Daily Papers

Recent progress in neural implicit functions has set new state-of-the-art in reconstructing high-fidelity 3D shapes from a collection of images. However, these approaches are limited to closed surfaces as they require the surface to be represented by a signed distance field. In this paper, we propose NeAT, a new neural rendering framework that can learn implicit surfaces with arbitrary topologies from multi-view images. In particular, NeAT represents the 3D surface as a level set of a signed distance function (SDF) with a validity branch for estimating the surface existence probability at the query positions. We also develop a novel neural volume rendering method, which uses SDF and validity to calculate the volume opacity and avoids rendering points with low validity. NeAT supports easy field-to-mesh conversion using the classic Marching Cubes algorithm. Extensive experiments on DTU, MGN, and Deep Fashion 3D datasets indicate that our approach is able to faithfully reconstruct both watertight and non-watertight surfaces. In particular, NeAT significantly outperforms the state-of-the-art methods in the task of open surface reconstruction both quantitatively and qualitatively.

Digital Surface Models (DSM) offer a wealth of height information for understanding the Earth's surface as well as monitoring the existence or change in natural and man-made structures. Classical height estimation requires multi-view geospatial imagery or LiDAR point clouds which can be expensive to acquire. Single-view height estimation using neural network based models shows promise however it can struggle with reconstructing high resolution features. The latest advancements in diffusion models for high resolution image synthesis and editing have yet to be utilized for remote sensing imagery, particularly height estimation. Our approach involves training a generative diffusion model to learn the joint distribution of optical and DSM images across both domains as a Markov chain. This is accomplished by minimizing a denoising score matching objective while being conditioned on the source image to generate realistic high resolution 3D surfaces. In this paper we experiment with conditional denoising diffusion probabilistic models (DDPM) for height estimation from a single remotely sensed image and show promising results on the Vaihingen benchmark dataset.

Reliable probability estimation is of crucial importance in many real-world applications where there is inherent (aleatoric) uncertainty. Probability-estimation models are trained on observed outcomes (e.g. whether it has rained or not, or whether a patient has died or not), because the ground-truth probabilities of the events of interest are typically unknown. The problem is therefore analogous to binary classification, with the difference that the objective is to estimate probabilities rather than predicting the specific outcome. This work investigates probability estimation from high-dimensional data using deep neural networks. There exist several methods to improve the probabilities generated by these models but they mostly focus on model (epistemic) uncertainty. For problems with inherent uncertainty, it is challenging to evaluate performance without access to ground-truth probabilities. To address this, we build a synthetic dataset to study and compare different computable metrics. We evaluate existing methods on the synthetic data as well as on three real-world probability estimation tasks, all of which involve inherent uncertainty: precipitation forecasting from radar images, predicting cancer patient survival from histopathology images, and predicting car crashes from dashcam videos. We also give a theoretical analysis of a model for high-dimensional probability estimation which reproduces several of the phenomena evinced in our experiments. Finally, we propose a new method for probability estimation using neural networks, which modifies the training process to promote output probabilities that are consistent with empirical probabilities computed from the data. The method outperforms existing approaches on most metrics on the simulated as well as real-world data.

Surface normal estimation from a single image is an important task in 3D scene understanding. In this paper, we address two limitations shared by the existing methods: the inability to estimate the aleatoric uncertainty and lack of detail in the prediction. The proposed network estimates the per-pixel surface normal probability distribution. We introduce a new parameterization for the distribution, such that its negative log-likelihood is the angular loss with learned attenuation. The expected value of the angular error is then used as a measure of the aleatoric uncertainty. We also present a novel decoder framework where pixel-wise multi-layer perceptrons are trained on a subset of pixels sampled based on the estimated uncertainty. The proposed uncertainty-guided sampling prevents the bias in training towards large planar surfaces and improves the quality of prediction, especially near object boundaries and on small structures. Experimental results show that the proposed method outperforms the state-of-the-art in ScanNet and NYUv2, and that the estimated uncertainty correlates well with the prediction error. Code is available at https://github.com/baegwangbin/surface_normal_uncertainty.

