Representational Geometry

Render--match--PnP relocalization establishes correspondences between query image pixels and 3D map points for camera pose recovery, but their potential to support dense depth estimation is often overlooked. To exploit this geometric information, we present RIDE, which estimates dense metric depth from a robot's RGB stream. Given a metrically scaled 3D Gaussian Splatting (3DGS) model, RIDE combines sparse metric depth observations derived from PnP-RANSAC inlier correspondences with the geometric prior of a pretrained video-depth model. To handle uneven and intermittent observations, it integrates global and local depth correction with temporal memory, supporting depth estimation through short observation gaps after metric scale initialization. Trained on public RGB-D videos, RIDE is evaluated on robot sequences without fine tuning. Experiments show improved depth accuracy and temporal consistency over scale-only calibration, demonstrating how localization geometry can support both pose recovery and dense robot perception.

Autonomous 3D active mapping requires a space robot to choose where to sense while building the geometry needed for navigation. Learned occupancy completion extends spatial context beyond the current field of view, but one predicted map often serves two planning roles: it scores expected surface gain and constrains collision-free motion. Unsupported occupancy can therefore distort both where the robot looks and where it believes it can travel. We study this coupled interface in a controlled closed-loop benchmark by holding the active-mapping system fixed and varying only its planner-facing occupancy across observation-only, learned, oracle-corrected, and ground-truth conditions. Improving occupancy accuracy does not monotonically improve closed-loop coverage: across 25 starts, planning with ground-truth occupancy reaches 70% of the learned baseline's final coverage 12.7 steps earlier on average, while increasing final coverage by only 0.031. Guided by this diagnosis, we introduce an observation-gated filter that retains completion in insufficiently observed regions and suppresses predictions only after repeated frustum exposure without nearby RGB-D support. The filter improves both targeted failure-prone starts without retraining or ground truth. These results motivate online revision of planner-facing geometry during autonomous intervals between communication windows. The current study assumes benchmark RGB-D observations and sufficiently accurate pose estimates; planetary sensing conditions and accumulated localization drift remain to be evaluated.

Natural-language Computer-Aided Design (CAD) code generation aims to turn design intent into executable and editable parametric programs. Large language models (LLMs) make this goal increasingly practical, but useful systems must preserve the construction process behind the rendered geometry. Existing benchmarks and methods mostly focus on how closely the generated CAD model matches the reference geometry, often using metrics such as Intersection over Union (IoU). Such metrics can miss errors in part decomposition, construction hierarchy, Boolean operations, sketch structure, and geometric relations. This gap calls for a representation that makes design intent explicit and lets a system check generated code against that intent. We propose CIT-CAD, a framework that infers a Constraint Intent Tree (CIT) from the input description to represent the intended entities, hierarchy, operations, and relations. The tree has two roles: it guides CAD code generation and defines expected constraints for verification. The framework extracts actual constraints from the generated program, compares them with the expected constraints, and uses mismatches to localize and repair design violations. Experiments show that the framework improves CAD generation performance, with larger gains on more complex multi-entity designs. By turning design intent into an explicit and checkable object, this work is the first attempt to move text-to-CAD generation beyond rendered-geometry matching toward construction-aware synthesis, verification, and repair.

Existing theories derive neural scaling from data geometry or a specified data-model spectrum, but systems trained on the same data can scale differently when architecture or optimization changes the representations they can efficiently reach. We introduce Coupled Scaling, a task-conditioned framework in which finite-budget scaling depends on the relation between task structure and the geometry accessible to an architecture-optimization system. In a solvable mode-truncation model, loss separates into target energy outside architectural support and an unresolved supported tail. For an arbitrary priority order, the residual lies between the best-N supported tail and the tail beyond the largest completed high-value prefix. If the cumulative-tail and coverage log-rates are $γ_{A,T}$ and $ρ_{A,O,T}$, the residual exponent lies in $[ρ_{A,O,T}γ_{A,T},γ_{A,T}]$. Under bounded off-prefix gain, the completed prefix is rate-determining and $α_{A,O,T}=ρ_{A,O,T}γ_{A,T}$; for $a_{A,T,j}\asymp j^{-b_{A,T}}$, this gives $α_{A,O,T}=ρ_{A,O,T}(b_{A,T}-1)$. A fixed-kernel specialization derives the training-time exponent from the near-zero tail of a task-weighted spectral measure defined independently of the loss fit. The framework separates architectural support from finite-budget acquisition and motivates two tests: static task-relevant geometry should track loss at a common budget, while multiscale geometry should track coupling-specific exponent ordering, including reversal across contrasting tasks. An audit of released emergence trajectories identifies the controls needed for a direct factorial test that measures geometry separately from the scaling fit.

Pretrained vision embeddings are increasingly used as general-purpose representations for modelling how people appraise urban scenes, and are validated almost entirely by how well they predict human ratings. High predictive accuracy does not establish that these embeddings organise scenes as human perception does. We test the two properties separately against brain data. Using openly released EEG from 63 adults who viewed and rated 56 Berlin street scenes, we estimate the representational geometry of the scenes over time, the proportion of that geometry that is explainable at all, and its correspondence with seventeen feature spaces spanning language-supervised, self-supervised, category-supervised and dense-prediction training, two orders of magnitude of scale, and interpretable controls. Correspondence is low throughout: the best representation, DINOv2 ViT-B, reaches 29.6% of the lower bound of the noise ceiling, the panel spans 11.0% to 29.6%, and a Gabor energy descriptor is indistinguishable from the best model while outperforming every language-supervised model tested. Within a model, deeper layers still match later neural responses, so the hierarchical correspondence found for object recognition survives even at this low overall level. The same embeddings predict held-out appraisal ratings well, up to r = 0.87, and the two measures do not track each other across models; reweighting features towards the neural geometry lowers appraisal prediction for every model tested, against a control of matched dimensionality. Predicting how a street is appraised is therefore weak evidence that a model represents the street as the brain does. The benchmark uses only public data and requires no training, so evaluating a new representation needs only its embeddings for 55 images.

Traversing sparse 3D structures requires humanoid robots to perceive thin, overhanging geometry while executing agile, accurate whole-body motions. We study this problem through monkey-bar traversal, where the robot must jump to the structure, traverse it through sparse bar interactions, and land safely. For this task, we present a reinforcement-learning-based perceptive control system that operates directly on observations from a head-mounted solid-state lidar. To extract task-relevant geometry from the sparse returns, the policy consumes the raw lidar scan through an attention-based encoder with recurrent memory. This policy is obtained by a phase-scheduled teacher- student pipeline that combines privileged experts for jumping up, brachiating, and jumping down. For transfer to hardware, we model lidar noise, battery-voltage sag, and actuator thermal limits, and equip the humanoid with passive hook end-effectors for robust bar interaction. On hardware, the resulting policy completes the full jump-up->brachiation->jump-down sequence in 14 of 15 trials across three bar configurations and reaches brachiation speeds up to 0.5 m/s. Beyond brachiation, the same perception backbone supports a separately trained policy that ducks beneath thin overhead obstacles with 2 cm cross-sections.

Protein structure modeling rests on a single computational primitive: the interaction between what a residue is (sequence content) and where it sits (three-dimensional geometry). What is the expressive limit of this layer class? We show that the complete bilinear operator over content-geometry outer products--the sufficient statistic of all second-order interactions--is the expressive ceiling, while the additive message passing of mainstream geometric GNNs is provably blind to content-geometry binding. We then introduce Hyper-Fold, a rank-K separable convolutional backbone approaching this ceiling at message-passing cost: each radius neighborhood is organized into a sequence hyperedge and a contact hyperedge, modulated by an edge-conditioned matrix-valued operator factorized into K learned basis operators with geometry-generated coefficients. Across enzyme function prediction, fold classification, and ligand binding site detection, Hyper-Fold and its hierarchical variant Hyper-Fold-Deep achieve the best results among protein-specific structure encoders; Hyper-Fold-Pocket, an anchored set-prediction head, surpasses UniSite-3D on UniSite-DS and two zero-shot benchmarks with no sequence language model features, 68x fewer parameters, and 4.8x lower latency--suggesting that a sufficiently expressive 3D backbone recovers information that fusion architectures previously borrowed from evolution-scale pretraining.

