Sphere

Generative models are increasingly central to many de novo discovery pipelines, in which designs are generated at scale and filtered through virtual screens to determine a set of candidates to experimentally validate. While Bayesian optimization (BO) is a natural fit for this setting, as it uses past evaluations to guide future proposals, the computational overhead required for its sequential decision-making becomes a bottleneck when virtual screens are relatively cheap. We make BO practical in this regime by exploiting the unique combination of a linear model constrained to a spherical domain where high-dimensional latents concentrate. We build off recent work justifying the use of linear surrogates, while deriving nearly closed-form solutions to the surrogate modelling and acquisition problems that exploit spherical symmetry. The result is at least a 100x speedup over state-of-the art baselines, with matching or improved performance across molecular and image generation benchmarks. Altogether, our method makes BO a practical drop-in for de novo pipelines where it was previously too slow to consider.

We propose to learn the unknown dynamics in the proximity of a small body directly from tracking data, representing them as a feed-forward neural network embedded in the system Hamiltonian. The equations of motion form a Neural Hamiltonian Ordinary Differential Equation, whose variational equations provide exact training gradients: estimation uses position and velocity arcs at realistic noise levels, without acceleration or potential labels, and a continual learning approach warm-starts the network as new data are acquired. The known part of the Hamiltonian carries whatever is available, from the central term and spin state to the constant-density model of the imaged shape. We assess the method against a normalized spherical harmonics expansion estimated from identical arcs through the same machinery, on scenarios built on the shapes of Itokawa, 67P, Bennu and Eros. The network remains usable inside the Brillouin sphere: it plans ballistic descents at Itokawa to \SI{4.6}{m} median touchdown error from tracking alone, against 5.1--\SI{48.9}{m} for harmonics of degree 4--12, and to \SI{0.9}{m} with the imaged shape as prior, a configuration that also recovers localised density anomalies invisible to any harmonics degree. The two representations are complementary, and we discuss their combined use across the phases of a small-body mission.

Detecting whether a transmitter is actively communicating with a receiver is a fundamental problem in molecular communications. A spherical receiver may measure not only the number and arrival times of absorbed molecules, but also their absorption locations on the receiver surface. These locations contain a directional signature that is lost in count-only detection. In this letter, we develop a molecular signal detector based on a Legendre polynomial expansion of the receiver-surface hit density and extend it to a Legendre-based Viterbi sequence detector. By exploiting the surface-level directional signature of molecular arrivals, the proposed detectors provide geometry-aware, efficient, and better-performing alternatives to count-only detection.

Spherical observations provide global visual context for 3D scene understanding. However, visual information is encoded in an angular domain, whereas the physical world is represented in Cartesian coordinates. This cross-space representation gap complicates geometric correspondence and semantic evidence aggregation. To delve into this challenge, we introduce Spheriverse, comprising 64,400 temporally aligned spherical image-LiDAR pairs organized into 644 sequences. The dataset spans diverse scenes, illumination, and weather conditions, with fine-grained semantic classes. We further establish benchmarks for semantic occupancy prediction, semantic mapping, and 3D object detection, evaluating 30+ methods through overall and scene-wise comparisons. For dense prediction, we propose SphereOcc, an occupancy framework that couples spherical geometry modeling with semantic evidence retrieval. Cartesian-Spherical Representation Remodeling (CSRR) incorporates spherical range-azimuth geometry into Cartesian voxel features through region-wise modulation. Spherical Evidence Re-querying (SER) then conditions queries on voxel content and range-height-azimuth geometry to adaptively retrieve relevant semantic evidence from source spherical image features. SphereOcc achieves 13.91% mIoU and 24.65% GeoIoU, yielding relative improvements of 13.9% and 9.3% over the respective best-performing methods, TPVFormer and SurroundOcc. It also ranks first in both metrics across all five scene categories, with consistent advantages across the evaluated spatial partitions and reduced fields of view. The established benchmark and source code will be available at https://feit-feiteng.github.io/Spheriverse.

360° salient object detection (SOD) aims to accurately segment salient regions across a full field of view. However, equirectangular projection (ERP) introduces severe spatial distortion when mapping the spherical domain onto a planar representation. Existing methods mainly focus on compensating projection distortion while overlooking the interaction between panoramic geometry and salient object structure during feature perception and prediction refinement. To this end, we propose SphereSOD, an ERP-native framework that couples panoramic geometry with evolving salient structures. Specifically, spherical geometry governs feature sampling and spatial weighting, while coarse-grained saliency and contour prediction influence context aggregation during the progressive decoding process. SphereSOD first initializes deformable sampling based on spherical projection geometry and then employs bounded, content-adaptive offsets, yielding features that are better aligned with the underlying panoramic geometry. Subsequently, the decoder performs structure-guided context aggregation and progressive refinement to recover complete salient regions and accurate boundaries. Extensive experiments on three public 360° SOD benchmarks demonstrate state-of-the-art performance and a favorable accuracy-efficiency trade-off, supporting structurepreserving inference directly in ERP space as a promising alternative to projection-heavy panoramic pipelines.

From horizon detection to fibre structures in X-ray imaging, many vision tasks recover lines via peak detection in Hough space $H=S^1\times\mathbb{R}$, the domain of orientation-offset pairs $(θ,ρ)$. Differentiable pipelines extract coordinates via \emph{soft-argmax}, a probability-weighted average that is only meaningful in a globally linear space. However, $(θ,ρ)$ and $(θ+π,-ρ)$ describe the same undirected line, so $H$ double-covers the space of undirected lines $H/\mathbb{Z}_2$: a Möbius strip, obtained by identifying each pair under $\mathbb{Z}_2$ action. Soft-argmax operates on the cover $H$, but since $H/\mathbb{Z}_2$ admits no linear structure, it tears geometrically adjacent lines apart. Thus we need a $\mathbb{Z}_2$-invariant embedding of lines into a linear space, on which soft-argmax is well-defined. We achieve this by parametrising lines via unit-norm homogeneous vectors $\ell=(1+ρ^2)^{-1/2}(\cosθ,\sinθ,-ρ)^{\top}\in\mathbb{R}^3$ and applying the Veronese map $v_2(\ell)=\ell\ell^{\top}$ that satisfies $v_2(\ell)=v_2(-\ell)$. This descends continuously to an embedding of the quotient $H/\mathbb{Z}_2$ into the linear space $\mathrm{Sym}^2(\mathbb{R}^3)$, where the antipodal ambiguity vanishes. Line extraction becomes a barycentre in $\mathrm{Sym}^2(\mathbb{R}^3)$, projected back via its leading eigenvector. We validate our \emph{Veronese soft-argmax} in a Hough transform-based network across all resolvable lines, confirming uniform and seam-free recovery. We further derive that the $L_2$-loss on isometrically weighted Veronese embeddings equals the squared chordal distance between lines in projective space, enabling a geometrically precise training objective.