Dynamic-scene reconstruction is almost always evaluated inside the observed time window, yet deployment settings such as AR overlays, robot interaction, and anticipatory planning need the future surface: the geometry at times beyond those captured. No standard benchmark measures this. We introduce FutureSurf, a controlled diagnostic benchmark and dataset for future-time surface reconstruction that trades scene diversity for exact future ground truth and falsification controls. A method trains on the observed first 75% of a sequence; we score its extracted per-frame surface on the held-out future by Chamfer distance, reporting absolute future CD as the primary score and the future/observed gap as a diagnostic. The dataset contains eight analytically defined controlled motions, including three falsification controls, with exact per-frame ground-truth meshes. We also provide a ground-truth-side recoverability oracle. The release includes split files, scoring code, a benchmark card, and Croissant metadata. On the controlled motions, the DG-Mesh backbone leaves a 2.7-4.1times gap even for futures predictable in principle (four of five recoverable from observed motion by a fixed rule), while the falsification controls behave as designed (the surface-invariant motion shows no gap). Beyond the contributed dataset, the gap persists across six animated DG-Mesh asset scenes and a second backbone, Deformable-3DGS (2.0-6.6times; both share a deformation-MLP temporal model). The benchmark also shows that future rendering quality and future-surface accuracy are statistically decoupled, so the novel-view-synthesis metrics the field reports do not track future geometry. The future error is structured, concentrating where the surface moves. The dataset, evaluation toolkit, and scoring code are available on Hugging Face and GitHub (https://github.com/Ricky-S/futuresurf).

When large language models (LMs) are applied in zero- or few-shot settings to discriminative tasks such as multiple-choice questions, their attentiveness (i.e., probability mass) is spread across many vocabulary tokens that are not valid choices. Such a spread across multiple surface forms with identical meaning is thought to cause an underestimation of a model's true performance, referred to as the "surface form competition" (SFC) hypothesis. This has motivated the introduction of various probability normalization methods. However, many core questions remain unanswered. How do we measure SFC or attentiveness? Are there direct ways of increasing attentiveness on valid choices? Does increasing attentiveness always improve task accuracy? We propose a mathematical formalism for studying this phenomenon, provide a metric for quantifying attentiveness, and identify a simple method for increasing it -- namely, in-context learning with even just one example containing answer choices. The formalism allows us to quantify SFC and bound its impact. Our experiments on three diverse datasets and six LMs reveal several surprising findings. For example, encouraging models to generate a valid answer choice can, in fact, be detrimental to task performance for some LMs, and prior probability normalization methods are less effective (sometimes even detrimental) to instruction-tuned LMs. We conclude with practical insights for effectively using prompted LMs for multiple-choice tasks.

In October 2007, a Research Proposal for the University of Sydney, Australia, the author suggested that biovie-physical phenomenon as `electrodynamic dependant biological vision', is governed by relativistic quantum laws and biovision. The phenomenon on the basis of `biovielectroluminescence', satisfies man/microbio/megabio/computer vision (MMMCV), as a robust candidate for physical and visual sciences. The general aim of this addendum is to present a refined text of Sections 1-3 of that proposal and highlighting the contents of its Appendix in form of a `Mechanisms' Section. We then briefly remind in an article aimed for December 2007, by appending two more equations into Section 3, a theoretical II-time scenario as a time model well-proposed for the phenomenon. The time model within the core of the proposal, plays a significant role in emphasizing the principle points on Objectives no. 1-8, Sub-hypothesis 3.1.2, mentioned in Article [arXiv:0710.0410]. It also expresses the time concept in terms of causing quantized energy f(|E|) of time |t|, emit in regard to shortening the probability of particle loci as predictable patterns of particle's un-occurred motion, a solution to Heisenberg's uncertainty principle (HUP) into a simplistic manner. We conclude that, practical frames via a time algorithm to this model, fixates such predictable patterns of motion of scenery bodies onto recordable observation points of a MMMCV system. It even suppresses/predicts superposition phenomena coming from a human subject and/or other bio-subjects for any decision making event, e.g., brainwave quantum patterns based on vision. Maintaining the existential probability of Riemann surfaces of II-time scenarios in the context of biovielectroluminescence, makes motion-prediction a possibility.