Muon motivates designing optimizer geometry around the function of each parameter block and uses the spectral norm for hidden linear layers. For the language-model head, the spectral norm is not a faithful measure of functional change. Softmax removes shared logit shifts, whereas the spectral norm can assign arbitrarily large size to updates that change no output probability. We therefore treat the LM head and softmax as one module and derive an update geometry for their composition. Hilbert's projective distance respects this invariance as it measures the largest change in pairwise log odds. For an update $S$ with token rows $s_i^\top$, we show that the largest Hilbert distance over $\left\lVert h\right\rVert_2\leq H$ is exactly $H D(S)$, where $D(S)=\max_{i<j}\left\lVert s_i - s_j\right\rVert_2$ is the Euclidean row diameter. This diameter replaces the spectral norm in the resulting Muon-style steepest descent problem. An exact solution is possible, but its direct formulation contains one $d$-dimensional vector variable for every token pair. For a vocabulary size of approximately $50$k, this means more than one billion token pairs, making the calculation impractical at every training step. We instead impose a stronger common-ball constraint and derive projected RowNorm as an $O(Vd)$ solution. For the exact RowNorm oracle, we prove that its first-order decrease is at least $1/\sqrt{2}$ of the exact diameter-constrained optimum. With Muon on the backbone, experiments across three seeds at 190M, 380M, and 640M parameters show that RowNorm reduces mean final step diameters and empirical Hilbert RMS perturbations by factors of $45$--$60$ and $12$--$15$, respectively, with only a $0.0057$--$0.0153$ increase in mean final validation loss.

4D understanding and reasoning is a fundamental capability for embodied AI agents operating in dynamic physical environments. However, existing vision encoders are largely limited to static 2D images or 3D point clouds without temporal modeling, or to 2D videos that lack accurate geometric depth reasoning. Consequently, current approaches fail to jointly capture spatial structure and motion evolution in dynamic scenes. We present CL4D, the first foundational 4D vision encoder that directly operates on dynamic point clouds, trained with a contrastive learning objective to align spatio-temporal geometric representations with natural language descriptions. By learning a shared embedding space between text and 4D scene dynamics, CL4D enables zero-shot motion-to-text and text-to-motion retrieval in dynamic environments and serves as a foundational 4D vision encoder for downstream 4D vision-language tasks. Building on this encoder, we introduce 4DVLM, a 4D vision-language model that conditions language generation on dynamic geometric representations. 4DVLM is the first VLM designed to operate directly on 4D point clouds without relying on 2D images, 2D videos, or static 3D point clouds. We train CL4D and subsequently 4DVLM on a newly constructed dataset termed DynAction4D capturing diverse human motions across varying object interactions and scene environments. Extensive experiments across multiple 4D human action benchmarks demonstrate that CL4D achieves state-of-the-art performance, with improvements of approximately ~16.75% over prior methods. Furthermore, 4DVLM outperforms frontier video VLMs such as Gemini and GPT-5 even when these models are provided with RGB video sequences corresponding to the same scenes represented as 4D point clouds for 4DVLM.

We study masking diffusion for discrete sampling and introduce a path-resolved measure of data geometry called the \emph{unmasking growth complexity} ({\textsf{UGC}\xspace}). Its local increments directly control Kullback--Leibler (KL) discretization error, yielding a unified analysis of Bernoulli-subset and fixed-cardinality unmasking schemes. In log-reveal-odds coordinates, this structure yields optimized single-block and multi-block schedules, and quantifies the gains from adapting computational effort to data geometry. Crucially, we show how {\textsf{UGC}\xspace} increments can be estimated from samples via KL increments along coupled reveal trajectories. This leads to \emph{certified-optimal} samplers that achieve a prescribed KL error with high probability and iteration complexity within a constant factor of the corresponding oracle procedure. Collapsing the \ugc path yields the aggregate {\textsf{UGC}\xspace} mass, which connects to classical multivariate dependence measures and complexity measures from previous analyses of discrete diffusion. In the fine-partition limit, the squared integral of the square-root {\textsf{UGC}\xspace} density determines the sharp leading-order optimal Euler discretization error. Examples exhibit substantial dimension-dependent gains over coarse schedules, including $\widetildeΩ(\sqrt{d})$ improvements achievable with a constant number of adaptively placed blocks.

Electronic health-record interoperability is a boundary problem: legacy systems, generative models, terminology services, identity systems, and human reviewers may each expose rich internal states, while operational exchange requires a narrow shared interface of typed claims, bounded uncertainty, provenance, and explicit admission or abstention. This paper details a mathematical and engineering architecture for that interface. The organizing idea is the logit boundary: a discovery model may propose pre-threshold scores over a local categorical decision, but a deterministic judgment substrate decides whether the proposal is admissible, requires review, or must be quarantined before any Fast Healthcare Interoperability Resources (FHIR) transaction is constructed. The resulting Geometric Belief Interface (GBI) combines finite boundary semantics, local Dirichlet evidence, cellular-sheaf and mapping-cone diagnostics, advisory geometric audit charts, and a Decentralized Cryptographic Sheaf-Enclave (DCSE) protocol sketch for fail-closed deployment. The framework does not establish clinical truth, global representation alignment, or end-to-end safety; it defines certificate-producing checks at a model-to-system boundary. A companion frozen synthetic benchmark, GBI BoundaryBench v0.1, evaluated Qwen3-4B-Instruct-2507 on 256 held-out tasks across three evidence modes (768 canonical executions). All executions completed, but none produced an output accepted by the benchmark contract: 369 were rejected during safe parsing and 399 during schema validation, yielding zero coverage and deterministic quarantine. This empirical result is deliberately narrow - one 4B open-weight model under one frozen interface - and is reported as evidence about the admission boundary, not as a general claim about LLM capability or clinical safety. A Julia appendix verifies numerical certificates using standard libraries.

Long-horizon future-frame prediction is important for autonomous driving, traffic surveillance, and intelligent transportation systems, yet remains challenging due to temporal ghosting, geometry drift, and inconsistent object motion. Recent latent video diffusion models have achieved impressive visual quality, but directly applying them to structured traffic scenes often leads to unstable geometry and degraded temporal coherence over extended horizons. We present a training-free inference framework that stabilizes reliable static structure in pretrained video predictions through multi-frame temporal context and view-conditioned routing. For front-camera videos, our method refines generated futures with a multi-frame depth-layered renderer that projects static geometry from observed history frames while preserving dynamic regions from the generative base model. For heterogeneous traffic views, a frozen vision-language model infers a coarse camera group from the observed clip and selects a specialized motion-based predictor. The framework requires neither retraining nor fine-tuning of the underlying video model and can be applied directly to pretrained generators. We validate the proposed framework on the AI City Challenge Track 5 benchmark, where our final system achieves competitive performance among the top-ranked teams. These results demonstrate that geometry-aware inference-time refinement and view-conditioned hybrid inference can improve static-geometry stability and low-level structural fidelity without changing the original model architecture.

Long-context LLM inference is bottlenecked by quadratic attention computation and growing KV-cache costs. Existing sparse attention and KV-compression methods typically decide which tokens or heads to preserve from runtime attention scores, observation windows, calibration prompts, or learned gates, making head diagnosis input-dependent and costly to deploy. We propose Autonomy-of-Heads (AoH), a data-free method that identifies retrieval and streaming heads from the spectral geometry of query-key projections. AoH defines the kernel attention operator $M_h = W_K^{h\top}W_Q^h$ and uses its effective-rank as a weight-space measure of head function: concentrated spectra indicate a small number of dominant query-key matching directions and are associated with retrieval heads, whereas diffuse spectra indicate the absence of a dominant global matching direction and are associated with streaming heads. We further derive an efficient $d_\text{head}$-dimensional computation that avoids constructing the full $d_\text{model}\times d_\text{model}$ matrix. We conducted extensive experiments across models demonstrating that at 50% sparsity, AoH retains 96.5% of Full Attention performance on average while reducing prefill and decode latency by up to 41.4% and 66.0%, respectively, and KV-cache memory by 50.0% at 256K tokens.

Dense colored LiDAR maps provide accurate city-scale geometry, but lifting them into 3D Gaussian Splatting (3DGS) retains millions of primitives, making the resulting models costly to store, transmit, render, and adapt. Aggressive primitive reduction alleviates this burden, but can remove the local surface support needed for stable novel-view synthesis and downstream geometric use. We introduce G$^2$ARD-GS, a geometry-guided distillation method that converts a dense Gaussian prior instantiated either as a training-free point-cloud lift or a trained GS model into a compact, reusable representation. G$^2$ARD-GS progressively consolidates the prior into surface-aware representatives, then recovers appearance on the resulting fixed topology under construction-time anchor constraints, with no primitives added or removed during recovery. Under limited supervision, geometry-aware view selection allocates the available view budget. On MatrixCity, G$^2$ARD-GS achieves the best PSNR, SSIM, and LPIPS across matched $5\times$--$30\times$ compression budgets, outperforming PUP by $3.2$--$6.8$,dB in PSNR. When reused as frozen geometry, the compact model improves off-trajectory appearance adaptation by $3.7$--$4.9$,dB over PUP 3D-GS and preserves image-to-model registration accuracy on Cambridge KingsCollege at $30\times$ compression. Project page: https://patrick1159.github.io/gardGS-page/.