Simulation of admittance-type models requires physically consistent dynamic models that are rarely available for off-the-shelf, everyday objects, limiting the fidelity of haptic interfaces that rely on such simulations. This paper presents the first complete workflow for producing physically consistent models of large real-world objects with various constraints and mechanisms, guaranteeing physical consistency of inertia and friction parameters. The workflow separates each object and identifies the handle and body in two stages, requiring no torque sensors at hinges, axles, or other constrained joints. Models are produced for a heavy, closer-actuated door and a wheelbarrow, representing objects of differing constraint types and model complexity. The door is modeled using four-bar linkage kinematics and a fluid dynamics-based lumped parameter model including opening, backcheck, swing, and latch zones. The wheelbarrow is modeled as a rigid body with a spherical wheel and no slip during rolling. Handle estimation RMS errors were below 0.64 N and 0.042 Nm across both objects. Door body estimation had RMS error of 2.19 Nm and wheelbarrow body estimation had RMS error of 6.77 Nm.

We present a theoretical foundation for inverse-distance attention, from its Euclidean prototype (Resolver) to its non-Euclidean realization (Riemann GeoResolver). The Euclidean part establishes three core theorems: (1) circuit separation---IDA achieves exact retrieval with $\mathcal{O}(1)$ resources while softmax requires $Ω((\log n)^2)$ width; (2) a Polyak--Lojasiewicz inequality with $Ω(e^{Δ^2/\sqrt{d}}/Δ^2)$ stronger constant than softmax, implying linear convergence, $\mathcal{O}(\log n)$ Lipschitz scaling under a low-rank/clustering assumption, $Θ(1)$ Hessian spread, and absence of spurious local minima; (3) a width-independent effective rank bound that limits noise memorization---softmax memorizes arbitrary labels when $d_h\ge n$, while IDA limits test error to $\mathcal{O}(η^2)$. The non-Euclidean extension then builds upon this prototype, replacing Euclidean distance with hyperbolic geodesic distance for storage and spherical geodesic distance for routing. The Riemann GeoResolver framework comprises ten integrated modules: four HIDA operators spanning $Θ(n^2)$ to $Θ(1)$ per token; Hyperbolic Curvature Compression (HCC) with provable error bounds; HyperGate with gradient lower-bound theorem; Spherical Inverse Distance Attention (SIDA) with sphere-analog PL inequalities; Dynamic Memory Genesis (DMG) with $\mathcal{O}(\log T)$ regret bounds; and Geodesic Sparse Routing (GSR) with quality and communication bounds. The Euclidean theorems are proved in full; the non-Euclidean extension theorems are proved with analogous arguments. This work establishes a theoretical arc: from Euclidean attention as a special case, to hyperbolic memory, to spherical retrieval.

Spherical Equivalent Refraction (SER) and Axial Length (AL) are core indicators for pediatric myopia screening, yet their measurements require dedicated biometry and cycloplegic refraction. Fundus photography offers an accessible imaging modality, as myopia-related posterior-pole changes are visible in 45$^\circ$ fundus images. However, these cues are often low-contrast, spatially diffuse, and multi-scale. Moreover, AL, Sphere (SPH), and Cylinder (CYL) share partially overlapping but non-identical anatomical correlates. We propose SpecF2M, a spectral-aware multi-task network for estimating AL and SER components from pediatric fundus photographs. SpecF2M integrates a deterministic anatomy-guided enhancement module, a hybrid spatial--spectral backbone combining MixCNN and Hybrid Spectral Learning (HSL) blocks, and an expert-routing head for component-level estimation of AL, SPH, and CYL. On a pediatric cohort of 4,359 eligible child visits and 6,966 fundus images, SpecF2M outperforms controlled CNN/ViT baselines for AL and SPH estimation, achieving MAEs of 0.5347 mm and 0.7062 D, respectively. Component-level analysis further reveals asymmetric task coupling, where CYL exhibits weaker association with fundus-derived myopic patterns than AL/SPH. These results support fundus-based, screening-oriented estimation of pediatric myopia indicators, while external validation remains necessary before deployment.

This paper answers the one-dimensional local root anti-concentration questions posed by Balcan, Pegden, and Sharma in the context of online optimization of piecewise-Lipschitz functions. For a homogeneous feature curve and coefficients whose density relative to the uniform law on a symmetric convex body $K$ is bounded by $A$, we show that the worst-case interval-hitting constant equals $A$ times a section-averaged projective incidence speed. For cube-supported coefficients, this speed is equivalent, up to universal constants, to the projective Lipschitz constant. This yields a sharp, dimension-free characterization and removes the previous $\sqrt N$ loss. For monic degree-$d$ polynomials under arbitrary coefficient laws, we prove that the interval-hitting constant is finite if and only if the ordered real-root laws have bounded densities, with a factor-$d$ comparison that is sharp. Conditional and joint coefficient-space area formulas, together with a two-chart certificate, make this criterion verifiable for dependent and singular coefficient laws. We also give two graph-learning applications that complete the transition-to-regret chain. A cost-sensitive Gaussian-RBF harmonic classifier uses the projective incidence theorem and achieves expected regret $\widetilde O((An^2D e^{BD}/\ell+1)\sqrt T)$. A common-offset polynomial-kernel model uses rigid translation of the ordered roots and achieves $\widetilde O((qn^2κ+1)\sqrt T)$ regret, even when the induced coefficient law is singular in the ambient coefficient space.

