Symmetry

Nonlinear activations can create equivariant interactions between irreducible representations that linear maps cannot. We use the Gaussian degree decomposition to extend ordinary polynomial degree to such nonlinear maps, and prove that for a fixed coordinatewise equivariant layer each degree factors into a polynomial determined by the linear maps and a scalar determined by the activation. This separates three distinct obstructions, coming from symmetry, coordinates, and activation.

The standard geometric derivation of neural scaling exponents takes the intrinsic dimension of a data manifold as its input. On modular addition in $\mathbb{Z}_p$ that derivation has no input. The exact algebraic solution is an orbit of $\mathbb{Z}_p$ acting by isometries. Transitivity alone makes the ratio statistic underlying the standard dimension estimator a point mass, so the estimator is undefined, and here the two nearest neighbour distances coincide exactly. Breaking the symmetry at scale $ε$ returns a number, but one that tracks $1/ε$ with no scale free plateau. We show that the failure is general, since on any finite orbit of a group acting by isometries the estimator reports the resolution at which the set is probed rather than a dimension. What replaces the power law is exponential in hidden width, $L(h)=L_\infty+A\exp(-c\,h^α)$, with $R^2$ between 0.982 and 0.995 against 0.857 to 0.906 for a power law admitting the same floor and fitted under the same protocol. Where the data supply is sufficient the rate belongs to the regulariser rather than to the group, since weight decay moves $c$ by a factor of 47 while group order moves it by 1.10, a residual below seed to seed resolution, for every fixed $α$ between 0.75 and 2. The critical width falls with group order rather than rising, against capacity counting that assigns a fixed number of neurons to each irreducible representation.

Many self-supervised methods for training foundation models at the Large Hadron Collider (LHC) rely on data augmentations to encourage the model to embed events into a representation space invariant to certain physical or detector symmetries. A common challenge arises from the large freedom in choosing a proper set of augmentations on which downstream performance depends. The implementation of augmentations involves either modifying existing events, potentially breaking the event fidelity, or simulating more event variants, which is computationally intensive. In this work, we present a data-driven method of pairing events by their similarity via the energy mover's distance (EMD), which measures how similar two events are in terms of the work required to transform one into the other. With this approach, distinct events are sampled and matched by their similarity to serve as views for learning invariance, keeping the physics content of each event intact without handcrafted distortions. We demonstrate this augmentation-free pairing method by pre-training on QCD jets via self-distillation and show that it can yield semantic jet embeddings with downstream discrimination power comparable to or better than an augmentation-based baseline.

Frozen vision-language models (VLMs) remain unreliable on fine-grained left-right claims, and raw claim likelihoods need not reliably rank verification errors. After a horizontal-reflection intervention is fixed, how should its induced likelihood measurements be combined into a selective verification signal? We introduce Relation-Orbit, a closed-form contrast with no learned fusion parameters that assigns eight normalized likelihoods to query-supporting and counterfactual roles determined by reflection, inverse relation, and entity exchange. A claim is asserted only when the signed contrast exceeds a threshold selected on held-out data using pointwise Clopper-Pearson upper confidence bounds. On VSR and GQA across four frozen VLMs, Relation-Orbit yields higher mean test coverage at a 10% selective-risk calibration target than an all-eight Orbit-Max baseline in all eight dataset-backbone settings; gains over a nearly abstain-all one-sided intervention score are reported separately. A separate LLaVA-1.5/COCO evaluation, reduced-orbit controls, and a two-sided partition diagnostic further characterize the structural advantage.

In social dilemma games, additional rewards or punishments have been studied as means of promoting cooperation. Therefore, it is important to investigate the ideal situation, in which such an additional payoff would change the game. In this study, we investigated the symmetric solution of the Bellman optimality equation for a repeated harmony game. The calculations showed that three types of symmetric solutions exist. One of them corresponds to the trivial All-C strategy, and another to the Win-stay Lose-shift strategy of the prisoners dilemma game. The nontrivial behavior of the strategy corresponding to the last solution is also discussed in detail. In addition, we numerically investigated which strategy the agents actually learn by the reinforcement learning algorithm.

The difficulty of training large language models (LLMs), together with their ubiquity, raises the threat of stegomalware, where malicious payloads are embedded into model weights. Recent work has demonstrated the use of permutation symmetry in model weights to mitigate these threats, but failed to show neutralization of stegomalware across all weights for LLMs. In this paper, we demonstrate the full potential of behavior-preserving symmetries as a defense against stegomalware, as well as the risks these symmetries pose when exploited by attackers. For stegomalware neutralization, we improve upon previous work, demonstrating that it is possible to select permutations which displace all model parameters. This contrasts with previous methods which left a significant percentage of weights unaltered in LLMs. When used in an attack, we show that permutation symmetries can encode malware into the weights of a model in a way that is theoretically lossless, requires no retraining after encoding, and needs no payload-specific information in the extraction script---a combination of characteristics not previously seen in any single method. While theoretically lossless, permutation can in practice alter model behavior due to the accumulation of numerical error. We therefore quantify the loss in model performance associated with applying these methods, for both attack and defense, showing it to be minimal.

Symmetries play a central role in reducing the complexity of reinforcement learning problems, yet most existing approaches rely on fixed group actions or predefined state abstractions. Classical reinforcement learning algorithms typically assume a globally structured Markov decision process with uniformly applicable actions and transitions, an assumption that limits their ability to exploit modularity and local, context-dependent regularities present in many realistic environments. We propose a reinforcement learning framework using groupoids to capture local, state-dependent symmetries and support the dy- namic discovery of equivalence structures during interaction. The agent maintains orbit representatives together with transporters that map raw states to canonical forms, enabling learning and decision-making to be performed in a symmetry-reduced space while preserving local distinctions. Empirical results demonstrate that the proposed groupoid-based approach improves sample efficiency and convergence in dense and large-scale environments exhibiting strong partial symmetries, yielding substantial performance gains over standard Q-learning. These findings show that dynamically exploiting local symmetry provides a practical and mathematically principled route to scalable and generalisable reinforcement learning.