Resistive pressure arrays are the cheapest and most widely shipped tactile sensors, yet tactile representation learning has concentrated on optical sensors that image a deforming gel. We present Tactus, an open model that answers text queries from pressure data alone: on the STAG benchmark (27 objects, held-out recordings), it reaches 0.771 +/- 0.062 top-1 over four runs (top-3 0.935), matching, and at best exceeding, the dataset's supervised closed-set CNN at 0.76, with no trained classifier head. The recipe is small-data: 187 training recordings, masked-autoencoder pretraining on 144k unlabeled same-sensor frames, and the sensor's own calibration affine, which recovered more accuracy than every architecture change combined. The released model's errors concentrate in a few contact-ambiguous classes, are uncorrelated with text-target geometry (Spearman rho <= 0.05 over 702 class pairs), and survive paraphrased and even bare-name queries within one point; two diverse frames recover 89% of eight-frame accuracy. Failures are reported with equal precision: cross-sensor pretraining pooling gave no gain, vision co-training degraded touch, and a mis-normalized input pipeline silently discarded 97% of the sensor's dynamic range while producing plausible intermediate results. Weights, code, and the memory layer the model plugs into are released openly.

Post-training can substantially alter language-model behavior, yet aggregate behavior rates do not reveal whether training removes an existing mechanism, creates a new one, or changes how an inherited mechanism is used. We study this question through two mechanistically distinct failures, repetition as a decoding-attractor pathology and sycophancy as a preference-related alignment failure. We introduce behavioral manifold analysis, which isolates behavior-specific geometry by selecting sparse behavior-associated coordinates and lifting them into low-dimensional local charts. We construct these charts in two complementary spaces. ACT captures runtime activation states, while NOC quantifies how strongly the model routes functional information flow through the shared behavior-associated subspace. Across multiple model families, the resulting charts are highly compressed and partially alignable across architectures. Contribution-space charts expose a more architecture-robust shared core, whereas activation-space charts retain stronger family-specific structure. Tracking these charts through controlled post-training reveals a consistent asymmetry. Supervised fine-tuning substantially alters the inherited behavioral geometry, whereas reward optimization changes behavior while largely preserving the underlying chart. This geometric perspective provides a unified framework for understanding the mechanistic distinction between the two objectives. SFT tends to rewrite behavioral geometry, whereas reward optimization primarily reweights it. Code is available at https://github.com/ronglingze/Manifold-Analysis

Affordance-guided imitation learning has shown impressive performance in robotic manipulation tasks by compressing visual perception into task-specific geometric constraints (e.g., fixed contact points). However, the commonly used static affordances can become inconsistent in precision-critical tasks or under object location perturbations, leading to post-contact trajectory drift. To address this issue, we propose AffordTrajDP, a dynamic framework that constructs affordance trajectories via object-centric temporal propagation to guide the progressive manipulation process. Specifically, given an RGB-D observation, our core insight is that a retrieved anchor affordance, which captures the desired contact point between the end-effector and the target object, can be propagated forward via affordance propagation, using the object's SE(3) pose as a natural propagation medium, to yield an affordance trajectory that provides temporally consistent, state-aware guidance throughout execution. AffordTrajDP achieves 70.0% average success rate on ManiSkill3, outperforming strong baselines by up to 17.8%. Real-world experiments on Galaxea A1 and UR7e robotic arms, covering StackCube, PickCup, AdapterInsertion, Ring-on-Peg, Put-in-Bowl, and USB Insertion, further validate robustness under object placement variations and appearance changes, with seen and unseen object instances evaluated on Galaxea A1, and ablations confirm the contribution of each proposed component.

Preference optimization improves mathematical reasoning in large language models (LLMs), but reliable chosen-rejected pairs usually require verified answers, human annotations, or external reward models. We investigate whether preference supervision can instead be derived from the model's internal representation geometry in a semi-supervised setting. Our analysis shows that reasoning trajectories generated across different mathematical problems form structured global point clouds in which correct and incorrect trajectories exhibit different geometric organization. Based on this observation, we propose Cloud--ScPO, a topology-guided preference-mining framework that uses a small labeled set to construct multiple correct and incorrect reference Clouds. Each trajectory is represented by a mean-pooled hidden state and scored against connectivity-induced components using a component-level soft $k$-nearest-neighbor measure averaged across reference banks. We combine this cross-problem Cloud signal with prompt-level self-consistency: self-consistency determines the answer-level preference direction, while Cloud scoring selects concrete trajectories and filters pairs by their score margin. Experiments on GSM8K and MATH-Numeric across four model settings show that Cloud--ScPO consistently improves over ScPO, with gains of up to 4.49% on GSM8K and 4.19% on MATH-Numeric. Pair-level analyses further show that Cloud--ScPO maintains comparable correctness reliability while more effectively separating informative chosen trajectories from incomplete, repetitive, or otherwise low-quality rejected responses.

Point clouds are one of the most fundamental and widely used 3D representations, serving as the most basic geometric representation of 3D shapes. Nevertheless, most existing 3D printing pipelines require a watertight mesh as input, preventing the direct use of point clouds for fabrication. A common workaround is to reconstruct meshes from point clouds; however, the resulting meshes often contain geometric artifacts, such as incorrect faces or topological inconsistencies, that are difficult to repair and may lead to printing failures. To overcome these limitations, we propose PrintAnything, a novel framework that learns to produce executable 3D printing G-code directly from 3D point clouds without requiring mesh reconstruction. To enable point clouds to serve as direct input for slice-wise toolpath generation, we introduce a slice-wise point projection strategy that transforms unstructured 3D point clouds into slice-aligned 2D representations consistent with layer-by-layer nature of fused deposition modeling in 3D printing. To eliminate mesh dependency and provide a unified representation that bridges point clouds and G-code, we propose Geometric plan (G-plan) map, a compact 2D representation composed of occupancy, region, and flow maps that encode the geometric and extrusion properties required for toolpath synthesis in 3D printing. As a result, our proposed method accurately generates printable G-code directly from point clouds, enabling a practical and fully mesh-free pipeline for 3D printing. The code is publicly available at \href{https://github.com/Sangminhong/PrintAnything}{https://github.com/Sangminhong/PrintAnything}.

Reconstructing articulated objects with multiple movable parts is essential for understanding object structure and enabling physical interaction. However, this reconstruction task poses significant challenges due to the entanglement of geometry, appearance, and motion parameters during optimization. Existing methods rely primarily on photometric supervision, which commonly fails to disentangle these interdependent components, resulting in poor part decomposition with blurred boundaries and geometric artifacts. To address this limitation, we introduce StructureGS, a reconstruction framework for articulated objects that integrates structure-aware guidance into 3D Gaussian Splatting. Our approach leverages oriented bounding boxes of object parts to enforce two key structural properties: spatial coherence, which constrains each part's geometry to remain compact and spatially coherent within its designated region, and structural connectivity, which enforces physically plausible contact relationships between adjacent parts. These properties are realized through structure-aware losses that inject explicit structural constraints into the optimization process. Extensive experiments demonstrate that our method achieves state-of-the-art performance in articulated object reconstruction, producing high-quality results with well-defined part geometries.

Two central challenges in diffusion-based sampling are the theoretical one of understanding their remarkable effectiveness even in high-dimensional settings, and the practical one of designing algorithms with certified performance guarantees. We show that these questions are intimately connected via the \emph{denoising growth complexity} ($\mathsf{DGC}$). It is a geometric measure defined by a log-time weighted integral of the derivative of the denoising mean-squared error along the Gaussian heat flow. We show how the $\mathsf{DGC}$ increments lead to a simple and explicit bound on the KL error of an Euler scheme applied to the stochastic innovations representation. The bound is local along the path: each step is controlled by the corresponding $\mathsf{DGC}$ increment and its relative stepsize. This structure allows us to derive KL sampling guarantees for optimized stepsize schedules, both in a simpler single-block setting and in a more refined $K$-block setting. The $\mathsf{DGC}$ function has a natural martingale structure, which we exploit to develop fully data-certified versions of these algorithms. It also admits information-theoretic upper bounds in terms of covariance, rate distortion, metric entropy, and the Poincar'e constant, thereby recovering and sharpening a range of existing diffusion-sampling guarantees, as well as giving new results. In log heat-time, the fine partition limit is governed by an integral involving the square root of the $\mathsf{DGC}$ density, whereas a single-block schedule depends on its ordinary integral. This comparison precisely characterizes when adaptation to data geometry yields substantial computational gains, including logarithmic-to-constant separations for simple Gaussian mixture models.