AI-generated video (AIGV) detection aims to distinguish real videos from AI-generated ones. In practice, detectors trained on existing data often fail to generalize to newly emerging generative models, making this task challenging. Therefore, continual learning (CL) is essential for improving the adaptability. However, CL frameworks for this task remain underexplored. To this end, we propose SphereVideo, a novel CL framework for AIGV detection built on two key observations. First, real videos exhibit a compact feature distribution. Based on this, we encourage real video features to cluster around a real prototype on a hypersphere while repelling AI-generated samples, thereby establishing a decision boundary. This prototype serves as a stable anchor for CL, regulating boundary evolution and mitigating catastrophic forgetting. Second, existing methods tend to rely solely on spatial artifacts as shortcuts. To enhance temporal modeling, we introduce a strategy that models the temporal dynamics of real data at both frame and clip levels. By strengthening real data modeling, this strategy further facilitates learning a real prototype and forming a stable decision boundary. Moreover, we construct a comprehensive and challenging benchmark. Extensive experiments demonstrate that SphereVideo achieves an improved plasticity-stability trade-off, outperforming prior methods by 3.08% on seen data and 4.00% on unseen AI-generated data.

Event cameras offer microsecond-level temporal resolution and high dynamic range, potentially facilitating motion-blur-free panoramic imaging from fast rotational scanning. Nonetheless, existing optimization-based methods remain computationally demanding, while prior learning-based reconstruction methods are largely designed for perspective imagery and lack geometry-aware support for panoramic outputs. We present E2Pano, a geometry-guided event-to-panorama pipeline with an end-to-end learnable photometric reconstruction stage. Our framework preserves real spherical coordinates from geometric mapping throughout the pipeline, employs a lightweight enhancement module with frequency-domain supervision to bridge the event-image domain gap, and leverages a spherical Transformer with 3D positional embeddings for photometric reconstruction. Experiments on synthetic data and captured rotational scans show improved reconstruction quality and lower photometric reconstruction cost than optimization-based baselines, together with encouraging transfer to real captures under our acquisition protocol despite training purely on synthetic data. Additionally, we construct PanoScan, a dataset with 4,370 synthetic and 30 real-world panoramic scenes paired with event streams. Our dataset and code will be released.

We analyze whether language models of size ~100B have a representation of the night sky map that is decodable from their residual stream. We find that most of the considered open-source models do have such a representation, and it often even surfaces to the top principal components on prompts that ask questions like ``what is close to this object in the night sky''. In all but one model this representation showed significant scores in LOO testing, containing up to 65-85% of variance ($R^2$-score) and having median angular error down to $12^\circ-21^\circ$. We verify that our representation is not a simple leak from a correlated flat representation. To our knowledge, this representation is the first example of a curved high-dimensional irreducible feature manifold. Codes used in the paper are published at https://github.com/l3erdnik/Decodable-sky

Rotary position embeddings (RoPE) modify attention scores through position-dependent rotations, but their effect on normalized token dynamics is not captured by the vanilla spherical self-attention model. We study the continuous-time dynamics obtained when queries and keys are rotated while values remain on the unit sphere. The resulting attention kernel is reversible and admits a sharp uniform softmax floor, yet the natural RoPE interaction energy has derivatives of both signs within one fixed nontrivial system. Every consensus state remains an equilibrium, and its transverse linearization is a reversible Markov operator whose kernel depends on the consensus point through its energy across RoPE planes. On a resonant single-frequency ring we derive an exact Bessel-aliasing spectrum, including non-coprime frequencies and the correct fixed-ring large-$β$ asymptotics. Globally, closed hemispheres are invariant, while pairwise non-obtuse configurations and strict open semicircles contract with explicit half-angle and single-point tail bounds. These regional estimates instantiate a kernel-generic positivity principle with the sharp RoPE softmax floor. RoPE also selects an explicit score-flattening twisted branch; the generic resonant family is non-hyperbolic and linearly unstable, whereas an odd antipodal family becomes a hyperbolic saddle after quotienting global rotation. In multiple dimensions, the local consensus gap can depend non-monotonically on the allocation of energy across frequency planes, so no universal ordering by frequency is valid. Independent matrix, finite-difference, and nonlinear-flow computations cross-check the theorem boundaries and the reported constants.

We study sparse random geometric graphs generated by connecting pairs of high-dimensional vectors whose inner product exceeds a threshold. The latent vectors are sampled either uniformly from the sphere or from a standard Gaussian distribution. Although every edge appears with probability $p$, the edges are dependent through their shared latent vectors. For the spherical model, at the connectivity scale $np=Ω(\log n)$, we prove $\|A-\mathbb E A\|=O\left(\sqrt{np\log n}+npτ\right)$, with high probability, where $τ$ is the cap threshold. This sharpens the spectral norm bound of Liu, Mohanty, Schramm, and Yang (2023) under weaker assumptions. An analogous result holds for the Gaussian model after removing the fluctuations of the vector norms, yielding improved global synchronization guarantees for the homogeneous Kuramoto model. We then recover the latent geometry from the leading eigenspace. When $np\gg\log n$, both the latent vector and relative Gram matrix errors vanish provided $d\ll np\log(1/p)/\log n$. The required lower dimension is only $d\gg\log(1/p)$ for the spherical model and $d\gg\log^2(1/p)\log n$ for the Gaussian model, improving the recovery guarantees of Li and Schramm (2023). Finally, we prove the first exact recovery result for the Gaussian mixture block model of Li and Schramm (2023). At the optimal connectivity scale $np=Ω(\log n)$, a polynomial-time semidefinite program exactly recovers all labels in a moderate-separation regime, whereas larger separation makes exact recovery impossible because isolated vertices appear with high probability. Our proofs combine orthogonal polynomial expansions, decoupling, and matrix concentration, avoiding the trace-moment arguments used in previous work.