Realistic virtual humans play a crucial role in numerous industries, such as metaverse, intelligent healthcare, and self-driving simulation. But creating them on a large scale with high levels of realism remains a challenge. The utilization of deep implicit function sparks a new era of image-based 3D clothed human reconstruction, enabling pixel-aligned shape recovery with fine details. Subsequently, the vast majority of works locate the surface by regressing the deterministic implicit value for each point. However, should all points be treated equally regardless of their proximity to the surface? In this paper, we propose replacing the implicit value with an adaptive uncertainty distribution, to differentiate between points based on their distance to the surface. This simple ``value to distribution'' transition yields significant improvements on nearly all the baselines. Furthermore, qualitative results demonstrate that the models trained using our uncertainty distribution loss, can capture more intricate wrinkles, and realistic limbs. Code and models are available for research purposes at https://github.com/psyai-net/D-IF_release.

In recent years, the neural implicit surface has emerged as a powerful representation for multi-view surface reconstruction due to its simplicity and state-of-the-art performance. However, reconstructing smooth and detailed surfaces in indoor scenes from multi-view images presents unique challenges. Indoor scenes typically contain large texture-less regions, making the photometric loss unreliable for optimizing the implicit surface. Previous work utilizes monocular geometry priors to improve the reconstruction in indoor scenes. However, monocular priors often contain substantial errors in thin structure regions due to domain gaps and the inherent inconsistencies when derived independently from different views. This paper presents DebSDF to address these challenges, focusing on the utilization of uncertainty in monocular priors and the bias in SDF-based volume rendering. We propose an uncertainty modeling technique that associates larger uncertainties with larger errors in the monocular priors. High-uncertainty priors are then excluded from optimization to prevent bias. This uncertainty measure also informs an importance-guided ray sampling and adaptive smoothness regularization, enhancing the learning of fine structures. We further introduce a bias-aware signed distance function to density transformation that takes into account the curvature and the angle between the view direction and the SDF normals to reconstruct fine details better. Our approach has been validated through extensive experiments on several challenging datasets, demonstrating improved qualitative and quantitative results in reconstructing thin structures in indoor scenes, thereby outperforming previous work.

Where are the intersection points of diagonals of a regular n-gon located? What is the distribution of the intersection point of two random chords of a circle? We investigate these and related new questions in geometric probability, extend a largely forgotten result of Karamata, and elucidate its connection to the Bertrand paradox.

Equation discovery from data is a core challenge in machine learning for science, requiring the recovery of concise symbolic expressions that govern complex physical and geometric phenomena. Recent approaches with large language models (LLMs) show promise in symbolic regression, but their success often hinges on memorized formulas or overly simplified functional forms. Existing benchmarks exacerbate this limitation: they focus on scalar functions, ignore domain grounding, and rely on brittle string-matching based metrics that fail to capture scientific equivalence. We introduce SurfaceBench, first comprehensive benchmark for symbolic surface discovery. SurfaceBench comprises 183 tasks across 15 categories of symbolic complexity, spanning explicit, implicit, and parametric equation representation forms. Each task includes ground-truth equations, variable semantics, and synthetically sampled three dimensional data. Unlike prior SR datasets, our tasks reflect surface-level structure, resist LLM memorization through novel symbolic compositions, and are grounded in scientific domains such as fluid dynamics, robotics, electromagnetics, and geometry. To evaluate equation discovery quality, we pair symbolic checks with geometry-aware metrics such as Chamfer and Hausdorff distances, capturing both algebraic fidelity and spatial reconstruction accuracy. Our experiments reveal that state-of-the-art frameworks, while occasionally successful on specific families, struggle to generalize across representation types and surface complexities. SurfaceBench thus establishes a challenging and diagnostic testbed that bridges symbolic reasoning with geometric reconstruction, enabling principled benchmarking of progress in compositional generalization, data-driven scientific induction, and geometry-aware reasoning with LLMs. We release the code here: https://github.com/Sanchit-404/surfacebench