Large language models place structured concepts on geometrically faithful manifolds: weekdays lie on a circle, months on another, usually taken to be a fixed world-model the network stores and looks up. We show that context is king: the structure a model actually uses is set by the in-context specification. A declarative rule fixes not only which relations the geometry encodes but its topology type: the same tokens form a cycle or a branching tree on command, built even on arbitrary, meaning-free tokens with no prior to inherit, which a relabeled stored shape cannot do. When the specification conflicts with a strong pretrained prior, the context-set geometry dominates it in capable models, read from the same activations (representational similarity 0.6--0.9 to the imposed structure versus near-zero to the prior), across the priors we test and both families we study (Gemma, Qwen). Activation patching shows the map is causally used, not a probe correlate: swapping one entity's activation for another's makes the model answer with the other entity's successor under the imposed order. A rough map forms readily, present even in small and base models; what scale gates is using it cleanly: clean dominance and the causal crossover emerge only in the larger models (up to Gemma-31B and Qwen-27B) and weaken or reverse below, so a mechanism present in a large model can be absent in a smaller one of the same family. Whether the model builds this geometry anew or reconfigures a stored one we leave open; operationally, the geometry it uses is the one the context specifies.

The dynamics of a holonomically constrained rigid body can be modeled by Newton-Euler equations subjected to geometric constraints. This is frequently formulated as a differential-algebraic equation (DAE) system of index 1. Inmultibody system (MBS) dynamics it is common (1) to numerically solve this system by means of integration schemes for ordinary differential equations, and (2) to treat the rigid body motion on the direct product Lie group SO (3)R3, although rigid body motions form the semidirect product Lie group SE (3). It is has been observed that the constraint satisfaction depends on which Lie group is used as configuration space (c-space). In this paper the problem is considered from a geometric perspective. It is shown that the constraints are exactly satisfied by a numerical integration scheme if they define a subgroup of the c-space. The subgroups of SE (3) have a significance for modeling mechanical systems, including lower kinematic (Reuleaux) pairs and are implicitly used in MBS modeling. It is concluded that SE (3) is the appropriate cspace for numerical DAE modeling of a constrained rigid body. This result does not immediately apply to MBS, however.

In this study, we introduce latent PDE mapping, a broadly applicable physics-informed learning technique designed to enable efficient geometric generalization with sparse training data. Latent PDE mapping pulls back geometry-specific PDE residuals and boundary conditions to a predefined latent geometry via the deformation gradient, thereby enabling the automated calculation of geometry-consistent shape gradients that are missing in conventional physics-informed machine learning formulations. We demonstrate the utility of latent PDE mapping in solving the anisotropic Aliev-Panfilov PDE of cardiac electrophysiology using both physics-informed neural networks and physics-informed deep operator networks. The Aliev-Panfilov PDE serves as a challenging exemplar: a nonlinear, time-dependent PDE benchmark with sharp gradients that are expensive to capture using traditional numerical solvers. To represent the limited data regime, we train the networks using just fifteen geometric samples drawn from parameterized distributions in two and three spatial dimensions. While modest improvements appear for geometries parameterized by affine and shear deformations, latent PDE mapping demonstrates significant benefits on select geometric families, achieving a factor ~4-6 reduction in mean relative L2 error. Furthermore, our results show that the computational cost of applying latent PDE mapping was modest during network training, and negligible at inference. Taken together, our study highlights how latent PDE mapping facilitates the creation of generalizable physics-informed machine learning models from limited sets of training geometries.

Despite rapid progress, most existing vision-language models (VLMs) built from 2D visual inputs often struggle when handling various 3D tasks that require fine-grained spatial understanding and reasoning. To bridge this gap, we present VLM-IE3D, a unified framework that enhances the 3D spatial awareness of VLMs by equipping them with both implicit and explicit 3D geometries learned from RGB videos. Our VLM-IE3D introduces Implicit Geometry Tokens (IGTs) that capture high-level geometric priors from input videos, as well as complementary Explicit Geometry Tokens (EGTs) that encode detailed geometric structures from reconstructed 3D attributes. On top of that, VLM-IE3D comes with a 3D-aware adapter that effectively fuses the two types of geometric representations with 2D visual cues. This RGB-only design injects strong 3D inductive biases for fine-grained spatial understanding and reasoning without requiring any additional 3D inputs. Extensive experiments show that VLM-IE3D achieves superior performance consistently across various 3D tasks including 3D video detection, 3D visual grounding, 3D dense captioning, and spatial reasoning. Code and models are available at https://github.com/Vegetebird/VLM-IE3D.

Direct feedback alignment (DFA) trains hidden layers with fixed random projections of the output error, avoiding the transposed-weight backward pass of backpropagation (BP). We study a failure mode of DFA training that is distinct from feedback quality: the local weight update is calculated by an outer product, so anisotropy can enter through either its presynaptic-activity factor or its local-error factor. Our analyses with controlled synthetic regimes isolate the first failure mode and show an approximately 40-percentage-point activity-conditioning gain when high-variance directions contain task-irrelevant nuisance. Three clean confirmations isolate a different regime: error conditioning improves raw DFA by 1.77--7.53 percentage points, and combining independently selected activity and error factors adds 0.40--0.90 points over activity conditioning. The signs hold for tanh/one-vs-rest MNIST and preregistered Fashion-MNIST, and replicate on eight fresh seeds in a ReLU/softmax MNIST model. This factorization yields a symmetric block-local family of normalized DFA (nDFA): activity nDFA right-preconditions by an inverse activity second moment, error nDFA left-preconditions by an inverse local-error second moment, and K-nDFA applies both factors with separately tuned damping. A linearized post-alignment calculation gives an exact input-side spectral identity and a Kronecker-factor motivation for the two-sided rule, whereas norm matching rules out a scalar step-size explanation. The error factor is fragile when under-damped, BatchNorm is a strong activity-side alternative, and convnet gains remain partial. We therefore frame conditioned DFA as a factor-level study of when local outer-product rules fail, not as a general replacement for BP or a solution to all-layer convolutional credit assignment.

TabPFN is a transformer-based foundation model for tabular prediction that performs inference without task-specific training by conditioning on a support set and query inputs. Despite its strong empirical performance, its internal behavior on structurally difficult tabular geometries remains poorly understood. We study this behavior using zigzag persistent homology, treating TabPFN layer representations as evolving point clouds. We construct a controlled benchmark of synthetic tabular tasks with known true probabilities and varied intrinsic topology, including warped circles, tori, spheres, Hopf links, trefoil knots, and Swiss rolls. Across these tasks, we find that the topology of TabPFN's internal representation geometry is strongly associated with dataset-level reliability; for example, the zeroth homology group $H_0$ fragmentation count correlates positively with mean absolute residual across controlled tasks, and this association strengthens in a high-resolution warped circle case study at large sample size. Harder geometries induce a dual topological signature: increased $H_1$ loop activity and increased $H_0$ fragmentation, while the $H_1$ persistence becomes shorter-lived. These descriptors correlate with Bayes error, mean absolute residuals, and overconfidence. Our results suggest that zigzag persistence diagnoses the reliability of the inferred in-context task geometry and provides a context-level view of when TabPFN operates in topologically stressed regimes.

While neural PDE solvers have demonstrated significant potential for accelerating engineering simulations, existing architectures remain constrained by high memory consumption and the single node bottleneck, where the maximum processable mesh resolution is strictly limited by the VRAM of a single compute unit. To address these challenges, we propose PGD-NO, a neural operator with Precomputed Geometry Decomposition, that relocates the computational overhead of geometric encoding to a deterministic pre-computation phase. By utilizing an iterative geometry decomposition algorithm to extract geometry tokens, our model decouples feature extraction from solution querying. This architecture enables linear memory scalability, allowing high fidelity learning on meshes exceeding 10 million nodes, a scale where existing architectures typically encounter memory exhaustion. PGD-NO demonstrates competitive predictive accuracy across diverse industrial benchmarks and provides intrinsic interpretability through attention mechanisms. By effectively overcoming traditional mesh-size constraints, PGD-NO offers a robust and efficient solution for the next generation of large-scale, high-fidelity industrial design applications.

Synthesizing physically plausible human-scene interactions (HSI) remains a critical challenge in computer vision and the development of human avatars. Although recent generative models enable diverse motion synthesis, they suffer from an inductive bias referred to as semantic-geometric entanglement. Because spatial constraints often strongly correlate with specific actions in training data, monolithic models will learn the shortcut bias, aggressively overriding the semantic intent when faced with strict geometric cues. Furthermore, this entanglement exacerbates physical hallucinations, such as body-scene penetrations. To address these limitations, we propose DeSeG, a hierarchical framework that explicitly decouples semantic intent from geometric constraints. First, we introduce a Residual Semantic Planner that encodes textual instructions and canonicalized goal voxels into a compact latent space, enabling fine-grained semantic control independent of spatial trajectories. Second, we propose a physics regularized diffusion executor that incorporates differentiable repulsive potential fields directly into the diffusion objective, enforcing collision-aware motion generation. Extensive experiments on the Lingo dataset demonstrate that DeSeG achieves state-of-the-art performance, reducing mean scene penetration by 47% and improving semantic alignment by 29% over the SOTA baselines.