Medical image classifiers are often trained within one source population, yet clinical deployment requires robustness to patients whose appearance, acquisition style, and disease prevalence differ from the source cohort. Existing fairness and robustness methods often require group supervision or treat appearance variation as an undifferentiated nuisance, which is insufficient when population-correlated low-level cues and lesion evidence share edge and texture structure. We study a strict source-only cross-population setting, where external populations are unseen during optimization, validation, scheduling, hyperparameter and model selection. We propose RED-Sphere, a plug-and-play robustness framework for image classification under unseen population shifts. It estimates shortcut-sensitive nuisance responses with an edge and feature energy prior, attenuates dominant responses through residual soft gating, regularizes masked nuisance views with counterfactual-inspired consistency and separation losses, and predicts labels with normalized spherical prototypes. It favours angular semantic evidence over source-correlated activation magnitude while preserving lesion structure. Although demonstrated on 2D Scanning Laser Ophthalmoscopy (SLO) fundus classification for Age-Related Macular Degeneration (AMD) and Diabetic Retinopathy (DR), RED-Sphere is not tied to retinal anatomy: the same principle can be adapted with modality-specific nuisance priors wherever appearance shortcuts and semantic evidence are entangled. Under a strict White-only Harvard-FairVision protocol, RED-Sphere improves held-out macro-F1 across all 20 task and backbone comparisons, with average gains of 1.28 and 2.98 F1 points on AMD and DR. Gains in AUC and PR-AUC, visual diagnostics, ablations, and sensitivity analyses further support stronger external semantic alignment and more stable angular disease geometry.

Rolling-shutter cameras introduce characteristic distortions when imaging fast moving objects, and these effects are typically treated as artifacts to be corrected. In this work, we instead leverage rolling-shutter distortions as a valuable source of temporal information to estimate the 3D translational and angular velocities of rapidly moving spherical objects from a single rolling-shutter frame. We design a robust and easily detectable spherical pattern and propose a correspondence-free formulation that recovers motion by enforcing geometric consistency in a back-projection framework. By exploiting the geometry of the sphere, translational and rotational motions are decoupled and estimated through a two-stage optimization process, enabling reliable velocity recovery even for textureless objects. Extensive experiments on both synthetic and real datasets demonstrate accurate and robust estimation of motion parameters under challenging high-speed conditions.

Multi-camera visual-inertial odometry (VIO) overcomes the inherent limitations of pure visual systems by expanding the field of view. However, existing algorithms are typically tailored for fixed camera setups and lack unified compatibility with heterogeneous multi-camera systems. Meanwhile, due to the absence of a unified cross-camera representation and association mechanism, current methods struggle to achieve a balance among robust cross-camera feature tracking, stable depth estimation, and reliable real-time performance. To address these issues, we present Sphere-VIO, a lightweight filter-based VIO framework with unified spherical representation for heterogeneous multi-camera systems. Specifically, we first propose a Unified Spherical Panorama Model (USPM) that supports all standard camera models and enables bidirectional fast mapping between multi-camera images and a shared spherical space without sequential stitching, simplifying cross-camera feature management and improving triangulation efficiency. Second, we design a parallel-accelerated depth-guided semi-direct tracking pipeline, namely Hierarchical Omnidirectional Feature Alignment (HOFA), with global spherical constraints for robust cross-camera matching, and fuse multi-camera depth observations into a standard depth filter for stable initialization. Finally, we develop a multi-camera-adapted ESKF backend that employs spherical bearing residuals and Schur complement marginalization to minimize computational overhead, enabling accurate real-time state estimation on resource-constrained devices. Extensive experiments on public benchmarks and a custom omnidirectional dataset show that Sphere-VIO achieves superior trade-offs between accuracy, robustness, efficiency, and cross-camera generality.

Panoramic images capture the full visual sphere in a frame, offering context unavailable to conventional cameras. Yet this completeness has an unavoidable geometric cost: the 2-sphere cannot be faithfully mapped to the plane, and every projection introduces distortions that challenge standard vision architectures. This survey traces panoramic scene understanding from projection-based adaptation and distortion-aware engineering to sphere-native modeling, reflecting increasing commitment to spherical geometry. Foundation models form a fourth family, geometry-aware tokenization, which adapts the input interface while reusing perspective-pretrained weights. We review these approaches across five task families: dense prediction, unified multi-task understanding, open-world perception, vision-language reasoning, and dynamic video analysis. Across tasks, the same shift toward spherical geometry recurs. In practice, however, the field has converged not on the strongest sphere-native operators, which are exactly rotation-equivariant but cannot reuse perspective-pretrained backbones and thus have not scaled, but on a compatibility-preserving middle ground combining moderate geometric awareness with large pretrained models. This commitment is uneven: deepest in dense prediction and shallowest in dynamic perception, where methods are spatially sphere-aware yet temporally planar. Foundation-model adaptation has advanced panoramic depth fastest, while layout, surface-normal, and video-level understanding remain largely unexplored. No panoramic foundation model has yet been pretrained on spherical data. We identify five evaluation gaps: spherical-area-weighted metrics, seam-consistency tests, polar-robustness stratification, cross-projection generalization, and standardized open-world protocols. We conclude with a six-point roadmap toward general-purpose panoramic intelligence.

Frontier AI training is increasingly shaped by access to dense, centrally controlled accelerator clusters. This creates a structural advantage for hyperscalers and large centralized laboratories, and makes open or independent AI efforts depend on scarce capital, privileged infrastructure, and data-center geography. We present Spheroid BlockTrain, a decentralized training protocol in which a model is partitioned into independently trainable blocks, each optimized on a local objective derived from the same global target and composed at inference into one model. On byte-level WikiText, BlockTrain reaches cross entropy 1.359 (perplexity 3.89), within about 0.04 CE of a same-setup end-to-end Transformer reference, while each active worker trains only one block and avoids full-model optimizer state. A shared six-worker block training run reaches CE 1.385 by averaging same-block updates into one assembled model. HTTP/TCP transport experiments move real serialized checkpoints and updates, including a public-IP three-host run that improves CE from 5.580 to 1.811 while moving 15.22 GB. For inference, the current BlockTrain path uses one block-stack traversal per full output and serves over direct TCP across three public-network GPU hosts up to a 75.80B-parameter logical fp16 shape, outperforming a matched plain-autoregressive TCP pipeline baseline because it emits a full sequence per WAN pipeline traversal rather than one token per traversal.