Creating plausible surfaces is an essential component in achieving a high degree of realism in rendering. To relieve artists, who create these surfaces in a time-consuming, manual process, automated retrieval of the spatially-varying Bidirectional Reflectance Distribution Function (SVBRDF) from a single mobile phone image is desirable. By leveraging a deep neural network, this casual capturing method can be achieved. The trained network can estimate per pixel normal, base color, metallic and roughness parameters from the Disney BRDF. The input image is taken with a mobile phone lit by the camera flash. The network is trained to compensate for environment lighting and thus learned to reduce artifacts introduced by other light sources. These losses contain a multi-scale discriminator with an additional perceptual loss, a rendering loss using a differentiable renderer, and a parameter loss. Besides the local precision, this loss formulation generates material texture maps which are globally more consistent. The network is set up as a generator network trained in an adversarial fashion to ensure that only plausible maps are produced. The estimated parameters not only reproduce the material faithfully in rendering but capture the style of hand-authored materials due to the more global loss terms compared to previous works without requiring additional post-processing. Both the resolution and the quality is improved.

The surface code family is a promising approach to implementing fault-tolerant quantum computations. Universal fault-tolerance requires error-corrected non-Clifford operations, in addition to Clifford gates, and for the former, it is imperative to experimentally demonstrate additional resources known as magic states. Another challenge is to efficiently embed surface codes into quantum hardware with connectivity constraints. This work simultaneously addresses both challenges by employing a qubit-efficient rotated heavy-hexagonal surface code for IBM quantum processors (ibm\_fez) and implementing the magic state injection protocol. Our work reports error thresholds for both logical bit- and phase-flip errors, of approx0.37% and approx0.31%, respectively, which are higher than the threshold values previously reported with traditional embedding. The post-selection-based preparation of logical magic states |H_Lrangle and |T_Lrangle achieve fidelities of 0.8806pm0.0002 and 0.8665pm0.0003, respectively, which are both above the magic state distillation threshold. Additionally, we report the minimum fidelity among injected arbitrary single logical qubit states as 0.8356pm0.0003. Our work demonstrates the potential for realising non-Clifford logical gates by producing high-fidelity logical magic states on IBM quantum devices.

This paper presents a learning-based method for transparent surface estimation from a single view polarization image. Existing shape from polarization(SfP) methods have the difficulty in estimating transparent shape since the inherent transmission interference heavily reduces the reliability of physics-based prior. To address this challenge, we propose the concept of physics-based prior, which is inspired by the characteristic that the transmission component in the polarization image has more noise than reflection. The confidence is used to determine the contribution of the interfered physics-based prior. Then, we build a network(TransSfP) with multi-branch architecture to avoid the destruction of relationships between different hierarchical inputs. To train and test our method, we construct a dataset for transparent shape from polarization with paired polarization images and ground-truth normal maps. Extensive experiments and comparisons demonstrate the superior accuracy of our method.

We propose a two-step framework for predicting the implied volatility surface over time without static arbitrage. In the first step, we select features to represent the surface and predict them over time. In the second step, we use the predicted features to construct the implied volatility surface using a deep neural network (DNN) model by incorporating constraints that prevent static arbitrage. We consider three methods to extract features from the implied volatility data: principal component analysis, variational autoencoder and sampling the surface, and we predict these features using LSTM. Using a long time series of implied volatility data for S\&P500 index options to train our models, we find two feature construction methods, sampling the surface and variational autoencoders combined with DNN for surface construction, are the best performers in out-of-sample prediction. In particular, they outperform a classical method substantially. Furthermore, the DNN model for surface construction not only removes static arbitrage, but also significantly reduces the prediction error compared with a standard interpolation method. Our framework can also be used to simulate the dynamics of the implied volatility surface without static arbitrage.