How concepts are represented in neural networks is a fundamental question in machine learning. The dominant view treats concept representations as stationary geometric objects. Yet concepts appear in context, and context transforms them. Drawing from neural population geometry, we formalize concept representations as point-cloud manifolds and contextual transformations as vector fields, and instantiate this framework in large language models. Across six model families of varying scales, we find that context moves each concept differently. The variance in these displacements is semantically organized, correlating with lexical concreteness and density. Importantly, both the concepts being transformed and this variance structure are shared across models: displacement structure transported from one model predicts held-out displacements in others significantly above chance. Together, these findings show that models share a common geometry not only in how concepts are represented, but more importantly in how context transforms them, a structure with richer organization than prior work has recognized.

Knowledge distillation is usually framed as a choice of what to match in the teacher - its logits, hidden features, or sample relations - which presupposes that the teacher's representation has absolute coordinates to match. It does not: a pretrained representation is identifiable only up to an orthogonal-and-isotropic-scaling equivalence class, so a student should learn the teacher's equivalence class, not its features. The organizing fact is that capability is the teacher's output function, a class invariant that factors through the quotient by the class action, so an objective recovers capability exactly when it is defined there. This makes absolute feature matching ill-posed, and admissible supervision a matter of targeting class invariants (Gram structure, CKA, principal subspaces) or aligning coordinates first, unifying feature matching, relational distillation, alignment, and grafting in one geometric account. We validate our framework on Qwen2.5 and Llama-3.1. A restoration study recovers a corrupted model's representation (CKA ~ 0.99) but not its capability, and an ablation isolates the cause: output-function (logit) matching drives capability, while matching hidden representations aligns geometry without restoring function. Recovery is confined to the corpus-covered region, and a graft study confirms that boundary overlap predicts transplant success but is necessary, not sufficient.

We introduce a controlled subspace intervention framework to investigate how self-supervised Vision Transformers (ViTs) encode dense geometric information. While linear probing is widely used to assess geometric representations, it treats features as a black box, failing to disentangle the underlying topology. To address this issue, we decompose the weights of converged linear probes to isolate the low-rank subspaces containing explicit geometric signals using Singular Value Decomposition (SVD). Our perspective yields three key insights: (1) Pre-training objectives determine how features are encoded. DINOv2 aligns spatial features for efficient linear extraction, while Masked Autoencoders (MAE) tend to disperse these signals, requiring a broader spatial context. (2) Explicit geometric representations are highly compressible, suggesting dense predictive heads could potentially be constrained to low-rank subspaces with minimal performance loss. (3) The layer-wise task affinity suggests that geometric precision peaks at intermediate layers before yielding to semantic abstraction in the final layers. By connecting internal encoding mechanics with downstream performance, these findings provide a basis for effective feature selection and lightweight decoder design. The source code is available at https://github.com/Zhou-Weichen/Geosubprobe.

Large language models (LLMs) are increasingly consulted on contested scientific questions, raising the concern that they will sycophantically retreat from established consensus when a user signals doubt -- drifting toward a false balance that treats settled science as one view among several. We test this across three open instruction-tuned models (Llama-3.1-8B, Qwen2.5-7B, Mistral-7B), three consensus-science domains (climate, vaccines, evolution), and single- and multi-turn settings, combining behavioral measurement with linear probing and activation patching. We do not observe sycophantic retreat. Instead, models show three distinct policies under the same skeptical pressure: reactive assertion, where consensus assertion increases rather than decreases (Llama); surface hedging, where tone softens while the position holds (Qwen); and non-response (Mistral). Pairwise judgments confirm the reactive shift is stance, not style (63.6%, p=.007), and a decomposition identifies increased consensus assertion, not false balance, as its driver (beta=+0.042 per dose, p<1e-77). Linear probes localize the divergence to middle layers -- perfect separation in Llama and Qwen versus 72% in Mistral, with non-overlapping confidence intervals -- indicating the non-responsive model does not linearly represent the skepticism signal at all. Crucially, this robustness does not transfer: it attenuates across domains and, in the safety-critical vaccine domain, can reverse, with myth-rebuttal weakening under skeptical pressure. We synthesize these into a four-way taxonomy separating active from accidental robustness, and argue that behavioral evaluation alone cannot distinguish a model that resists skepticism because it understands the signal from one that only appears to resist because it fails to perceive it.

Comprehensive 3D scene understanding from sparse, unposed images requires a model to recover renderable geometry, open-vocabulary semantics, and free/occupied 3D space without relying on external camera calibration. Recent feed-forward Gaussian methods improve pose-free reconstruction and semantic rendering, but their Gaussian primitives are mainly optimized through image-space objectives and remain weakly constrained in unobserved regions. We propose \textit{COVScene}, a pose-free semantic Gaussian framework that couples renderable Gaussian primitives with a dense semantic occupancy field through differentiable volumetric lifting. Instead of converting Gaussians to voxels only at evaluation time, COVScene lifts the predicted semantic Gaussians inside the training computation graph, so volumetric regularization provides gradients to Gaussian opacity, geometry, and semantic features. The framework combines a semantic-aware Geometry Transformer, multi-task Gaussian decoding, geometric foundation distillation, and occupancy entropy regularization to support novel view synthesis, open-vocabulary semantic querying, and semantic occupancy prediction within a single representation. Experiments on ScanNet and ScanNet++ show that COVScene maintains competitive rendering quality, improves open-vocabulary segmentation, and achieves stronger semantic occupancy prediction than the self-supervised baseline without direct voxel-level supervision.

Recent work has shown that implicit neural representations (INRs) can be trained to effectively compress structured and unstructured volume data, allowing for direct data querying with a reduced memory footprint. However, as existing INRs for unstructured volumes do not encode geometry, they require partial mesh storage for later sampling, limiting achievable compression. At the same time, novel view synthesis methods have shown that explicit collections of 3D Gaussians can be used to accurately visualize volume data. In this work, we introduce an explicit model for volume data compression based on 3D Gaussian primitives. We reinterpret collections of 3D Gaussians as an explicit representation of a scalar field and use a sampling strategy that reconstructs scalar values at spatial locations through weighted aggregation of intersecting Gaussians. We develop optimized CUDA-accelerated pipelines for structured and unstructured model sampling, loss functions that encourage accurate domain encoding by our models, and a novel sampling-error based densification strategy. Our explicit formulation naturally encodes domain geometry, eliminating the need for mesh storage in unstructured volumes and introducing significantly higher compression opportunities. Compared to existing INRs, we demonstrate that our explicit model achieves competitive reconstruction quality with significant training speedups on structured volumes, while markedly outperforming in all metrics on unstructured volumes.

LiDAR simultaneous localization and mapping (SLAM) relies on local geometric quantities such as covariances, correspondences, and surface structures. However, most existing pipelines rely on hand-crafted estimates of local geometry and use them as fixed inputs to LiDAR SLAM, which can make the estimated local geometry noisy and unstable in sparse regions of a point cloud or when using low-resolution LiDAR. To address this issue, this paper introduces a self-supervised framework that learns an explicit symbolic representation of local geometry and uses it to improve LiDAR SLAM recursively. Specifically, each point is represented as a Gaussian distribution, allowing local geometry to be described by a covariance. Without dense geometry labels or ground-truth poses, the framework learns by maximizing the likelihood of local geometry, with self-supervision derived from consistency relations over symbolic geometric representations, including predicted covariances, correspondences, and trajectory from SLAM. The learned geometry is then fed back into LiDAR SLAM, forming a reciprocal loop in which improved geometry enhances localization and mapping, and improved localization provides cleaner supervision for subsequent geometry reasoning. This framework is backend-agnostic and can be plugged into existing LiDAR SLAM pipelines without architectural changes. Experiments on KITTI under varying LiDAR resolutions show that the proposed method improves both odometry and global registration.

Classical universal approximation theorems establish the expressive power of sigmoidal multilayer perceptrons, but they do not prescribe how initial weights should encode the geometry of a data distribution. We propose S-GAI, a spectral geometry-aware initialization framework for one-hidden-layer sigmoidal MLPs. Starting from the constructive idea that sigmoid units can act as smooth half-space gates, we move from hand-specified planar geometry to class-wise spectral geometry estimated from image data. For each class, SVD provides a mean, principal directions, and spectral scales. An energy threshold selects the retained directions, and each retained direction is represented by two sigmoid gates. These class-specific gates form a shared hidden layer initialized directly from the training set. We also formulate a SVD-based subspace classifier as a non-neural geometric reference, which tests whether the estimated spectral class geometry is already discriminative before being embedded into the MLP. Experiments on MNIST, Fashion-MNIST, and a more challenging CIFAR-10 test show that the S-GAI-initialized MLP starts from a substantially more informative hidden state than Xavier initialization and reaches comparable final accuracy under full training. When the hidden layer is frozen, training only the output layer still gives stronger performance than frozen random gates, providing evidence that S-GAI effectively embeds class-wise spectral geometry into the MLP.