We present SpheriCity, an expert-grounded conversational prototype designed to support trustworthy knowledge sensemaking from sustainability reports. City-level circularity assessment reports contain rich information about materials, infrastructure, and policy interventions, yet their length and heterogeneous structure make cross-document synthesis and comparison difficult for practitioners and researchers working on circular economy initiatives. While large language models (LLM) promise faster knowledge access and synthesis, their opaque reasoning, hallucinations, and lack of source transparency introduce risks for trust and interpretability, and require verification in high-stakes sustainability contexts. SpheriCity addresses these challenges through a provenance-first conversational agent that foregrounds evidence traceability, structured synthesis, and interaction scaffolds to support exploratory querying and cross-document synthesis across sustainability reports. We conducted a formative expert review with six sustainability experts using representative queries spanning cross-city comparison, policy summarization, and recommendation-oriented tasks. Experts evaluated responses across dimensions and provided qualitative reflections on the system's usefulness for sustainability knowledge work. Our results reveal that transparent sourcing, contextual explanation, interpretability, and alignment with expert workflow strongly shape expert trust and judgments of system usefulness. This work contributes (1) a conversational prototype for sustainability knowledge sensemaking, (2) an expert-grounded evaluation framework for assessing AI responses in high-stakes knowledge domains, and (3) design insights into how provenance, uncertainty communication, and integration in workflow influence expert users' trust in AI assistance for sustainability decision support.

In recent years, list replicability has emerged as a framework for formalizing reproducibility in learning theory. A central question is how the required list size relates to the accuracy parameter and natural complexity measures of the hypothesis class. To achieve sharp bounds on list replicability, we prove a novel topological sphere covering theorem, derived from the Borsuk-Ulam theorem. Specifically, if the $d$-sphere is covered by open sets, each of which lies in an open hemisphere, then $d+1$ of these sets must have a common intersection. Using this result, we obtain a sharp bound on the relationship between list size and accuracy for VC classes. We also show that for large-margin half-spaces, provided the margin is not too large, the optimal list size equals the ambient dimension. However, when the margin is taken to be very large, we devise a replicable algorithm achieving the minimal list size of $\lceil d/2 \rceil + 1$.

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}.

We propose a topological framework for comparing trained Graph Neural Networks (GNNs) by mapping the Stochastic Block Models (SBMs) induced on the graphon-signal space of a Message Passing Neural Network (MPNN) onto the unit $n$-sphere $\sphere^{n-1}\subset\R^n$. The construction rests on three classical pillars: the \emph{compactness} of the cut-distance graphon space $(\Wo,\cutdist)$ \citep{lovasz2006limits,lovasz2012large}, the Frieze--Kannan \emph{weak regularity lemma} together with its graphon-signal extension due to \citet{levie2023graphon}, and the Lipschitz continuity of MPNNs with respect to the cut-distance. We show that, for any prescribed tolerance $\varepsilon>0$, a trained MPNN $Φ$ acting on a sufficiently large graph factors (up to $\varepsilon$) through a step-graphon-signal of bounded complexity, and we construct an explicit measure-preserving map $Ψ_n\colon[0,1]\to\sphere^{n-1}$ that places the SBM regions on disjoint spherical caps. This produces a problem-agnostic, low-dimensional ``fingerprint'' of a trained GNN that is amenable to visual inspection and to nearest-neighbour search across model zoos, enabling \emph{transfer-learning candidate retrieval} without retraining. We discuss the obstruction posed by concentration of measure in high dimension -- a phenomenon directly relevant to LLM-scale embeddings. We close with five concrete future research directions: hyperbolic and Grassmannian alternatives to the spherical model, Gromov--Wasserstein distances on graphon-signals as an isometry-free alternative to the $n$-sphere map, an information-geometric (Fisher) reformulation of the SBM manifold, persistent-homology fingerprints of layer-wise embedding clouds, and a spectral-distance baseline derived from the graphon eigendecomposition.

We consider the black-box optimization problem on a sphere. Two information-geometric optimization flows (IGO flows) are designed with rigorous calculation of natural search gradients based on hyperbolic (information) geometry of Poincar' e and Bergman balls. We demonstrate that ensembles of generalized Kuramoto oscillators on spheres compute natural search gradients and realize IGO algorithms on both manifolds. The relationship between natural gradient policies in Bergman balls and quantum decision making is pointed out.

A fundamental open question in self-supervised learning (SSL) is the explicit characterization of the optimal geometry of the learned representations. Recently, LeJEPA identified isotropic Gaussian embeddings as optimal for minimizing downstream prediction risk in Euclidean spaces. However, the corresponding problem for distributions supported on lower-dimensional manifolds, such as the hypersphere, remains unexplored. In this work, we demonstrate that extending this minimax analysis to smooth distributions on Riemannian manifolds fundamentally changes the optimal solution. We show that, under a worst-case formulation, both k-nearest neighbors and kernel ridge regression induce hyperspherical uniformity. More precisely, we show that uniform distributions on manifolds are optimal for k-nearest neighbors, and that the uniform distribution on the sphere is optimal for kernel ridge regression with both the exponential dot-product kernel and the linear kernel. This theoretical insight reveals a fundamental limitation of Gaussian embeddings: their non-uniform density induces anisotropic k-NN neighborhoods, severely biasing the estimator. To correct this, we introduce SPHERE-JEPA, a theoretically grounded SSL framework. We adapt LeJEPA's Cram{é}r-Wold projection mechanism to enforce hyperspherical uniformity rather than a Gaussian prior. Empirically, SPHERE-JEPA yields significant improvements, boosting texture retrieval mAP by over 6%, while consistently matching or outperforming LeJEPA on standard benchmarks-including a +1.8% linear probing gain on ImageNet-1K (ViT-B/14).