Crystal-orientation maps are physical fields defined only up to crystal symmetry; electron backscatter diffraction (EBSD) resolves them experimentally, but acquisition-time constraints limit spatial resolution. Unlike conventional images, EBSD data lie on the quotient space $\mathrm{SO}(3)/G$, where $G$ is the crystal-symmetry group. Standard Euclidean interpolation can therefore mix symmetry-equivalent representations and blur grain boundaries. We introduce the Symmetry-Group-Aware Super-Resolution Attention Network (SG-SRAN), which incorporates crystal symmetry and boundary preservation by design. A frozen, locally isometric encoder maps equivalent orientations to a common latent representation in which Euclidean distance approximates misorientation. Super-resolution is performed in this space, with each high-resolution token restricted to a feature-consistent local support to prevent cross-boundary mixing. A dictionary-based decoder then recovers valid orientations. Across FCC and HCP benchmarks, SG-SRAN matches 15-16 million parameter backbones using only 27-49k trainable parameters, while achieving the lowest p68 errors, highest inverse-pole-figure fidelity, and zero-shot transfer to unseen alloys.

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

Fully homomorphic encryption (FHE) enables inference on private data without revealing it to the server, but evaluating an entire input under FHE is expensive. We study \emph{selective homomorphic inference}, where only a sensitive region of interest (ROI) is encrypted, and computations independent of that region are performed in plaintext. Selective evaluation produces the same output as full FHE on the same model, without retraining. Its efficiency depends on how quickly encrypted dependencies spread through the network. For small encrypted ROIs, locality-preserving architectures can achieve order-of-magnitude homomorphic-evaluation speedups, whereas architectures with early global mixing provide essentially no speedup. These results identify locality as the key architectural property governing the benefit of selective homomorphic inference.

Sparsity is a powerful structural resource in optimization and statistics. We develop frameworks for leveraging sparsity in sampling problems over the Hamming slice $\mathcal{X}_k^d:=\{\mathbf{x}\in\{\pm 1\}^d:|\{i:\mathbf{x}_i=1\}|=k\}$, in high-dimensional regimes where $k\ll d$ (i.e., where $\mathcal{X}_k^d$ is \emph{highly magnetized}). We use our frameworks to design improved samplers for canonical problems in the study of \emph{Ising models} and \emph{Bayesian sparse linear regression}. Our first main result considers the \emph{Sherrington--Kirkpatrick} (SK) model restricted to fixed-magnetization slices $\mathcal{X}_k^d$. We give a polynomial-time sampler for fixed-magnetization SK models at any inverse temperature $β>0$, under arbitrary external fields, provided that $k\le c_βd$ for an appropriate constant $c_β$. By combining this result with an annealing strategy for estimating normalizing constants, we obtain polynomial-time samplers for the SK model at arbitrarily low temperatures under a sufficiently strong external field of strength $h$. In the large-$β$ limit, our framework permits sampling at field strengths within constant factors of the \emph{Almeida--Thouless line} delineating the replica-symmetric and replica-symmetry-breaking regions ([dAT78]), improving polynomially over the field strength $h(β)$ required by the recent work of [BAR26]. Our second main result concerns the measurement complexity of polynomial-time Bayesian sparse linear regression. Recent work by [KSTZ25] shows how to sample from the canonical \emph{Gaussian spike-and-slab posterior} with expected sparsity $k$, at any signal-to-noise ratio, given $n\gtrsim k^3\log^3 d$ Gaussian measurements. We improve this requirement to $n\gtrsim k^{3/2}\log^2 d+k\log^3 d$, using a common sparsity-aware framework underlying both our results.

Equivariant networks are commonly trained with Adam, yet recent work reports that matrix-structured optimizers such as Muon can perform better on these architectures without explaining why. We identify one source of this difference inside equivariant linear layers. Each irrep block learns a channel-mixing matrix $W_l$ shared across its $2l+1$ components, giving the expanded map $W_l \otimes I_{2l+1}$. For a single application of the layer, the gradient of $W_l$ sums $2l+1$ outer product contributions and has rank at most $2l+1$. Adam rescales stored weights individually without using the irrep boundaries, so one learning rate can produce different spectral step sizes across blocks within a layer. We address this mismatch by normalizing each block update separately, without introducing a new hyperparameter. This changes only the scale of the update, leaving Adam's moment estimates and its direction within each block unchanged. We evaluate the mechanism in a controlled $\mathrm{SO}(3)$-equivariant model with a matched dense control and in an e3nn interatomic potential model trained on rMD17 and MD22. The toy setup isolates a mismatch that grows with width while the dense control shows no corresponding growth. In the interatomic potential model, block normalization and tuning Adam's momentum coefficients independently improve performance, but neither alone matches Muon. Combined, they make Adam competitive with Muon on all datasets, indicating that blockwise step control and momentum accumulation account for much of Muon's advantage.

The evolution of DNA sequences can be viewed as stochastic dynamics on a high-dimensional discrete space, but it is unclear when empirical transition biases reduce to an effective energy landscape versus retain irreducible non-equilibrium circulation. Human context-dependent mutation probabilities offer a direct test: every single-nucleotide substitution in a local context has a reverse substitution, so the logarithm of the forward-to-reverse probability ratio defines an antisymmetric field-the human mutation field. We show this field has a dominant gradient component and a smaller but reproducible curl component. Using seven-base human germline substitution probabilities, we infer an effective mutational landscape with a Siamese neural network constrained to predict only energy differences. This model predicts forward-to-reverse log-ratios for held-out mutations with a correlation of about 0.93, close to both an unconstrained predictive reference (0.948) and the empirical reversible ceiling from Hodge projection (about 0.96). Although trained only on mutation probabilities, the inferred landscape largely recovers short-word genomic composition and Chargaff reverse-complement symmetry for sequences up to length four. Deviations from equilibrium structure reveal a small but detectable nonequilibrium component: a residual irreversible circulation violating the Kolmogorov cycle condition for detailed balance, reproducible across African, Asian, and European populations, and strongest in CpG-linked cycles and CpG-transition edges, consistent with methylcytosine deamination. These results give a thermodynamic decomposition of the human mutation field: most mutation bias is organized by a local equilibrium-like energy landscape aligned with genome composition, while the residual circulation points to specific directional mutational mechanisms.