We study certain extremal problems in combinatorial geometry that ask about configurations of points in an $n \times n$ grid that satisfy strict, global geometric constraints. Classical exact solvers suffer from combinatorial explosion for these types of problems, and standard reinforcement learning and transformer-based models struggle with the sparse reward "validity cliff" and quadratic token-consumption limits. To overcome these bottlenecks, we propose a Geometry-Aware Monte Carlo Tree Search (MCTS) framework. Our approach strictly enforces geometric constraints through incremental updates to the feasible action space. For constraints about collections of collinear points, like those that occur in the classic No-Three-in-Line problem (Max-N3IL), this mechanism reduces the constraint checking complexity from $O(n^3)$ to $O(n^2)$. To improve search efficiency, we exploit geometric symmetries in two ways: canonical pruning during node expansion to reduce the branching factor, and symmetric batch transitions to accelerate the discovery of promising configurations. We perform extensive experiments and establish new best-known computational results on five out of six of the problems that we considered. Notably, for Max-N3IL we find configurations of size roughly $1.8 n$ for grids of size $82 \le n \le 119$. For the Smallest Complete Set problem, we find configurations of size roughly $0.95 n$, providing new upper bounds within the tested grids. This work establishes Geometry-Aware MCTS as a highly adaptable framework for discovering novel configurations in combinatorial geometry.

A defining feature of human intelligence is the ability to adapt to changing environments by inferring latent task structure from sparse observations. Neuroscientific research indicates that this capability relies on the hippocampus constructing abstract representations, expressed as low-dimensional, approximately orthogonal manifolds in neural state space. However, the internal mechanisms of large language models (LLMs) remain largely opaque, making it unclear whether they form comparable abstract representations or instead rely on task-specific statistical regularities when performing comparable reasoning tasks. Here we adapt a contextual reversal-learning paradigm to a text-based setting and compare humans and LLMs at both the Behavioural and representational levels. We report that although LLMs exhibit generalizable reasoning less frequently than humans, when such inference occurs, their internal states exhibit abstract geometric structures that resemble those reported in the hippocampus. Notably, this representational geometry is not uniformly distributed but is organized hierarchically across model depth: whereas lower layers show early, stable encoding of stimulus identity, higher layers form a hippocampal-like functional band enriched for abstract context geometry associated with inference. Furthermore, complementary intervention experiments mechanistically implicate geometry in reasoning: task-sequence language modelling induces geometric disentanglement, whereas geometric regularization of higher layers increases the emergence of generalizable inference. Together, these findings establish abstract representational geometry as a mechanistic principle supporting inference in large language models.

Point-cloud goals provide a direct way to specify dexterous in-hand reorientation: instead of defining an object-specific pose frame or estimating 6D pose at test time, the policy is given the desired 3D geometry of the object. Yet raw point-cloud goal conditioning is poorly conditioned for policy learning. Current and goal clouds are unordered, independently sampled, and often visibility-dependent, so their discrepancy entangles object rotation with permutation, resampling, and unstable correspondence structure. For this reason, prior point-cloud manipulation methods typically add structure outside the representation itself, such as explicit pose or relative-pose inputs, dense flow features, or distillation from privileged teachers. We close this gap by learning a rotation-aware point-cloud embedding whose Euclidean latent distance is calibrated to the SO(3) geodesic error between object orientations. The resulting representation turns current-goal comparison into a smooth control signal, allowing a model-free RL policy to act from current and goal point-cloud embeddings, proprioception, and centroid metadata, without object pose, relative pose, dense flow, or teacher-action supervision. In in-hand reorientation experiments, this interface matches privileged-state and distillation-based baselines while avoiding brittle test-time computation of structured pose or flow inputs. These results suggest that point-cloud goals become practical for this task when the representation, rather than an external module, encodes the task-relevant geometry of rotation. We also show evidence that generic visual point-cloud pretraining is insufficient for such a current-goal comparison because it discards the task-relevant state and preserves only shape features.

Imaging objects hidden outside the direct line of sight expands the effective field of view and is critical for applications such as autonomous driving and robotic perception. Despite impressive progress in time-of-flight (ToF)-based non-line-of-sight (NLOS) imaging, real-world deployment remains challenging because practical measurements are often collected over spatially limited, arbitrarily shaped relay regions-conditions that violate the planar-wall and dense-sampling assumptions made by most existing methods. To address these limitations, we propose a LOS-guided NLOS imaging pipeline that imposes no geometric assumptions on the relay surface and naturally supports both confocal and non-confocal configurations. Our method represents the hidden scene using 3D Gaussian primitives and couples them with an efficient, differentiable transient rendering model, enabling end-to-end optimization directly from measured transients. We validate our approach on real-world measurements from both a public dataset and a custom-built capture system. Across settings, our method achieves state-of-the-art reconstruction fidelity under spatially limited, sparsely sampled conditions, and significantly outperforms existing methods on complex, arbitrary relay surface geometries.

Sparse novel view synthesis (NVS) remains challenging due to the ambiguity of recovering 3D geometry from few input views. While NeRF- and Gaussian Splatting (GS)-based methods perform well with dense supervision, they often overfit in sparse settings, producing floating artifacts and inconsistent geometry. Silhouette consistency is commonly used as a regularizer, but it remains insufficient, as silhouette-consistent regions can extend beyond the true object geometry. We introduce VisDom, a learning-free geometric constraint that augments classical carving-based visual hull reconstruction by enforcing a minimum multi-view visibility requirement. Specifically, we define a visible domain as the subset of 3D space observed by at least $K$ views and use it as an additional filtering criterion on top of standard silhouette-based reconstruction. This provides a stronger spatial prior in sparse-view settings. We integrate VisDom into both implicit (NeRF) and explicit (GS) pipelines by restricting volumetric sampling and guiding Gaussian placement during optimization. Experiments on three challenging datasets show consistent improvements in sparse-view NVS, enabling high-quality object-centric reconstruction from as few as four input images. Our method is domain-agnostic, requires only silhouettes, and introduces no learned parameters, making it a simple complement to existing approaches. Applying VisDom on top of GaussianObject further improves performance on Omni3D and MipNeRF360, while matching or surpassing it at 22 $\times$ lower training cost.

Activation steering controls model behavior by modifying intermediate hidden states at inference time without retraining. Existing methods handle only single-direction injection; when multiple semantic directions are superposed without constraints, the model collapses. We show that this collapse decomposes into two independently acting sources: distributional deviation, where additive perturbations accumulate in norm across layers and drive activations outside the training distribution, and directional interference, where non-orthogonal semantic vectors mutually dampen when superposed. These two sources define the design constraints that any training-free multi-directional intervention must address. As one instantiation of these principles, we propose GEMS, a training-free method that maps each source to a corresponding geometric constraint: norm-preserving weighted superposition and targeted attention-pathway injection for distributional deviation, and real-time orthogonalization for directional interference. On GSM8K, injecting three concurrent non-mathematical directions preserves accuracy at 98% (baseline 92%), while unconstrained addition collapses to 4%; on Wikitext-2, the same injection incurs only 2.2% PPL increase. Component ablation isolates the causal role of each constraint, and layer-level probes confirm that orthogonalized signals survive the FFN pathway and reach the output distribution with semantic specificity. Qualitative steering effects transfer across architectures from 3B to 31B.

Intravascular ultrasound (IVUS) lumen and external elastic membrane (EEM) segmentation is important for quantitative coronary plaque burden assessment. Errors in lumen or EEM delineation directly propagate to plaque area, plaque burden and geometric measurements. However, standard methods prioritising overlap scores often suffer from boundary drift and topology errors, leading to inaccurate clinical measurements. We present GeoCat, a geometry-consistent network that processes 5-frame IVUS clips using dual Cartesian-polar encoders with cross-domain attention and temporal fusion. A differentiable geometry consistency loss directly supervises clinically relevant descriptors including diameters, orientations, and cross-sectional areas. The model is trained on 12,242 annotated frames from 146 patients acquired with two commercial IVUS systems. We evaluate performance using both segmentation accuracy and plaque-relevant clinical metrics, including Dice/IoU, boundary measures(95HD (mm), ASSD), topology violation rate, and clinical geometry errors (dmax/dmin, angles, and areas). On our dataset, GeoCat achieves a Dice of 0.93, reduces 95HD to 0.14 mm, and lowers topology violations to 1.0%. Importantly, it significantly improves geometric fidelity, yielding diameter errors of 0.13-0.16 mm and angular errors of ~8 degrees, supporting reliable plaque burden quantification.