We study autoencoder and variational-autoencoder latent spaces through the lens of spin-glass theory. The paper has two components. First, we formalize a latent-space spin-glass dictionary: for a fixed decoder, the reconstruction term together with a hyperspherical coordinates prior induces a Hamiltonian on the latent sphere, where latent coordinates play the role of continuous spins and the prior acts as an external magnetic field. This allows us to import operational spin-glass diagnostics -- overlap distributions, susceptibility, and block-spin coarse-graining -- to detect ordered, disordered, and edge-of-stability phases in trained latent representations. Second, we show that deliberately driving the latent system toward the edge-of-stability of the topological trivialization regime has concrete downstream consequences. In generation, hyperspherical compression improves the reconstruction-generation trade-off on CIFAR-10 and CelebA64, yielding lower self-FID while preserving or improving reconstruction. In anomaly detection, the same semi-ordered latent geometry improves both fully unsupervised and conditional OOD detection, including real-world Mars Rover and Galaxy Zoo datasets, as well as CIFAR-10/100 and Imagenette-based OOD benchmarks. We therefore advocate a phase-aware evaluation paradigm for AEs/VAEs, in which spin-glass observables complement standard ML metrics and expose the latent regimes that underlie downstream success or failure in many cases.

The generation of immersive and navigable 3D environments is increasingly prevalent with the growing adoption of virtual reality and 3D content. However, recent methods face a fundamental limitation: they cannot produce 3D worlds that simultaneously (i) are navigable over long-range spatial extents and (ii) cover the complete omnidirectional field of view ($360^\circ$ horizontally and $180^\circ$ vertically). To address this challenge, we introduce SphericalDreamer, a method for generating fully immersive and long-range 3D outdoor environments from textual prompts. Our approach is built on the generation of multiple panoramic images, which are subsequently lifted into 3D and fused together while maintaining visual and geometric consistency. SphericalDreamer produces highly detailed, fully immersive 3D environments, while substantially improving scale and navigability compared to prior approaches.

We develop a complete theory of projective cross-ratios in n-dimensional Plane-Based Geometric Algebra (PGA), R(n,0,1), covering geometric objects of every grade: finite and ideal points, hyperplanes, and intermediate flats. For each object type and configuration, we establish an explicit cross-ratio formula, prove that it recovers the appropriate classical invariant, and identify the canonical pairwise measurement operator. A systematic duality analysis further revealed that all eight configurations organize into four dual pairs under the Hodge dual, and that all measurement operators reduce to either the commutator or the commutator dual, depending solely on the geometric configuration rather than on object grade. In each case the formula recovers the appropriate classical invariant: signed distance ratios for parallel configurations and sine cross-ratios for secant ones. These results establish the cross-ratio as a grade-agnostic projective invariant within PGA, and provide a constructive foundation for defining n-dimensional homographies directly from prescribed invariants.

Optimal transport provides a powerful framework for comparing measures while respecting the geometry of their support, but comes with an expensive computational cost, hindering its potential application to real world use cases. On manifolds, convolutional algorithms based on the heat kernel have been proposed to alleviate this cost, but their theoretical properties remain largely unexplored. We establish that the heat kernel cost converges to the optimal transport cost as time vanishes in the balanced and unbalanced cases. In the specific case of the 2-sphere $\mathbb{S}^2$, we ensure that the associated Sinkhorn divergences retains the desirable geometric and analytic properties of classical optimal transport discrepancies. Moreover, we leverage the harmonic structure of the sphere to derive a fast Sinkhorn algorithm, requiring only $\mathcal{O}(n)$ memory and $\mathcal{O}(n^{3/2})$ time per iteration, with fully dense GPU-friendly operations. We validate its computational efficiency on synthetic data, and discuss its potential use in the evaluation of global climate models, providing both spatial and seasonal insights into models performances.

Partial differential equation (PDE) solvers underpin scientific computing, but real-world deployment is bounded by compute. Classical Monte Carlo solvers such as Walk-on-Spheres (WoS) are unbiased and geometry-agnostic but are slow. Learned solvers are fast but biased and brittle under distribution shift. We present \textbf{MC$^2$}, a hybrid WoS-Neural Network (WoS-NN) PDE solver that treats a low-budget Monte Carlo solution as a structured estimator of the true field and learns a single-pass neural correction to recover a high-fidelity solution. MC$^2$ matches the accuracy of solutions using over $1000\times$ more Monte Carlo compute, outperforming all evaluated classical, denoising, and neural-operator baselines. To enable reproducible study of finite-compute PDE solving, we additionally release \textbf{PDEZoo}, the largest standardized elliptic PDE benchmark to date: 2M PDEs spanning five elliptic families and unlimited geometric compositions, with analytic ground truth and multi-budget Monte Carlo trajectories. Together \textbf{MC$^2$} and \textbf{PDEZoo} (1) empirically establish that finite-sample Monte Carlo error is structured, learnable, and correctable in a single forward pass, (2) show that we can solve PDEs $\sim$\textbf{1000x} faster than with just WoS, and (3) provide the evaluation infrastructure the field has so far lacked.

Video anomaly detection (VAD) aims to automatically identify events that deviate from normal patterns in untrimmed surveillance videos. Existing methods universally depend on large-scale annotations or task-specific training procedures, severely limiting their rapid deployment to novel scenes. We observe that intermediate-layer features of pre-trained multimodal large language models (MLLMs) already encode rich anomaly semantics, yet existing approaches rely on the language output pathway and fail to exploit the geometric discriminability latent in these representations. Based on this finding, we propose SphereVAD, a fully training-free, zero-shot VAD framework that recasts anomaly discrimination as von Mises-Fisher (vMF) likelihood-ratio geodesic inference on the unit hypersphere, unleashing latent discriminability through principled geometric reasoning rather than learning new representations. Specifically, SphereVAD first applies Frechet mean centering to unfold feature distributions and eliminate domain biases, then employs Holistic Scene Attention (HSA) to reinforce feature consistency using cross-video priors, and finally performs vMF-guided Spherical Geodesic Pulling (SGP) to align ambiguous segments with directional prototypes on the spherical manifold. This training-free pipeline requires only minimal synthetic images for calibration. SphereVAD establishes new state-of-the-art results among training-free approaches on three major benchmarks and remains competitive with fully supervised baselines. Code will be available upon acceptance.