The emergence of agentic AI and LLM-based multi-agent systems (MAS) presents unprecedented opportunities for automating complex tasks, while simultaneously raising critical concerns about the preservation of fundamental human-centered values, such as privacy, fairness, and safety. Although software engineering has traditionally focused on functional correctness, the adoption of LLMs and AI agents into complex socio-technical systems has intensified the need for responsible software engineering and robust value alignment. In MAS, architectural design decisions, such as coordination mechanisms, communication protocols, and system topologies, play a central role in shaping system behavior and the outcomes they produce. This paper argues that architectural choices influence not only the functionality and performance of MAS but can also promote value-oriented system behavior. Therefore, we investigate how different architectural designs support different human-centered values, discussing the following value-preserving architectural patterns: (i) a privacy-aware architecture with a federated topology, (ii) a distributed architecture to promote pluralism and diversity, and (iii) a guard-agent architecture to detect and mitigate unfairness. Finally, we introduce representative use cases to illustrate the proposed architectures in real-world scenarios. By linking architectural design with human-centered values, this work lays the foundation for a unified set of architectural patterns and guidelines towards the design of trustworthy MAS.

We study the ordered Bernoulli-word kernel f(p,n,k)=p^k(1-p)^(n-k) and the geometry generated by its inverse-integer level sets. The binary level 2^(-n) selects p=1/2 as the unique real split-independent anchor. Under complement-preserving complex continuation, the pair becomes z=1/2+iu and 1-z=1/2-iu, producing a conjugation-symmetric vertical geometry before any zeta-function input is introduced. The quadratic coordinate Q(z)=z(1-z)=1/4+u^2 has a sharp minimum at the central point and admits an exact integer quantization. For critical-line zero ordinates gamma_k, the induced levels L_k=1/4+gamma_k^2 are decomposed exactly as L_k=N_k+delta_k, where N_k is the nearest integer and delta_k is a periodic first-Bernoulli residual. Circularization gives Z_k=exp(2 pi i delta_k), isolating gamma_k^2 mod 1 as the residual phase variable. Unique factorization resolves the integer shells into prime-generator coordinates, while a distinct complex exponent s lifts the same construction to the Dirichlet atoms m^(-s), linking the Dirichlet-series and Euler-product assemblies. Exact identities, classical zeta connections, numerical controls, and open conditional Weyl tests are kept explicitly separate. No proof of the Riemann Hypothesis is claimed.

In the natural sciences, symmetries and cause-effect relationships are ubiquitous. Yet for complex machine-learning tasks, like world-modeling in reinforcement learning, they appear difficult to harness. We propose a formal description of statistical systems based on symmetries in data leaving causal mechanisms invariant. The result is an abstract, simple and general mathematical language for causal reasoning. This paper provides formal descriptions of models and queries, setting up this language, and the formal infrastructure and strategies for their mathematically rigorous identification from data within this formalism. This approach reproduces and matches standard theoretical results on IID data and transport of experimental and non-experimental data. But its main purpose is to unify and substantially extend the scope of causal reasoning, in going beyond IID data and in approaching complex causal queries not captured by do- or soft-interventions. This new perspective on causally relevant aspects of data-modeling additionally sheds new light on well-known structures like c-components or hedges but also includes aspects of missing data and is inherently well-suited for the description of transfer and robustness properties.

Artificial neural networks are often regarded as powerful yet opaque black boxes. Here, we demonstrate that learning in deep neural networks generates local symmetries known in graph theory as fibrations and coverings. We prove that covering symmetries are stable attractors of stochastic gradient descent. Consistent with this theory, we report the emergence of covering symmetries across major network architectures, including multilayer, convolutional, recurrent, and transformer networks. Exploiting these symmetries enables drastic model compression - reducing networks to 17% of their original size without sacrificing performance. Furthermore, controlled breaking of covering symmetry overcomes the loss of plasticity, achieving state-of-the-art performance in continual learning. The theoretical results provide a new foundation for AI systems based on symmetries that convert black boxes into interpretable colored graphs and enable more efficient inference and lifelong learning.

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

Historically originating from Hilbert's 13th problem, the Kolmogorov-Arnold representation theorem (KART) has recently experienced a major revitalisation through its applications to neural networks, specifically Kolmogorov-Arnold Networks (KANs). While the exact representation is well established, its stability under continuous adversarial perturbations of the hidden layer remains a critical open question. In this paper, we investigate the robustness of KART against bounded adversarial translations. We provide an explicit, self-contained, and constructive proof of an approximate representation using fixed, piecewise linear inner functions. Crucially, our construction employs a single outer function that remains invariant for all summands and is independent of the specific adversarial translation, provided its maximum bound is known a priori.

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

Crystal symmetry dictates whether a physical response tensor must vanish, establishing a direct test for machine learning predictions independent of property calculations. We derive the parity gap, a group-theoretic metric quantifying the piezoelectric tensor freedom permitted by a crystal's proper rotation subgroup $SO(3)$ but eliminated by inversion symmetry in $O(3)$. Across state-of-the-art equivariant neural network architectures, unconstrained $SO(3)$ models systematically predict forbidden non-zero responses matching the parity gap of each centrosymmetric crystal class, while polar distortion paths dynamically map output responses to the loss of inversion symmetry. Regression controls confirm that enforcing full $O(3)$ parity incurs no consistent accuracy cost across predictive tasks. Crucially, while training interventions using explicit zero labels reduce violation magnitudes, they leave residual forbidden outputs. Exact physical compliance instead requires structural enforcement through $O(3)$ representation design or explicit output antisymmetrization. The parity gap thus provides a unified framework to distinguish empirical error reduction from exact structural compliance with physical law.

We propose Symmetric Nonlinear Motion-guided Generative Video Frame Interpolation (SNM-VFI), a training-free framework for motion-controllable generative video frame interpolation with pre-trained optical flow and video diffusion models. Unlike conventional diffusion-based VFI methods that synthesize intermediate frames from random noise, SNM-VFI guides the generative process with correspondence-aware frames produced by a symmetric nonlinear motion model. Specifically, we first utilize a pre-trained optical flow model to construct multi-frame nonlinear flow-based intermediate frames and confidence maps. These flow-guided frames are then encoded as latent priors to initialize and iteratively guide a pre-trained Video Diffusion model, enabling the diffusion model to preserve dense motion correspondence while improving perceptual realism. To further enhance output quality, we employ confidence maps to fuse structurally reliable flow-based predictions with diffusion-generated details in uncertain regions such as occlusions and object boundaries. Extensive evaluations on challenging benchmarks, including DAVIS, Sintel, and KITTI, demonstrate that SNM-VFI achieves strong perceptual quality, competitive reconstruction accuracy, and robust temporal coherence across diverse motion scenarios.