We study the properties of common loss surfaces through their Hessian matrix. In particular, in the context of deep learning, we empirically show that the spectrum of the Hessian is composed of two parts: (1) the bulk centered near zero, (2) and outliers away from the bulk. We present numerical evidence and mathematical justifications to the following conjectures laid out by Sagun et al. (2016): Fixing data, increasing the number of parameters merely scales the bulk of the spectrum; fixing the dimension and changing the data (for instance adding more clusters or making the data less separable) only affects the outliers. We believe that our observations have striking implications for non-convex optimization in high dimensions. First, the flatness of such landscapes (which can be measured by the singularity of the Hessian) implies that classical notions of basins of attraction may be quite misleading. And that the discussion of wide/narrow basins may be in need of a new perspective around over-parametrization and redundancy that are able to create large connected components at the bottom of the landscape. Second, the dependence of small number of large eigenvalues to the data distribution can be linked to the spectrum of the covariance matrix of gradients of model outputs. With this in mind, we may reevaluate the connections within the data-architecture-algorithm framework of a model, hoping that it would shed light into the geometry of high-dimensional and non-convex spaces in modern applications. In particular, we present a case that links the two observations: small and large batch gradient descent appear to converge to different basins of attraction but we show that they are in fact connected through their flat region and so belong to the same basin.

The creation of photorealistic virtual worlds requires the accurate modeling of 3D surface geometry for a wide range of objects. For this, meshes are appealing since they 1) enable fast physics-based rendering with realistic material and lighting, 2) support physical simulation, and 3) are memory-efficient for modern graphics pipelines. Recent work on reconstructing and statistically modeling 3D shape, however, has critiqued meshes as being topologically inflexible. To capture a wide range of object shapes, any 3D representation must be able to model solid, watertight, shapes as well as thin, open, surfaces. Recent work has focused on the former, and methods for reconstructing open surfaces do not support fast reconstruction with material and lighting or unconditional generative modelling. Inspired by the observation that open surfaces can be seen as islands floating on watertight surfaces, we parameterize open surfaces by defining a manifold signed distance field on watertight templates. With this parameterization, we further develop a grid-based and differentiable representation that parameterizes both watertight and non-watertight meshes of arbitrary topology. Our new representation, called Ghost-on-the-Shell (G-Shell), enables two important applications: differentiable rasterization-based reconstruction from multiview images and generative modelling of non-watertight meshes. We empirically demonstrate that G-Shell achieves state-of-the-art performance on non-watertight mesh reconstruction and generation tasks, while also performing effectively for watertight meshes.

Neural representations of 3D data have been widely adopted across various applications, particularly in recent work leveraging coordinate-based networks to model scalar or vector fields. However, these approaches face inherent challenges, such as handling thin structures and non-watertight geometries, which limit their flexibility and accuracy. In contrast, we propose a novel geometric data representation that models geometry as distributions-a powerful representation that makes no assumptions about surface genus, connectivity, or boundary conditions. Our approach uses diffusion models with a novel network architecture to learn surface point distributions, capturing fine-grained geometric details. We evaluate our representation qualitatively and quantitatively across various object types, demonstrating its effectiveness in achieving high geometric fidelity. Additionally, we explore applications using our representation, such as textured mesh representation, neural surface compression, dynamic object modeling, and rendering, highlighting its potential to advance 3D geometric learning.

We provide a theoretical framework to analyze the properties of frontal collisions of two growing interfaces considering different short range interactions between them. Due to their roughness, the collision events spread in time and form rough domain boundaries, which defines collision interfaces in time and space. We show that statistical properties of such interfaces depend on the kinetics of the growing interfaces before collision, but are independent of the details of their interaction and of their fluctuations during the collision. Those properties exhibit dynamic scaling with exponents related to the growth kinetics, but their distributions may be non-universal. These results are supported by simulations of lattice models with irreversible dynamics and local interactions. Relations to first passage processes are discussed and a possible application to grain boundary formation in two-dimensional materials is suggested.

We present As-Plausible-as-Possible (APAP) mesh deformation technique that leverages 2D diffusion priors to preserve the plausibility of a mesh under user-controlled deformation. Our framework uses per-face Jacobians to represent mesh deformations, where mesh vertex coordinates are computed via a differentiable Poisson Solve. The deformed mesh is rendered, and the resulting 2D image is used in the Score Distillation Sampling (SDS) process, which enables extracting meaningful plausibility priors from a pretrained 2D diffusion model. To better preserve the identity of the edited mesh, we fine-tune our 2D diffusion model with LoRA. Gradients extracted by SDS and a user-prescribed handle displacement are then backpropagated to the per-face Jacobians, and we use iterative gradient descent to compute the final deformation that balances between the user edit and the output plausibility. We evaluate our method with 2D and 3D meshes and demonstrate qualitative and quantitative improvements when using plausibility priors over geometry-preservation or distortion-minimization priors used by previous techniques. Our project page is at: https://as-plausible-aspossible.github.io/