In deep reinforcement learning (DRL), an agent is trained from a stream of experience. In a continual learning setting, such agents can suffer from plasticity loss: their ability to learn new skills from new experiences diminishes over training. Recently, Mixture-of-Experts (MoE) networks have been reported to enable scaling laws and facilitate the learning of diverse skills. However, in continual reinforcement learning settings, their performance can degenerate as learning proceeds, indicating a loss of plasticity. To address this, building on Neural Tangent Kernel (NTK) theory, we formalize the plasticity loss in MoE policies as a loss of spectral plasticity. We then derive a tractable proxy for spectral plasticity, one expressible in terms of individual expert feature matrices. Leveraging this proxy, we introduce SPHERE, a practical Parseval penalty tailored for MoE-based policies that alleviates the loss of spectral plasticity. On MetaWorld and HumanoidBench, SPHERE improves average success under continual RL by 133% and 50% over an unregularized MoE baseline, while maintaining higher spectral plasticity throughout training.

When deploying reinforcement learning policies to physical robots, actuator rate constraints -- hard limits on how fast each joint can move per control step -- are unavoidable. These limits vary substantially across joints due to differences in motor inertia, power bandwidth, and transmission stiffness, creating pronounced heterogeneity that existing methods fail to handle geometrically: the per-joint feasible region forms a high-dimensional box in action-increment space, yet QP projection and spherical parameterization methods impose isotropic ball-shaped constraints, exponentially under-covering the true feasible set as heterogeneity grows. This paper proposes Dynamic Decoupled Spherical Radial Squashing (DD-SRad), which resolves this mismatch by computing a position-adaptive radius independently for each actuator, achieving tight alignment with the true per-joint feasible region. DD-SRad satisfies per-step hard constraints with probability~1, preserves well-conditioned gradients throughout training, and admits exact policy gradient backpropagation with zero runtime solver overhead. MuJoCo benchmark experiments demonstrate the highest task return at zero constraint violation -- matching the unconstrained upper bound -- with 30%--50% improvement in constraint-space coverage over spherical baselines. High-fidelity IsaacLab simulations with Unitree H1 and G1 humanoid robots confirm end-to-end optimality parameterized directly from official joint specifications, validating a systematic pathway from hardware datasheets to safe deployment.

Organoids are complex, three dimensional, self-organizing cell cultures which manifest organ-like features and represent a powerful platform for studying human disease and developing treatment options. Organoid development is characterized by dynamic morphological and cellular organization, which mimic some aspects of organ development. To study these rapid changes over the course of organoid development, advanced imaging and analytical tools are critical to accurately monitor the trajectory of organoid growth and investigate disease processes. In this work, we focus on computer vision and machine learning techniques to automatically measure the size and shape of developing spheroids derived from pluripotent stem cells (iPSCs), which are typically the starting material for generating organoid cultures. To facilitate this task, we introduce a composite method that combines the Segment Anything Model (SAM), a general-purpose foundation model, with an existing domain-specific tool. This composite method is evaluated together with several existing tools by testing them on organoid image data and comparing with the results of manual image segmentation. We find that no single existing tool is able to segment the test images with sufficient accuracy across all test conditions, but the newly introduced composite method produces consistent and accurate results for all but a very small fraction of the most challenging images. Finally, we compare the accuracy of this method to the variability between manual segmentations by independent annotators (inter-observer variability) and find that by one measure it performs at the level of inter-observer variability and by others it performs very close to it.

Large-scale 3D point clouds can consist of hundreds of millions of points. Even after downsampling, these point clouds are too large for modern 3D neural networks. In order to develop a semantic understanding of the scene, the point clouds are divided into smaller subclouds that can be processed. Typically, this division is done using spherical crops, resulting in a loss of surrounding geometric context. To address this issue, we propose alternative methods that produce subclouds with larger crop sizes while maintaining a similar number of points. Specifically, we compare exponential, Gaussian, and linear cropping methods with the spherical method. We evaluated three 3D deep learning model architectures using multiple indoor and outdoor environment datasets. Our results demonstrate that altering the cropping strategy can enhance model performance, especially for large-scale outdoor scenes, yielding new state-of-the-art results. Code is available at https://github.com/mvg-inatech/point_cloud_cropping

The emergence of deep neural networks capable of revealing high-fidelity scene details from sparse 3D point clouds has raised significant privacy concerns in visual localization involving private maps. Lifting map points to randomly oriented 3D lines is a well-known approach for obstructing undesired recovery of the scene images, but these lines are vulnerable to a density-based attack that can recover the point cloud geometry by observing the neighborhood statistics of lines. With the aim of nullifying this attack, we present a new privacy-preserving scene representation called \emph{sphere cloud}, which is constructed by lifting all points to 3D lines crossing the centroid of the map, resembling points on the unit sphere. Since lines are most dense at the map centroid, the sphere cloud mislead the density-based attack algorithm to incorrectly yield points at the centroid, effectively neutralizing the attack. Nevertheless, this advantage comes at the cost of i) a new type of attack that may directly recover images from this cloud representation and ii) unresolved translation scale for camera pose estimation. To address these issues, we introduce a simple yet effective cloud construction strategy to thwart new attack and propose an efficient localization framework to guide the translation scale by utilizing absolute depth maps acquired from on-device time-of-flight (ToF) sensors. Experimental results on public RGB-D datasets demonstrate sphere cloud achieves competitive privacy-preserving ability and localization runtime while not excessively compensating the pose estimation accuracy compared to other depth-guided localization methods.

Real-world knowledge is often organized as hierarchies such as product taxonomies, medical ontologies, and label trees, yet learning hierarchical representations is challenging due to asymmetric structure and noisy semantics. We introduce Polaris, a polar hyperspherical embedding framework that separates semanticity from hierarchy using angular geometry and radius, enabling the learning of meaning and structure without interference. To map latent representation onto the sphere, we project it to the tangent space at the north pole, apply the exponential map, and learn unit-norm representations using spherical linear layers. Polaris then combines robust local constraints, global regularization that prevents geometric collapse, and uncertainty-aware asymmetric objectives that encourage directional containment. At inference time, Polaris uses structure-guided retrieval to efficiently narrow down candidate parents before final ranking. We evaluate Polaris on different settings of taxonomy expansion - spanning trees, multi-parent DAGs, and multimodal hierarchies, showing consistent improvements of up to ~19 points in top-K retrieval and up to ~60% reduction in mean rank over fourteen strong baselines.