Generating crystals has recently attracted significant interest due to their broad applications in materials science. However, existing generative models struggle to produce complete crystallographic specifications, limiting their ability to capture global symmetry and structural dependencies. In particular, current state-of-the-art approaches generate crystals only up to site symmetries and rely on sampling space groups from empirical distributions during generation. Inspired by \emph{spontaneous symmetry breaking} in physics, where crystals break symmetries under external conditions, we propose a novel diffusion-based framework that generates full structure specifications by reversing from the lowest-symmetry priors. Our method leverages a Markovian jump-diffusion process to model these symmetry-breaking dynamics, enabling it to traverse different space groups in a physically motivated manner. Our model, dubbed \emph{Symmetry-breaking Crystal Diffusion} (SbCD), introduces a principled approach to explicitly incorporate inter-space-group transitions into the generative process. In de novo generation experiments on MP20 and MPTS-52, SbCD outperforms its symmetry-preserving counterpart by a substantial margin, offering a promising perspective for generative modeling of crystalline materials.

Neural networks trained by gradient descent on a smooth cost function can nevertheless learn in steps: the cost holds on long plateaus and then drops abruptly. Meanwhile, training losses instead follow smooth power laws. Variants of both behaviors occur in architectures with very different microscopic structures, which is the signature of a few relevant collective variables. We show that a symmetry fixes what those variables are: a network layer is a sum over interchangeable units, so relabeling the units leaves it unchanged; given smoothness and the condition that a unit's gradient vanish at the origin, symmetry then enforces a universal leading form for the expansion about the near-zero weights present at the start of training, the quadratic $\Tr[WW^{\top}A(x)]$, in which every architectural detail is confined to a single structure matrix" $A(x)$ that we compute for each architecture. Perceptrons, attention layers, mixtures of experts, and convolutions become one model at different $A$. Its training dynamics then close on the order parameter" $M=WW^{\top}$ and, whenever the data matrices share an eigenbasis, reduce to a Lotka--Volterra equation whose modes switch on one after another. The smaller the initial weights, the further apart the switch-on times, and the plateaus appear as a singular limit of a smooth flow; when many modes are unresolved the same events merge into a power law in training time whose exponent the theory predicts. We confirm both numerically across training methods and architectures.

Equivariant Neural Networks (ENNs) have empowered numerous applications in scientific fields. Despite their remarkable capacity for representing geometric structures, ENNs suffer from degraded expressivity when processing symmetric inputs: the output representations are invariant to transformations that extend beyond the input's symmetries. The mathematical essence of this phenomenon is that a symmetric input, after being processed by an equivariant map, experiences an increase in symmetry. While prior research has documented symmetry increase in specific cases, a rigorous understanding of its underlying causes and general reduction strategies remains lacking. In this paper, we provide a detailed and in-depth characterization of symmetry increase together with a principled framework for its reduction: (i) For any given feature space and input symmetry group, we prove that the increased symmetry admits an infimum determined by the structure of the feature space; (ii) Building on this foundation, we develop a computable algorithm to derive this infimum, and propose practical guidelines for feature design to prevent harmful symmetry increases. (iii) Under standard regularity assumptions, we demonstrate that for most equivariant maps, our guidelines effectively reduce symmetry increase. To complement our theoretical findings, we provide visualizations and experiments on both synthetic datasets and the real-world QM9 dataset. The results validate our theoretical predictions.

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

Reasoning shortcuts are solutions of a neurosymbolic system's rules that produce correct predictions through unintended concepts. A recent framework of Takemura, Inoue, and Nishino analyzes them through an automorphism group of value relabelings and asks, as its central open question, when rules pin concepts down. We first show that the framework's key definition, one shared permutation applied at every position, does not apply as stated to any of the four heterogeneous benchmarks it was evaluated on, and that the most direct embedding, padding domains to a common size, produces confident false pathology: 90.91% of solution pairs reported unexplained on CLE4EVR, where every well-defined member of the hierarchy we introduce reports 0%, and the padded verdict's content rotates with configuration-file ordering. Re-measuring eleven rule families under fifteen pre-specified predictions (thirteen confirmed), unexplained-pair rates span 0% to 99.9999% and track provable structure: six theorems give sufficient conditions for transitivity and its failure, including a Free Slot Lemma certifying Kandinsky's pathology from syntax alone. For circuit-given rules, deciding symmetry-inertness of a coordinate is coNP-complete; nontrivial-automorphism existence is coNP-hard under randomized reductions, lies in $Σ_2^p$, is not $Σ_2^p$-complete unless PH collapses, and on monotone circuits is coNP-complete outright. In the Boolean case transitivity is classified exactly: automorphisms explain everything iff the solution set is an affine coset. Weakly supervised models place all 94 observed shortcuts at the one level the componentwise theory flags and none at the 48 it certifies transitive; twelve typed-ambiguous levels produce none, separating what symmetry permits from what optimization selects, and a dual-head control replicates the geography. All numbers trace to released artifacts.

Dynamical systems model trajectory data generated by fixed underlying dynamics, with applications ranging from biology to physics. Especially in scientific settings, dynamical systems are not generic but often exhibit symmetries imposed by physical laws, formalized through equivariance with respect to group actions. The identification problem concerns recovering the parameters of a system from observed trajectories. In this work, we study adaptive symmetry discovery for dynamical system identification and address how a system can be identified from a single trajectory when it is equivariant with respect to an unknown symmetry group. To this end, we first show that for known symmetries, the system can be identified from a significantly shorter single trajectory than in the generic setting, and we precisely characterize this improvement. We then consider the automatic symmetry discovery setting, proposing a method to learn the symmetry group directly from a single trajectory and incorporate it into the identification procedure, achieving the same optimal trajectory length as in the known-symmetry case. Our analysis relies on tools from group representation theory and the expander properties of Cayley graphs, and may be of independent interest for the study of symmetries in dynamical systems.