Deep neural representations of 3D shapes as implicit functions have been shown to produce high fidelity models surpassing the resolution-memory trade-off faced by the explicit representations using meshes and point clouds. However, most such approaches focus on representing closed shapes. Unsigned distance function (UDF) based approaches have been proposed recently as a promising alternative to represent both open and closed shapes. However, since the gradients of UDFs vanish on the surface, it is challenging to estimate local (differential) geometric properties like the normals and tangent planes which are needed for many downstream applications in vision and graphics. There are additional challenges in computing these properties efficiently with a low-memory footprint. This paper presents a novel approach that models such surfaces using a new class of implicit representations called the closest surface-point (CSP) representation. We show that CSP allows us to represent complex surfaces of any topology (open or closed) with high fidelity. It also allows for accurate and efficient computation of local geometric properties. We further demonstrate that it leads to efficient implementation of downstream algorithms like sphere-tracing for rendering the 3D surface as well as to create explicit mesh-based representations. Extensive experimental evaluation on the ShapeNet dataset validate the above contributions with results surpassing the state-of-the-art.

Detecting surface anomalies of industrial materials poses a significant challenge within a myriad of industrial manufacturing processes. In recent times, various methodologies have emerged, capitalizing on the advantages of employing a network pre-trained on natural images for the extraction of representative features. Subsequently, these features are subjected to processing through a diverse range of techniques including memory banks, normalizing flow, and knowledge distillation, which have exhibited exceptional accuracy. This paper revisits approaches based on pre-trained features by introducing a novel method centered on target-specific embedding. To capture the most representative features of the texture under consideration, we employ a variant of a contrastive training procedure that incorporates both artificially generated defective samples and anomaly-free samples during training. Exploiting the intrinsic properties of surfaces, we derived a meaningful representation from the defect-free samples during training, facilitating a straightforward yet effective calculation of anomaly scores. The experiments conducted on the MVTEC AD and TILDA datasets demonstrate the competitiveness of our approach compared to state-of-the-art methods.

Visual surface anomaly detection aims to detect local image regions that significantly deviate from normal appearance. Recent surface anomaly detection methods rely on generative models to accurately reconstruct the normal areas and to fail on anomalies. These methods are trained only on anomaly-free images, and often require hand-crafted post-processing steps to localize the anomalies, which prohibits optimizing the feature extraction for maximal detection capability. In addition to reconstructive approach, we cast surface anomaly detection primarily as a discriminative problem and propose a discriminatively trained reconstruction anomaly embedding model (DRAEM). The proposed method learns a joint representation of an anomalous image and its anomaly-free reconstruction, while simultaneously learning a decision boundary between normal and anomalous examples. The method enables direct anomaly localization without the need for additional complicated post-processing of the network output and can be trained using simple and general anomaly simulations. On the challenging MVTec anomaly detection dataset, DRAEM outperforms the current state-of-the-art unsupervised methods by a large margin and even delivers detection performance close to the fully-supervised methods on the widely used DAGM surface-defect detection dataset, while substantially outperforming them in localization accuracy.

Counterfactual explanations are attracting significant attention due to the flourishing applications of machine learning models in consequential domains. A counterfactual plan consists of multiple possibilities to modify a given instance so that the model's prediction will be altered. As the predictive model can be updated subject to the future arrival of new data, a counterfactual plan may become ineffective or infeasible with respect to the future values of the model parameters. In this work, we study the counterfactual plans under model uncertainty, in which the distribution of the model parameters is partially prescribed using only the first- and second-moment information. First, we propose an uncertainty quantification tool to compute the lower and upper bounds of the probability of validity for any given counterfactual plan. We then provide corrective methods to adjust the counterfactual plan to improve the validity measure. The numerical experiments validate our bounds and demonstrate that our correction increases the robustness of the counterfactual plans in different real-world datasets.