In 2016, Viazovska famously solved the sphere packing problem in dimension $8$, using modular forms to construct a 'magic' function satisfying optimality conditions determined by Cohn and Elkies in 2003. In March 2024, Hariharan and Viazovska launched a project to formalize this solution and related mathematical facts in the Lean Theorem Prover. A significant milestone was achieved in February 2026: the result was formally verified, with the final stages of the verification done by Math, Inc.'s autoformalization model 'Gauss'. We discuss the techniques used to achieve this milestone, reflect on the unique collaboration between humans and Gauss, and discuss project objectives that remain.

IceCube is a cubic-kilometer-scale neutrino detector located at the geographic South Pole. A precise directional reconstruction of IceCube neutrinos is vital for associations with astronomical objects. In this context, we discuss neural posterior estimation of the neutrino direction via a transformer encoder that maps to a normalizing flow on the 2-sphere. It achieves a new state-of-the-art angular resolution for the two main event morphologies in IceCube - tracks and showers - while being significantly faster than traditional B-spline-based likelihood reconstructions. All-sky scans can be performed within seconds rather than hours, and take constant computation time, regardless of whether the posterior extent is arc-minutes or spans the whole sky. We utilize a combination of $C^2$-smooth rational-quadratic splines, scale transformations and rotations to define a novel spherical normalizing-flow distribution whose parameters are predicted as a whole as the output of the transformer encoder. We test several structural choices diverting from the vanilla transformer architecture. In particular, we find dual residual streams, nonlinear QKV projection and a separate class token with its own cross-attention processing to boost test-time performance. The angular resolution for both showers and tracks improves substantially over the whole trained energy range from 100 GeV to 100 PeV. At 100 TeV deposited energy, for example, the median angular resolution improves by a factor of $1.3$ for throughgoing tracks, by a factor of $1.7$ for showers and by a factor of $2.5$ for starting tracks compared to state-of-the art likelihood reconstructions based on B-splines. While previous machine-learning (ML) efforts have managed to obtain competitive shower resolutions, this is the first time an ML-based method outperforms likelihood-based muon reconstructions above 100 GeV.

The Maxey-Riley-Gatignol equations (MaRGE) model the motion of spherical inertial particles in a fluid. They contain the Basset force, an integral term which models history effects due to the formation of wakes and boundary layer effects. This causes the force that acts on a particle to depend on its past trajectory and complicates the numerical solution of MaRGE. Therefore, the Basset force is often neglected, despite substantial evidence that it has both quantitative and qualitative impact on the movement patterns of modelled particles. Using the concept of universal differential equations, we propose an approximation of the history term via neural networks which approximates MaRGE by a system of ordinary differential equations that can be solved with standard numerical solvers like Runge-Kutta methods.

Contrastive learning produces coherent semantic feature embeddings by encouraging positive samples to cluster closely while separating negative samples. However, existing contrastive learning methods lack a principled construction of geometric sets in the semantic feature space with distribution-free guarantees at any user-specified coverage level. We extend conformal prediction to this setting by introducing covering sets equipped with learnable generalized hyper-ball constraints. We propose a method that constructs conformal sets guaranteeing user-specified coverage of positive samples while maximizing negative sample exclusion. We theoretically motivate volume minimization as a proxy for negative exclusion, enabling our approach to operate effectively even when negative pairs are unavailable. The positive inclusion guarantee inherits the distribution-free coverage property of conformal prediction, while negative exclusion is maximized through learned set geometry optimized on a held-out training split. Experiments on simulated and real-world image datasets demonstrate improved inclusion-exclusion trade-offs compared to standard distance-based conformal baselines.

Omnidirectional images are increasingly used in robotics and vision due to their wide field of view. However, extending 3D Gaussian Splatting (3DGS) to panoramic camera models remains challenging, as existing formulations are designed for perspective projections and naive adaptations often introduce distortion and geometric inconsistencies. We present Spherical-GOF, an omnidirectional Gaussian rendering framework built upon Gaussian Opacity Fields (GOF). Unlike projection-based rasterization, Spherical-GOF performs GOF ray sampling directly on the unit sphere in spherical ray space, enabling consistent ray-Gaussian interactions for panoramic rendering. To make the spherical ray casting efficient and robust, we derive a conservative spherical bounding rule for fast ray-Gaussian culling and introduce a spherical filtering scheme that adapts Gaussian footprints to distortion-varying panoramic pixel sampling. Extensive experiments on standard panoramic benchmarks (OmniBlender and OmniPhotos) demonstrate competitive photometric quality and substantially improved geometric consistency. Compared with the strongest baseline, Spherical-GOF reduces depth reprojection error by 57% and improves cycle inlier ratio by 21%. Qualitative results show cleaner depth and more coherent normal maps, with strong robustness to global panorama rotations. We further validate generalization on OmniRob, a real-world robotic omnidirectional dataset introduced in this work, featuring UAV and quadruped platforms. The source code and the OmniRob dataset will be released at https://github.com/1170632760/Spherical-GOF.

We establish two related but logically distinct results for shallow ReLU$^k$ neural networks on the unit sphere $\SS^d$. First, for an arbitrary set of inner neural-network parameters, the best $\mathcal{L}^2(\SS^d)$ approximation of a fixed target function with smoothness $r>\tfrac{d+2k+1}{2}$ admits an asymptotic lower bound given by a constant multiple of $n^{-1/2}\underline{h}^{k+1/2}$, where $\underline{h}$ denotes the antipodal separation distance of the normalized inner-parameter set. This lower bound depends explicitly on the parameter configuration through $\underline{h}$ and applies without additional assumptions on the parameters. Second, for antipodally quasi-uniform parameters, $\underline{h}\simeq n^{-1/d}$, and the lower bound establishes the exact saturation order $n^{-\frac{d+2k+1}{2d}}$ for such parameter families: a target function with regularity greater than $\frac{d+2k+1}{2}$ and satisfying the required parity condition can be approximated at this rate, whereas approximation at any strictly faster rate forces the target function to be zero. Our results therefore place linearized neural-network approximation within the classical saturation framework and show that, although ReLU$^k$ network spaces can outperform finite elements of the same degree, this advantage is intrinsically limited.