Continuous hemodynamic monitoring guides treatment decisions in surgery and intensive care. However, gold-standard signals are only measured in severe cases due to risks associated with invasive measurement. In this work, we introduce HIPNO (Hemodynamic Inference via Physics-informed Neural Operators) to recover hemodynamic state from ubiquitous, non-invasive signals and expand access to advanced monitoring. HIPNO addresses a problem of scale symmetry in physics-informed hemodynamic inference, where different combinations of flow, resistance, and compliance can generate the same observed pressure. We identify the symmetry group of the observation model and parameterize the network in its quotient space. For the 3-element Windkessel model, the quotient coordinates are the compliance-normalized flow $U=Q/C$, the decay time constant $τ_{WK}=R_2 C$, and the characteristic-impedance coordinate $κ=R_1 C$. Across 945499 intraoperative windows from 2562 patients, HIPNO predicts $τ_{wave}$, a proxy for vascular decay derived from pressure, with 32% lower error on the log scale than a population baseline while preserving mean arterial pressure accuracy. Because vascular decay and flow drive occupy separate coordinates, counterfactual perturbations produce the expected directional responses in at least 90% of windows in almost all prespecified scenarios, a separation unavailable to pressure-only baselines. The coordinates are also used as inputs to a calibration model for monitored cardiac output. Finally, the formulation identifies the external compliance or flow reference required to recover absolute physical scale.

In this paper, we address the certification of datadriven feedforward control for periodic tracking of unknown nonlinear systems under partial state measurements. To this end, we adopt an invertible neural network (INN) as a surrogate for the unknown system. This choice allows us to bypass solving a nonconvex inversion problem, eliminating the associated inversion errors and reducing tracking error certification to a surrogate modeling problem. We then apply conformal prediction to provide finite-sample probabilistic guarantees on the surrogate modeling error which, through the derived tracking error bound, yield marginal certificates on feedforward tracking error. Finally, we demonstrate the approach on a DC-motor-driven mechanical load with nonlinear friction.

Ptychography neural networks suffer from scaling inconsistencies when generalizing out of distribution, limiting their real world viability. We address this scaling mismatch using a factorization strategy which decouples the learned object texture from measurement scaling, enabling a single trained network to produce measurement-consistent reconstructions across varying illumination conditions. This requires predicting the learned object in real and imaginary units instead of the canonical amplitude and phase representation. We additionally introduce a synthetic object sampling strategy that minimizes phase distribution mismatch between synthetic training data and experimental targets. These improvements yield up to a 5x reduction in Fourier error over the previous PtychoPINN-torch baseline across 5 experimental datasets spanning multiple beamlines and facilities.

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

The wide adoption of Vision Transformers (ViTs) in safety-critical applications raises reliability concerns related to hardware faults. Algorithm-Based Fault Tolerance (ABFT) methods have emerged as lightweight and symmetric protection mechanisms for DNNs. However, they are particularly challenging for ViTs due to their significant computational requirements. This work comprehensively evaluates the reliability of ViTs, emphasizing the need for symmetric protection in their layers. Furthermore, we present CheckOne, a novel, cost-effective method for fault detection and mitigation in ViTs that significantly reduces the computational cost compared to conventional ABFT. Through extensive experiments with multiple ViTs, CheckOne mitigates critical faults by up to $26\times$ and achieves an average 3.8x higher performance than ABFT in ViTs.

Symmetry is central to modern machine learning and physics: invariances and equivariances improve sample efficiency, robustness, and out-of-distribution generalization, while symmetry principles guide scientific modeling. Yet for stochastic dynamical systems the relevant continuous symmetries are rarely known, and symmetry discovery for SDEs has remained essentially unexplored. We introduce \textit{LieStoNet}, an end-to-end, \emph{template-free} framework for discovering Lie-point symmetries of SDEs directly from spatiotemporal trajectories, without prespecifying symmetry groups, templates, or canonical coordinates. Building on the seminal SDE Lie-symmetry theory of Gaeta and Quintero (1999), which formalizes Lie-point SDE symmetries and their relation to Fokker-Planck symmetries, LieStoNet learns neural surrogates for drift and diffusion from increments, then learns projectable generators by enforcing the SDE determining equations, separately regularizing for closure under Lie brackets, adherence to the Lie algebra axioms (bilinearity, antisymmetry, Jacobi), and a non-redundant independent basis. The surrogate also defines an associated Fokker-Planck equation, enabling optional discovery of its Lie-point symmetries in parallel. Across multiple canonical SDEs with known analytic symmetries, LieStoNet recovers generators consistent with the ground-truth symmetry algebra, providing interpretable symmetry discovery for noisy dynamics. Code is available at \href{https://github.com/sumit-sinha-seas/LieStoNet_Final.git}{this link}.

Data augmentations define the invariances learned by self-supervised learning (SSL). Standard augmentation pipelines were designed for natural images, yet scientific imaging modalities are governed by physical measurement processes with distinct symmetry and acquisition constraints. Enforcing invariances that contradict these constraints can distort learned representations and limit downstream performance, but practitioners moving from machine learning into a new scientific modality currently have little guidance beyond transferring natural-image pipelines unexamined. We address this gap with a principled, reproducible procedure for augmentation design in scientific SSL: we formalise the physics-aligned augmentation set as a union of measurement-consistent symmetries and acquisition-driven perturbations, and we give a concrete, largely label-free workflow---enumerate candidates, label each by the measurement operator, validate with representation-geometry diagnostics, and confirm by single-factor ablation---for selecting them. We instantiate the procedure for real-space electron microscopy and reciprocal-space 4D-STEM diffraction, and evaluate it across five SSL paradigms (DINOv2, SimCLR, MAE, VICRegL, I-JEPA) on classification and crystal-orientation regression. Physics-aligned augmentations substantially improve downstream performance for objectives relying on cross-view consistency, reduce geodesic error and improve robustness under realistic acquisition variability (detector gain, resolution loss), and systematically reshape representation geometry. While our experiments use electron microscopy, the procedure is modality-agnostic and applies to other measurement-driven domains such as medical and remote-sensing imaging. These results position augmentation design as a primary, and controllable, source of inductive bias in scientific self-supervised learning.