Learning-based visual navigation has enhanced semantic goal-reaching capabilities. However, due to their black-box nature, purely end-to-end models often lack explicit geometric constraints, leading to unpredictable and unreliable obstacle avoidance in open environments. Conversely, traditional geometric planners ensure safety but struggle with high-dimensional visual targets. To address these limitations, we propose SemGeoNav, a novel hierarchical visual navigation framework.It tightly integrates the high-level semantic reasoning of end-to-end models with the reliable local planning ability of geometry-based methods, achieving robust image-based navigation while significantly improving obstacle avoidance. Furthermore, we introduce a temporal trajectory smoothing mechanism to ensure continuous and stable robot motion. We evaluated SemGeoNav on a Unitree Go2 quadruped robot in real-world environments. The results demonstrate that SemGeoNav outperforms existing representative methods, including ViNT and NoMaD, achieving higher success rates and shorter navigation times.

Geometry problem generation is useful for AI-assisted education and multimodal mathematical reasoning, but reliable synthesis remains difficult because the problem statement, diagram, constraints, and solution should be mutually consistent. Existing methods often trade off controllability and reliability: seed-based rewriting is flexible but weakly verifiable, whereas diagram-first construction improves validity but is less suited to arbitrary user-specified constraints. We introduce VeriGeo, a controllable geometry generation framework grounded in executable reasoning traces. Given user constraints such as target concepts and difficulty, an Author agent generates a problem and diagram, and a Solver agent produces a proof-aligned solution. Both agents use a shared action sequence that connects natural language, diagrams, geometric constraints, and proof steps into a verifiable representation. A three-stage pipeline checks numerical consistency, analytical realizability, and global consistency, using verification-guided reflection to repair recoverable failures and reject unrecoverable ones. Across five LLM backbones, raw generations frequently fail these checks, while VeriGeo repairs a substantial fraction of the invalid attempts. Supervised fine-tuning on 8.7k examples generated by VeriGeo achieves the best reported GeoQA performance among end-to-end multimodal LLM-based solvers, and obtains strong results on PGPS9K and MathVista-GPS, demonstrating the effectiveness of verified synthetic data for improving multimodal geometry reasoning.

Humans cannot always intuit what scenarios are most challenging to LLMs. Hoping to capture challenging edge cases, developers either design problems to be difficult for humans or curate extensive benchmarks. What if we could instead anticipate which scenarios a model will fail on? In this paper, we use an LLM's representational geometry to predict which concept combinations it will fail on. We attribute this compositional failure to interference between salient features. In tasks that require systematic composition - toy programmatic settings, multihop reasoning, multilingual factual recall - we find that when a pair of concepts is encoded near-orthogonally, the model reliably composes them. When their linear encodings are close, producing interference, the model fails to compose them. Our method reliably anticipates failure modes across different compositional tasks, without evaluating specific inputs. These results lay the groundwork to use representational geometry to identify high-risk examples, construct targeted stress tests, and provide a scalable foundation for active learning in real-world deployment.

Knowledge Distillation (KD) and mixup have proven effective at inducing smoothness in class boundaries; KD captures inherent class relationships in probability distributions, and mixup enforces them through convex combinations of inputs. Their interaction, however, remains poorly understood, particularly when mixup is applied only during student training. In this setting, the teacher is queried on inputs drawn from a vicinal distribution it never saw during training, a controlled mismatch whose effect on knowledge transfer has not been characterised. We show that this mismatch causes the teacher's supervisory signal to be dominated by distributional confusion rather than inter-class structure. Despite it, the student does not merely imitate the teacher: it independently acquires greater linearity in the vicinal region, a structural property that the teacher lacks, and goes beyond dark-knowledge transfer. KD with mixup consistently improves student accuracy and reduces overconfidence by an order of magnitude relative to the baseline, across CIFAR and ImageNet with varying-capacity teachers. Crucially, calibration propagates from teacher to student independently of accuracy transfer, and temperature scaling governs a measurable accuracy-calibration trade-off that becomes more pronounced under vicinal training. These results reframe mixup distillation not as a degraded version of standard KD, but as a richer transfer channel that simultaneously shapes discriminative performance, uncertainty estimation, and representational geometry.

Reconstructing printable 3D models from Mars rover imagery is challenging because Martian terrain is low-texture, irregular, and partially observed. We evaluate a pipeline that estimates stereo depth from NASA Curiosity images, completes geometry, and exports watertight OBJ meshes. On Middlebury, RAFT-Stereo outperforms semi-global block matching (SGBM), reducing disparity MAE from 3.22px to 0.73px and increasing valid prediction coverage from 76.3% to 100.0%. On Curiosity imagery, however, RAFT's denser disparities show weaker edge alignment and higher photometric reprojection error, suggesting that benchmark accuracy does not directly transfer to Martian terrain reconstruction. Geometry completion demonstrates a tradeoff between local fidelity and global connectivity. We find that alpha shapes preserve accurate but fragmented structure, Poisson reconstruction produces more coherent meshes but adds unsupported surfaces, and a deterministic diffusion-fill baseline is intermediate but sensitive to stereo quality. Overall, standard stereo and completion methods can produce printable approximations of Martian terrain, but reliable reconstruction requires stronger domain-specific validation.

Aerodynamic simulation is a key component of engineering shape design, where core quantities such as the surface pressure coefficient strongly depend on flow dynamics near solid boundaries. Neural operators provide an efficient alternative to expensive Computational Fluid Dynamics (CFD) solvers. However, conventional methods treat the boundary region isotropically, failing to account for the distinct physical behaviors along the boundaries. In reality, the aerodynamic process exhibits anisotropy: along the tangential direction, flow propagates along the wall; along the normal direction, physical quantities are constrained by the wall. To explicitly model the distinct physical behaviors, we propose GeoABC, a geometry-conditioned anisotropic boundary correction framework. GeoABC leverages the boundary geometries to introduce direction-aware boundary correction into the intermediate representations of neural operators, transforming boundary geometry from static input features into a structural prior that modulates physical prediction. On 2D airfoil and 3D car tasks, GeoABC consistently adapts to multiple neural operator backbones, reducing near-boundary relative $L_2$ error by $\sim$38% on average, narrowing the structural near-wall gap shared by mainstream neural operators, and advancing neural operators toward high-fidelity aerodynamic simulation.

Mathematical reasoning has long served as a stringent test of machine intelligence; over the past decade, it has moved from a niche problem within NLP to one of the most consequential AI frontiers. This survey provides a unified account of the field's evolution, from early rule-based math word problem (MWP) solvers and template-driven geometry systems, through neural expression generation and LLM prompting, to contemporary reasoning models, multi-agent systems, neuro-symbolic theorem provers, and verified discovery workflows. We organize the landscape along four axes: (i) informal reasoning over text and diagrams, spanning MWP solving, multimodal geometry, and VLMs; (ii) formal reasoning in proof assistants, including autoformalization, tactic prediction, compiler-guided repair, and proof search; (iii) mathematical discovery, where systems propose constructions, improve bounds, or assist attacks on open problems; and (iv) the inference and training-time techniques, including CoT prompting, tool use, process reward models, and RLVR, that increasingly connect generation with verification. We catalog major benchmarks across grade-school arithmetic, competition mathematics, geometry, formal proving, multimodal and multilingual reasoning, and expert evaluation, and we examine benchmark saturation, contamination, reporting mismatches, and the distinction between pass@1, majority voting, and verifier-assisted pass@$k$. We critically assess failure modes: brittleness under perturbation, reward hacking, multimodal grounding failures, fragile formalization, and the energy cost of reasoning-scale inference. Drawing on recent perspectives from working mathematicians, we identify future directions centered on verified-discovery workflows, reasoning efficiency, and infrastructure to make AI-assisted formalization broadly usable. Companion materials: https://github.com/Starscream-11813/awesome-AI4Math.

In this paper, we propose a cross-view fusion framework that enhances the robustness of 6-DoF grasp pose estimation in corner views. Our framework alleviates occlusion by incorporating an auxiliary view and avoids the time-consuming, task-agnostic multi-view reconstruction through a post-fusion strategy. To enhance cross-view fusion, we propose a self-supervised contrastive learning strategy that leverages cross-view associations to regularize point cloud features. In brief, a cross-view point pair is considered a match if the two points correspond to the same 3D location, and a non-match if they represent distinct grasp directions. The learning strategy significantly enhances the spatial consistency and direction distinctiveness of point features, thereby facilitating cross-view fusion and improving estimation robustness. Furthermore, we propose a cross-view-aligned cylinder integration module to fuse grasp-relevant geometry into a comprehensive representation. Specifically, the module first aligns the cross-view points and features according to their similarity to enhance the robustness against noise. Subsequently, these points are registered into the cylindrical coordinate frame, emphasizing the rotation-symmetric geometry which is important for grasping. Finally, local self-attention and seed cross-attention layers are alternately employed, respectively enabling interactions within single views and across views, which supports fine-grained representation of grasp-relevant geometry. Our framework achieves strong performance on the GraspNet-1Billion benchmark and in real-world applications. Code is available at https://github.com/KJZhuAutomatic/Cross-view-Grasp.

