity and validation-first atlas construction. On Amara Spatial generated meshes, the full atlas construction path gives packed-atlas coverage on a 25-asset set and 1000/1000 strict locally valid atlases with zero UV flips in a large-scale Rust atlas run after fallback routing.

Probabilistic inference over spatially embedded variables requires beliefs that respect $SE(3)$ 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 $SE(3)$. 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 $SE(3)$. On GEOM-QM9 and GEOM-Drugs, ENBP achieves 98.9% conformational coverage at 0.090 $\mathring{A}$ error with sub-second latency -- over $100\times$ 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 (${\sim}10^{-7}$ vs.\ $10^{-1}$ for augmented baselines).

Diffusion MRI (dMRI) tractography is the only noninvasive approach for mapping white-matter pathways in the living human brain. It represents each brain as a tractogram: a large, unordered set of three-dimensional streamlines that includes information about both local streamline geometry and whole-brain anatomical organization. This structure makes tractograms a natural but challenging target for representation learning. Existing methods treat streamline classification and subject-level prediction as separate problems: streamline classifiers focus on geometric patterns, whereas subject-level prediction often depends on hand-crafted features. As a result, current methods do not learn reusable representations that connect streamline anatomy with whole-brain inter-subject variation. Here we introduce TractFM, a tractogram foundation model that learns reusable representations directly from whole-brain streamline sets. TractFM combines a local streamline encoder with a permutation-equivariant tractogram encoder, allowing all streamlines from a subject to be contextualized jointly in a single forward pass. Pretraining on dense anatomical tract parcellation, i.e., assigning anatomical labels to individual streamlines, yields two complementary representations: contextualized streamline-level embeddings for tract parcellation and compact subject-level descriptors for downstream prediction of subject phenotypes. Across three tractography algorithms and five dMRI datasets, TractFM transfers to both streamline-level and subject-level tasks. Its frozen representations achieve accurate tract parcellation and predict age and sex across independent datasets. These results show that whole-brain geometric context, learned once, can generalize across tractography pipelines, datasets, and prediction tasks.

Many striking phenomena in deep learning, such as linear mode connectivity and the structured behavior of training dynamics, are closely tied to parameter symmetries: transformations that leave the realized function unchanged. Despite growing attention to parameter symmetries, the exact interplay between parameters, data, and representations remains underexplored. To investigate this, we develop a theoretical framework of effective function classes, i.e., the set of functions a neuron can realize on its input support, and the norm cost of realizing them. We then formalize effective symmetry breaking via neuron identifiability across independent training runs. Our analysis shows that neural networks can admit large families of approximately equivalent solutions even in structurally asymmetric models. We further show that neuron identifiability enables representation merging without prior alignment, and characterize when such merging admits a linear low-loss path. These findings highlight the role of effective function classes in affecting the loss landscape.

Single-view 3D generative models have achieved impressive visual quality, yet they are not designed to satisfy structural or functional requirements, and in practice, often fall short. Symmetry is one such requirement: violations, even subtle ones, on symmetry can render a model physically unusable. We present SymTRELLIS, a method that enforces arbitrary finite point group symmetries (rotational, reflectional, and polyhedral) during the flow-based 3D generation of TRELLIS.2, without retraining the underlying VAE or flow model. Our key idea is to approximate the latent-space action of spatial transformations as a learned linear operator on voxel latents, implemented as a lightweight spatial-transform latent mapper trained on generic, non-symmetric 3D data. At generation time, we enforce symmetry by averaging predicted flow velocities across all symmetry-equivalent transformations at each ODE step, a process we call velocity symmetrization. The symmetry specification can be estimated automatically from an initial TRELLIS.2 generation or supplied by the user, enabling deliberate fold manipulation beyond what the input image suggests. On a curated benchmark of 266 strictly symmetric objects spanning 2- to 20-fold rotations and polyhedral symmetry groups, SymTRELLIS substantially reduces all symmetry error metrics compared to TRELLIS.2, Hunyuan3D-2.1, and TripoSG, while maintaining reconstruction accuracy comparable to the base model.

Machine learning systems deployed in high stakes socioeconomic settings routinely display bias. We formalize bias as a symmetry breaking operation: a classifier is fair if its outputs remain invariant under the counterfactual operation of switching a sensitive attribute, with merit features held fixed. We implement loss based regularization as a symmetry restoring mechanism and evaluate the framework on four synthetic datasets with varying levels of noise, correlation, and bias. The framework achieves upwards of 90% violation reduction, with accuracy costs around 5%. This framework does not require causal graph knowledge, is computationally lightweight, and generalizes to any sensitive attribute definable as a bit-flip, making it suitable for contexts where local sources of discrimination remain absent from mainstream benchmarks.

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

Robot storytelling offers a unique blend of technological innovation and creative expression that engages children in unprecedented ways. However, the technical aspects are often too complicated for children. We propose an interactive system that facilitates robot storytelling with tangible and natural language interactions. Children arrange the playground with their own stuff and create narratives with an LLM agent. The created narratives are transformed into a motion sequence based on the map and characters, and the motions are executed by self-navigating swarm robots. This system enhances robot storytelling with flexible scenarios, enabling young children to create robot dramas with everyday objects.

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

We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution layers, and we show how for suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. We additionally describe groupoid invariant global pooling as a generalization of group invariant global pooling. Furthermore, we show that each of the aforementioned layers is a special case of recently introduced admissible category-equivariant layers by demonstrating that they define continuous natural transformations between continuous feature functors.

This work studies neural architectures for classifying symmetric positive-definite matrices, focusing on congruence-like layers, in which the input matrix is multiplied on the left and right by a (possibly rectangular) weight matrix $W$ and its transpose. Such layers lie at the core of the celebrated SPDNet and have also been employed independently for dimensionality reduction on positive-definite data. We show that the (semi)-orthogonality constraint commonly imposed on $W$ limits the expressivity of these layers: for certain activation functions, the resulting architecture collapses to a one-hidden-layer equivalent. This lack of expressivity follows from a loss of spectral diversity in congruence-like layers for semi-orthogonal $W$ and is a direct consequence of Poincaré's separation theorem. We then examine the choice of the final classifier, comparing several Riemannian classifiers and discussing their compatibility with the feature maps produced by congruence-like layers.

Object pose estimation is a fundamental problem for an agent system to perceive or manipulate objects in images or videos. However, current instance-level methods struggle with generalization to unseen objects. Category-level methods seek to address this, but remain constrained by the complexities of learning in the non-linear Sim(3) space and intra-class variations. To address these challenges, We propose an effective method for category-level object pose estimation with two key innovations: (1) A translation/size estimator, featuring a semantic-guided symmetry-aware module that leverages robust generalization capabilities of a large vision model (LVM) to infer symmetry points, resulting in accurate translation and size without shape priors. This result serves as a precomputed cue for rotation estimation, thereby reducing the difficulty of learning in the non-linear Sim(3) space and laying a robust foundation for tackling the inherently more challenging rotation estimation. (2) A feature fusion module, based on our proposed spherical large-kernel inception convolution, fuses semantic features from the LVM with systematically computed geometric features to extract essential pose features from intra-class variations by modeling long-range dependencies without excessive computational cost. Built on these innovations, we achieve SOTA on benchmarks and real-world scenes, while developing a robust robotic picking system capable of handling diverse objects. Our code will be available at the project page: {\hypersetup{urlcolor=blue}https://panfei-cheng.github.io/SSH-Pose}.