Over the last decade, neural networks have been applied to an increasingly diverse range of applications, including data with rich geometric, topological, or symmetry-related structure. As a result, researchers have increasingly drawn inspiration from topology, algebra, and geometry. Despite this rich algorithmic development, the supporting software ecosystem remains fragmented. Many important methods exist only as research prototypes in unmaintained repositories. We address this by introducing Topology, Algebra, and Geometry Torch (TAGTorch), an open-source, PyTorch-based library that unifies tools inspired by topology, algebra, and geometry, including data-preprocessing methods, architectures, training techniques, and model analysis tools. We describe the design philosophy of TAGTorch and then discuss its current architecture and capabilities, highlighting areas where it can fill gaps in the current software ecosystem. We conclude with a discussion of our future development priorities for the library.

Large language models (LLMs) are increasingly required to integrate multiple sources of information that may be inconsistent or conflicting. However, there is still a lack of controllable and attributable methods for analyzing how models resolve conflicts between competing specifications. We propose a controlled experimental framework for studying model preferences under conflicting specifications. By constructing specifications with explicit conflicts, the framework enables model choices between competing specifications to be directly observed and analyzed. A symmetry-based design further reduces confounding factors, allowing preferences across representation types to be compared systematically. We evaluate the framework on an executable mathematical benchmark with 550 conflict instances spanning 11 function families, comparing four representation types: pure natural language, formal language, naturalized formal language, and input--output examples. Results show systematic preference patterns rather than random behavior, with a consistent ordering: $\text{Formal} \approx \text{Naturalized Formal} > \text{Pure Natural Language} > \text{Input--Output Examples}$. Example effects further depend on model capability and function family. We extend the framework to heterogeneous specification conflicts in Boolean algebra, code generation, and the clinical domain, demonstrating its applicability across diverse tasks and specification forms. The framework provides a unified approach for measuring how LLMs resolve conflicts between competing sources of information.

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

Deep reinforcement policy learning directly in physical robots (on-robot learning) remains bottlenecked by slow wall-clock training times. We present SymmGrid, a trajectory level augmentation framework inspired by parallelized symmetries that super-scales group transformations to significantly accelerate on-robot learning in both egocentric and exocentric visual setups. We model a Markov Decision Process (MDP) under a symmetry tree, in which state-action pairs have admissible parallelized invariant transformations that yield a geometric grid structure. The state is modelled with ego- or exocentric images and proprioception information. The latter require special treatment, in the form of homographies, to warp visual scenes in line with their corresponding spatial transformations. These parallelized transformations produce a large set of unique symmetric equivalences that populate the replay buffer with diverse and consistent experiences that speed up learning and improve performance. We present extensive training and evaluations performed directly on real robot manipulation contact tasks including peg-insertions, cable routing, and object relocations. Relative to SOTA, SymmGrid achieved wall-clock training convergence speed-ups of 1.37-2.17x, evaluation success rate improvements of 1.09x-1.27x, fastest training convergence times of 16.6, 10.9, and 79.3 minutes respectively. For trajectory wide assessments, we used normalized area under the curve (nAUC) ratios. SymmGrid achieved improvements of up to 2.59x. These results confirm that simple branch symmetries can have an outsized result due to super-scaling and bring us closer to sub-10 minute on-robot learning training in manipulation tasks suitable for arms and humanoids. The project page is available at symmgrid-robot.github.io

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

A gradient-based GNN explainer given a molecule with two chemically equivalent nitro groups assigns them attribution scores that are equal to the last bit. It cannot do otherwise: message passing is exactly permutation equivariant, so any automorphism of the input leaves every attribution invariant. Yet the standard report, the top-k edges, names one of the two, and which one is settled by the order of an array. We show this is a structural obstruction rather than an implementation slip. When no minimal valid explanation is fixed by the input's automorphism group, no rule can be single-valued, minimal and symmetry-respecting at once. For the exact-k reports used in practice we give a parameter-free criterion, mechanised in Lean 4 with no axiom dependencies, that decides from the graph alone whether every score-optimal report of that size must split an orbit. Across 21298 instance-budget decisions the criterion agrees with a mechanical model-equivalence check without exception, and no severing case we found admitted a neutral alternative. The obstruction is common. Nontrivial automorphisms occur in 93.4% of Mutagenicity, the dataset the seminal explainability papers use, so the measure-zero dismissal of symmetric inputs, sound on the continuous domains it was made for, collapses here. At the sparsity budget those papers report, 24.0% of molecules with two interchangeable nitro groups (6 of 25) surface exactly one of them, every one arbitrary under mechanical verification. A model's blindness also manufactures symmetry: every MUTAG molecule contains atoms chemistry separates and the network provably cannot, and a matched control shows the resolution is set by what the model reads rather than how it is parameterised. Reporting orbits removes the arbitrariness at 0.11 ms and 0.43 extra edges per graph.

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

We investigate a normalizing-flow approach for reconstructing parton distribution functions (PDFs) from synthetic matrix-element data. Our framework combines Gaussian Process priors with invertible neural networks to learn a posterior distribution over PDFs consistent with limited Ioffe-time data. We demonstrate that the architecture preserves physical constraints and extrapolation properties.

We formulate a statistical physics framework to model a networked stochastic dynamical system exhibiting bistability, driven by additive noise and social conformity. We apply this model to understand and mitigate AI-induced delusional spiraling-a phenomenon where algorithmic sycophancy from Large Language Models continuously reinforces inaccurate beliefs within a socially interacting society. By partitioning the network into a majority of regular agents and a minority of "aware" nodes (Teachers) placed at topological hubs, we use a degree-weighted mean-field approximation to reduce high-dimensional coupled Langevin equations into a single macroscopic drift equation. We provide a closed-form analytical derivation for the deterministic critical tipping time through a saddle-node bifurcation. We validate this analytical boundary using finite-size scaling and demonstrate a universal data collapse across diverse network topologies. Finally, we optimize an intervention strategy under a strict budget constraint that balances the topological footprint against driving velocity. We prove mathematically that under certain conditions, a highly concentrated, rapid intervention targeting massive hubs strictly outperforms a distributed, slow approach to rescue the network.