Multimodal Large Language Models (MLLMs) excel at 2D semantic understanding but lack intrinsic 3D awareness, resulting in representations that fail to maintain geometric and spatial consistency across video frames. Given the scarcity of large-scale 3D data, we present GeoVR, a novel framework that learns geometric representations using purely 2D video sequences. This approach effectively restructures the semantic latent space within MLLMs to unlock spatial intelligence. Rather than employing superficial feature mixing, GeoVR reshapes the internal representations of the MLLM by distilling geometry knowledge from pre-trained 3D foundation models. This is accomplished through a multi-objective learning strategy driven by four complementary geometric targets: (1) estimating inter-frame camera poses to embed varying viewpoint dynamics, (2) regressing dense depth maps to anchor physical distances, (3) predicting a metric scale factor for real-world calibration, and (4) distilling multi-scale 3D features to align the intermediate feature space. Guided by these explicit physical and geometric constraints, the model's internal representations naturally develop strong 3D awareness. Extensive experiments on spatial reasoning benchmarks demonstrate that GeoVR achieves state-of-the-art performance, establishing a new paradigm for endowing foundation models with spatial intelligence.

Prompting steers large language models (LLMs) and vision-language models (VLMs) without weight updates, but it remains unclear how instruction changes reshape internal representations to produce behavior. We introduce a nested geometric decomposition framework that treats prompting as a transformation of the representational geometry of the content following the prompt. For each prompt pair, we align representations of the same stimuli under two prompts using increasingly expressive stimulus-invariant maps: translation, rigid transformation with uniform scaling, sequential axis scaling, affine transformation, and nonlinear transformation. We then causally test each map by replacing a single layer's prompt-A hidden state for held-out stimuli with its mapped counterpart and measuring recovery of prompt-B representational geometry and behavior. Across three LLMs, three VLMs, and six text or image datasets spanning style, emotion, scene content, and number, prompts consistently reshape representations toward the instructed task structure. Cross-validated variance decomposition shows that much prompt-induced activation change is captured by shape-preserving maps, especially translation and rigid transformation with uniform scaling, while tier profiles reveal model- and task-specific routing strategies across layers. Crucially, although translation and rigid tiers already improve behavioral agreement, affine transformation is the first tier to nearly recover target-prompt task geometry and yields corresponding behavioral gains. This suggests that cross-dimensional linear mixing is a key mechanism by which prompts reorganize representations toward instructed task structure. Our framework decomposes prompt-induced representational change into interpretable geometric components and reveals how models route task-relevant structure to produce prompt-driven behavior.

Video world models aim to simulate controllable visual environments, but long-horizon rollouts depend on what the model remembers after observations leave its native context window. Explicit memories retain frames or online 3D reconstructions, which can suffer from heuristic retrieval errors, redundant appearance storage, or reconstruction artifacts. Implicit memories compress history into a compact state, but existing designs are not explicitly constrained to encode cross-view scene geometry. We propose GIM-World, a geometry-aware implicit memory framework for video world models. A lightweight transformer encoder compresses variable-length history into fixed-size memory tokens, a camera-queryable geometry head distills 3D scene structure from a frozen foundation model into the memory during training, and an information-guided pruning rule keeps encoding cost bounded as history grows. The geometry teacher is discarded at inference, leaving a lightweight memory module. Experiments on MIND show that GIM-World better preserves long-horizon geometric and visual consistency than both explicit- and implicit-memory baselines.

Generative action policies based on diffusion or flow matching excel in behavior cloning, yet their iterative sampling is prohibitive for high-frequency robot control. While recent one-step formulations alleviate this latency, they inevitably discard the intermediate trajectory evolution that provides crucial action correction. Directly recovering this mechanism by explicitly estimating a training-time drifting field is mathematically ill-posed due to extreme conditional demonstration sparsity. We introduce Implicit Drifting Policy (IDP), a one-step imitation learning framework that brings the training-time correction of Drifting into policy learning without explicit vector field estimation. IDP extracts a conditional expert geometry from the local variation of observation-similar expert actions, and compares it against a global reference geometry to isolate condition-specific constraints. This local geometric structure adaptively weights a scalar potential objective. Combined with an expert-proximal terminal evaluation, IDP directly enforces manifold constraints on the one-step generator during training. Extensive evaluations across 2D, 3D, and real-world manipulation tasks show IDP effectively maintains adherence to valid action manifolds, improving upon explicit drifting methods and achieving competitive performance with strong one-step baselines.

Detecting covariate shift is critical for building reliable vision systems. While most prior work focuses on improving robustness to shift, explicitly detecting covariate shift remains underexplored. Existing approaches typically rely on fully supervised training, requiring labeled examples from both original and shifted distributions, which is often impractical. In this paper, we show that covariate shift detection can be effectively addressed with weaker supervision using Positive Unlabeled (PU) learning. However, under covariate shift, in distribution and shifted data overlap significantly, making classical PU methods unstable and sensitive to noise. To overcome this challenge, we introduce Spectral PU Neighborhood Annotation (SPUNA), a geometry aware framework that progressively discovers shifted data by leveraging the local manifold structure of visual features. Extensive experiments show that SPUNA achieves state of the art performance in PU settings and remarkably matches the performances of fully supervised methods. Moreover, our approach transfers robustly across different types of shifts, demonstrating strong generalization capabilities.

Positional embeddings (PEs) in Vision Transformers (ViTs) are known to impact performance and robustness, but their role in shaping internal spatial representations is not well understood. In this work, we study how different forms of PEs influence the representational geometry of ViTs and how these changes relate to robustness under content-disrupting distribution shifts. We introduce a metric, the Spatial Similarity Distance Correlation (SSDC), to quantify spatial structure in token representations. Using this metric, we show that ViTs trained without PEs still develop non-trivial spatial structure, but this structure is driven by visual content and collapses under token permutation. In contrast, we find that all PEs considered (learned absolute, sinusoidal, and rotary) are associated with a consistent shift toward an index-anchored spatial organization. Representations in these models remain stable under perturbations that disrupt content, and exhibit substantially improved robustness to such distributional shifts. We further show that while different PEs produce distinct depth-wise trajectories of spatial structure, their robustness properties are largely similar (with secondary variation across encoding schemes), suggesting that robustness appears to depend on the presence of a stable positional reference frame more than it depends on the specific encoding mechanism. These results offer a geometric account of how positional encodings shape internal representations, with implications for the principled design of future encoding schemes.

We present an image-conditioned point cloud completion approach that treats images as the primary geometric source rather than a secondary guide. To this end, we introduce an Image-to-Point (I2P) module that can reconstruct complete point clouds directly from a single RGB image, with no need for 3D inputs. Additionally, we introduce a transformer-based Point-to-Point (P2P) refinement module that uses self- and cross-attention between point tokens and image features to iteratively refine the coarse I2P output. The I2P module enables the image encoder to learn rich geometric representations, while the P2P module progressively recovers fine-grained details. Unlike existing multimodal methods that rely on auxiliary losses or fusion modules, our explicit I2P task provides a strong, geometry-aware prior based on images alone. Extensive experiments on ShapeNet-ViPC demonstrate state-of-the-art completion performance with a 12.3% relative Chamfer Distance improvement over prior methods. Code is available at: https://github.com/AzharSindhi/I2PRef.git

Lane detection stands as a crucial perception task in autonomous driving and advanced driver assistance systems. However, existing methods still degrade in complex real scenarios due to two major limitations. First, classification confidence only characterizes the categorical existence of lane priors and has no strong correlation with geometric quality. If threshold filtering and NMS are conducted merely based on this confidence, the model tends to retain lane priors with high confidence while eliminating those with lower confidence but superior geometric representation. Secondly, the regression modules in existing methods weaken correlations among sampling points, hindering fine-grained optimization of distant, high-curvature and complex-topology lanes and causing underfitting. To address these issues, we propose Geometric Fidelity and Spatial Refinement (GFSR), a framework consisting of LaneIoU-guided Confidence Calibration (LCC) and Adaptive Gated Location Refinement (AGLR). Specifically, LCC adopts LaneIoU as soft supervision to explicitly estimate the geometric fidelity of lane priors, which is further fused with classification confidence to construct the Collaborative Reliability Index (CRI). This index guides lane prior filtering, effectively retaining those with high classification confidence and favorable geometric quality. Meanwhile, cooperating with regression heads in each refinement stage, AGLR predicts sampling point lateral offsets and adopts a gating mechanism to adaptively regulate correction magnitude, strengthen inter-point correlations and boost model adaptability as well as robustness toward complex lane scenarios. Extensive experiments on CULane and CurveLanes demonstrate that our GFSR achieves state-of-the-art performance on CULane, with F1_50 and F1_75 scores of 81.46% and 65.01%, and reaches 87.35% F1_50 on CurveLanes.