Delayed generalization, or grokking, remains poorly understood despite extensive empirical study. We identify an exactly solvable late-time relaxation mechanism for grokking in linear models trained with full-batch heavy-ball optimization and weight decay, together with a locally quadratic extension to nonlinear neural networks. Our analysis reveals a distinguished population-active component of the empirical null space, which we call the grokking subspace. Along this subspace, the training predictions remain unchanged, leaving weight decay as the sole restoring force and giving rise to a slow dissipative relaxation governed by an exact discrete-time and continuous-time law. We show that only this subspace contributes to the slow asymptotic decay of the population risk and derive explicit iteration-scale predictions for the grokking time, recovering the familiar $(1-β)/(ηλ)$ scaling in the weak-regularization regime. The theory further predicts distinct effects of optimizer choice, distinguishing coupled $L_2$ regularization from decoupled weight decay, and yields causal predictions for interventions that modify the grokking component. We verify all theoretical identities without fitted parameters in a synthetic model where every subspace and relaxation rate is computable in closed form. We further observe genuine delayed generalization in modular addition, where the measured delay follows the predicted scaling and the late-time relaxation agrees closely with the theoretical clock.

We prove the identifiability of deep generative models (DGMs) with piecewise-affine (PWA) decoders and Gaussian mixture model (GMM) priors, in a purely unsupervised setting. We introduce three algebraic contrast principles for symmetry breaking: domain contrast, which trivializes the mixture symmetry group; mechanism contrast, which ensures every decoder branch is witnessed by a unique boundary; and interaction contrast, which forbids parameter conspiracies between latent components and decoder branches. Together they exploit the interplay between the discrete combinatorics of the PWA map and the continuous symmetry structure of the latent GMM. Continuity is replaced by algebraic symmetry conditions; injectivity is decoupled from structural identification and required only for pointwise inversion. Our results form a hierarchy: from law identifiability (LID; latent distribution up to a global affine map) through map identifiability (MID; decoder up to the same map) to posterior and pointwise identifiability. The ICA-form ambiguity emerges under conditions on diagonal component covariances. Assumptions are only on the data-generating process, not on learning methods, except for the interaction contrast. To our knowledge this is the first to make algebraic symmetry-breaking the engine of nonlinear identifiability, the first to admit discontinuous decoders, and the first to handle fully non-injective decoders, where every observation admits multiple latent codes.

Point cloud registration critically depends on local features that are both distinctive and robust to arbitrary 3D rotations. Existing learning-based methods typically approximate rotation invariance via fragile local reference frames or extensive data augmentation, providing only empirical invariance and often degrading under unseen rotational transformations. In this paper, we propose SHReg, a strictly rotation-equivariant point cloud registration framework grounded in the representation theory of $SO(3)$. By representing local geometric features as irreducible representations of $SO(3)$, SHReg guarantees exact equivariance under arbitrary rotations without relying on local reference frames. Built upon a spherical-harmonics-based equivariant backbone, SHReg jointly learns rotation-invariant descriptors for robust correspondence matching and rotation-equivariant features that preserve fine-grained orientation information. The preserved equivariant structure enables each correspondence to directly hypothesize a rigid transformation, reducing reliance on large-scale hypothesis sampling in conventional RANSAC-based pipelines and leading to improved robustness under challenging rotational variations. Extensive experiments on 3DMatch, 3DLoMatch, and KITTI demonstrate that SHReg consistently outperforms state-of-the-art methods in registration accuracy, particularly under large rotational perturbations.

This paper introduces a compact reformulation of a modular end-to-end neural network for global minimum-variance portfolio optimization that decouples model complexity from both look-back window length and universe size. A five-parameter hyperbolic weighted moving average combined with a saturating exponential replaces the original 2,400-parameter lag-transformation layer, and a bidirectional gated-recurrent-unit eigencleaning module together with a streamlined marginal-volatility network reduce total learnable parameters from 39,586 to just 2,175. In out-of-sample tests against state-of-the-art nonlinear-shrinkage and risk-parity benchmarks, the compact network attains the lowest realized portfolio variance without compromising expected return. Under long-only constraints, the variance reduction supports substantially higher leverage while maintaining comparable drawdown control. Validation in a high-fidelity trading simulator that incorporates realistic margin-call dynamics confirms enhanced over-leverage resilience. These findings demonstrate that end-to-end variance-minimization architectures can achieve substantial parameter efficiency and robust capital-efficiency gains without sacrificing risk-adjusted performance.

Domain Generalization (DG) aims to learn representations robust to distribution shifts. Recent geometric alignment methods, such as CPCANet, extract domain-invariant structures through batch-wise Common Principal Component Analysis (CPCA). However, CPCANet suffers from rank-deficient covariance estimation due to the small-sample-size issue in mini-batch training. To address this limitation, we propose Projection Pursuit CPCANet (PP-CPCANet), a covariance-free framework that learns a global orthogonal basis on the Stiefel manifold and jointly optimizes it with network parameters via the Cayley transform. We further introduce a symmetry-breaking detached-median PP dispersion objective to extract common principal components (CPCs) with dense and robust optimization signals. Experiments on four DG benchmarks show that PP-CPCANet achieves SOTA performance while maintaining stable training.

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

Transformers are known to have internal continuous symmetries that leave outputs invariant, while modifying quantization. GaugeQuant leverages this in-training by introducing a LogSumExp term to the loss that breaks the symmetries, thus selecting a basis that minimizes activation outliers. A stop-gradient operator ensures that only rotation matrices are updated, yielding the language modeling objective completely unaltered. Our requires no specific calibration data, no quantization simulation, and adds negligible training overhead. With the LLaMA-2 7B model under W4A4 quantization with group size 128, perplexity drops from 8.22 to 6.73, competing with post-training methods that require frozen models and calibration datasets. Under W4A16, perplexity drops from 11.16 to 5.45. Code is available at https://github.com/MPedraBento/gauge-quant.

Overparameterized models often have continuous parameter symmetries, so different parameters define the same predictor. We show that PAC--Bayesian analysis should be performed on the quotient predictor space: pushing a prior and posterior to the quotient preserves the empirical and population Gibbs risks while removing the nonnegative KL contribution caused solely by how the two distributions differ among parameterizations of the same predictor. Quotienting alone does not determine which prior to use. We construct a canonical choice of one parameterization for each predictor and account for the geometric volume of its equivalent parameterizations. This transforms a neutral reference prior into a data-independent prior that reflects the model's implicit bias. It approximates the ideal but inadmissible posterior-matched prior, which would minimize the KL term by depending on the training data. The resulting certificate is tighter exactly when this geometry-induced prior has smaller KL divergence from the learned quotient posterior than the neutral prior. We test this prediction in Fourier regression with a Hadamard parameterization and in Query-Key attention, using ordinary SGD without an explicit regularizer. The implicit-bias prior reduces the mean quotient-space KL by $40.69\%$ and the mean PAC--Bayes certificate by $21.40\%$ in the Fourier-Hadamard experiment. The smaller, prior-scale-dependent improvement in Query-Key attention confirms the predicted conditional nature of the effect.

Global LiDAR-to-BIM initialization must place a robot within an as-designed building model without a prior pose. In repetitive interiors, the principal failure mode is not low-confidence registration but confident aliasing: a submap can match several BIM regions with comparable scores, producing a high-scoring pose displaced by symmetry. We present Move First, Commit Later, a selective layer that treats a registration front-end as an evidence source and decides whether to commit. Candidates from multiple submaps are mapped to a common $\mathrm{SE}(2)$ anchor; a top-1 consensus $B_m$, invariant to non-champion multiplicity, aggregates cross-submap evidence; and topology serves only as a binary feasibility gate. The decision is typed--COMMIT, DEFER, or AMBIGUOUS($τ$), reporting the detected symmetry period--and reversible: symmetry-breaking motion upgrades AMBIGUOUS to COMMIT. On a real multi-room building and a controlled symmetric simulation, the layer commits correctly in every tested trajectory-scale configuration and otherwise abstains with a typed state, whereas forced-choice policies on the same front-end select wrong rooms in most cases. Committed anchors are within 0.02--0.36 m of an independent laser-tracker position reference and within $1.9^\circ$ of a BIM-registration orientation proxy. The evaluation covers one building and one front-end; the layer is designed to be front-end modular.

Most existing watermarking techniques are primarily designed for low-resolution images, with few methods tailored for high-resolution images. Moreover, the embedding capacity is often limited to fixed lengths (e.g., 30, 100, 256 bits, etc.), which struggles to meet practical demands. To address these issues, this paper proposes a high-capacity robust watermarking method for high-resolution images, capable of embedding a watermark of 4 KB (32,768 bits) into images with a resolution of 1024*1024, achieving an embedding rate of 0.0313 bpp. Specifically, this paper adopts a block-wise strategy to effectively embed the watermark into local regions, enabling the network to train and learn normally even under low-resource conditions. The encoder and decoder structures respectively employ a reversible symmetric architecture with three convolutional and three deconvolutional layers, ensuring consistency in the coupling and decoupling of the watermark and image features. Additionally, the loss function combines global and local losses with weighted contributions. By incorporating constraints on the visual quality and robustness of local block regions, the overall imperceptibility and robustness of the image are further enhanced. Extensive experimental results verify that the proposed method is effective and feasible in high-resolution image scenarios with high-capacity watermarking, while demonstrating strong robustness against various noise attacks.

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

In robotic motion planning, equivariance to rigid body transformations is crucial for robust spatial generalization. However, current learning-based planners face a critical dilemma: they either lack inherent equivariance, treating transformed tasks as novel scenarios, or enforce it via computationally expensive specialized architectures that bottleneck real-time inference. To break this trade-off, we propose BridgeFlow, a fast and strictly SE(2)-equivariant generative motion planning framework. Rather than relying on heavy equivariant networks, BridgeFlow achieves exact spatial equivariance via a lightweight task-centric canonicalization module, enabling generalization using standard architectures. To further accelerate inference, we pair a Brownian bridge informative prior with context-aware mini-batch optimal transport. This constructs a straightened vector field that minimizes transport costs and stabilizes training. Furthermore, environmental awareness is explicitly embedded via Classifier-Free Guidance. Evaluations in dense 2D environments and on a 7-DoF Franka manipulator demonstrate that BridgeFlow achieves up to a 15x inference speedup and a 2x higher valid trajectory rate over state-of-the-art diffusion baselines, alongside robust generalization to entirely unseen environments and arbitrary spatial transformations.

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

This paper establishes a direct analogue of the classical Coding Theorem in the setting of symmetry groups. We consider computable bijections on the set of binary strings, called symmetries and define the symmetry prior of a string as the probability that a randomly chosen symmetry from a given group has the string as its unique fixed point. We show that for any fix-retractable symmetry group, a group admitting a computable section that selects an isolating symmetry for every string, the symmetry prior is a universal lower semi-computable semi-measure. In this case, the Geometric Coding Theorem holds. We also develop a Galois connection between subgroups of G and subsets of binary strings, characterizing closed points and maximal closed subgroups, and explore the join-semilattice of dense subgroups. Our results unify algorithmic information theory with group theory and provide a framework for studying symmetry-induced complexity measures. This paper is the first in a series on Computational Algorithmic Statistics (CAS).

We investigate conditional invertible neural networks (cINNs) as probabilistic inverse-dynamics models for multirotor control. For a planar X8 coaxial multicopter, we learn $p(u \mid s_t, c_t)$ from an incremental nonlinear dynamic inversion (INDI) teacher using rational-quadratic spline coupling and invertible linear mixing. Open-loop reproduction reaches $R^2 = 0.944$, mean CRPS 0.0915, and log-probability-error correlation $ρ= -0.60$. Over 15 closed-loop scenarios, position RMSE matches INDI (9.7 vs. 9.5 m), with 47 percent tracking acceptably; failures separate into attitude divergence under aggressive steps and phase lag under high-frequency references, isolating command bandwidth and data coverage as dominant failure mechanisms.

We present cayleyR, an R package for solving permutation puzzles by detecting cycle intersections in Cayley graphs. The core algorithm performs an iterative bidirectional search: from both the initial and target permutation states, random operation sequences generate cycles in the Cayley graph of the symmetric group Sn; their intersection yields a connecting path. When no direct intersection is found, a distance-guided bridge selection narrows the gap, and the process repeats. The package targets the TopSpin(n,k) puzzle, whose state space is a Cayley graph of Sn generated by a cyclic shift and a prefix reversal. We describe the mathematical framework, the algorithm, and its implementation, which combines a C++ hash-indexed state store with optional Vulkan GPU acceleration. The software is publicly available on CRAN.