The first direct detection of neutrinos at the LHC not only marks the beginning of a novel collider neutrino program at CERN but also motivates considering additional neutrino detectors to fully exploit the associated physics potential. We investigate the feasibility and physics potential of neutrino experiments located at the surface-level. A topographic desk study was performed to identify all points at which the LHC's neutrino beams exit the earth. The closest location lies about 9 km east of the CMS interaction point, at the bottom of Lake Geneva. Several detectors to be placed at this location are considered, including a water Cherenkov detector and an emulsion detector. The detector concepts are introduced, and projections for their contribution to the LHC forward neutrino program and searches for dark sector particles are presented. However, the dilution of the neutrino flux over distance reduces the neutrino yield significantly, limiting the physics potential of surface-level detectors compared to ones closer to the interaction point, including the proposed FPF.

We study orbits in a family of Markoff-like surfaces with extra off-diagonal terms over prime fields F_p. It is shown that, for a typical surface of this form, every non-trivial orbit has size divisible by p. This extends a theorem of W.Y. Chen from the Markoff surface itself to others in this family. The proof closely follows and elaborates on a recent argument of D.E. Martin. We expect that there is just one orbit generically. For some special parameters, we prove that there are at least two or four orbits. Cayley's cubic surface plays a role in parametrising the exceptional cases and dictating the number of solutions mod p.

High-quality view synthesis relies on volume rendering, splatting, or surface rendering. While surface rendering is typically the fastest, it struggles to accurately model fuzzy geometry like hair. In turn, alpha-blending techniques excel at representing fuzzy materials but require an unbounded number of samples per ray (P1). Further overheads are induced by empty space skipping in volume rendering (P2) and sorting input primitives in splatting (P3). We present a novel representation for real-time view synthesis where the (P1) number of sampling locations is small and bounded, (P2) sampling locations are efficiently found via rasterization, and (P3) rendering is sorting-free. We achieve this by representing objects as semi-transparent multi-layer meshes rendered in a fixed order. First, we model surface layers as signed distance function (SDF) shells with optimal spacing learned during training. Then, we bake them as meshes and fit UV textures. Unlike single-surface methods, our multi-layer representation effectively models fuzzy objects. In contrast to volume and splatting-based methods, our approach enables real-time rendering on low-power laptops and smartphones.

In this paper, we present Surf-D, a novel method for generating high-quality 3D shapes as Surfaces with arbitrary topologies using Diffusion models. Specifically, we adopt Unsigned Distance Field (UDF) as the surface representation, as it excels in handling arbitrary topologies, enabling the generation of complex shapes. While the prior methods explored shape generation with different representations, they suffer from limited topologies and geometry details. Moreover, it's non-trivial to directly extend prior diffusion models to UDF because they lack spatial continuity due to the discrete volume structure. However, UDF requires accurate gradients for mesh extraction and learning. To tackle the issues, we first leverage a point-based auto-encoder to learn a compact latent space, which supports gradient querying for any input point through differentiation to effectively capture intricate geometry at a high resolution. Since the learning difficulty for various shapes can differ, a curriculum learning strategy is employed to efficiently embed various surfaces, enhancing the whole embedding process. With pretrained shape latent space, we employ a latent diffusion model to acquire the distribution of various shapes. Our approach demonstrates superior performance in shape generation across multiple modalities and conducts extensive experiments in unconditional generation, category conditional generation, 3D reconstruction from images, and text-to-shape tasks.

Physical interactions follow a long-tailed distribution: a set of common and regular interactions dominates human experience and visual data, while a broad spectrum of rare and irregular interactions remains underrepresented. Although recent visual world models, including image and video generation models, achieve impressive realism on existing benchmarks, they primarily focus on simulating common physical interactions. This raises a central question: Do current visual world models internalize and generalize physical principles? In this work, we introduce Tailor-Bench, a benchmark that challenges world models to simulate irregular physical interactions. To enable systematic evaluation, we design three scenario modes that progressively challenge model reasoning: Regular scenarios reflect common tool-task pairs, Unconventional scenarios replace conventional tools with attribute-compatible substitutes to test affordance generalization, and Impossible scenarios introduce attribute-violating tools to probe constraint awareness. Additionally, we design two complementary settings under a unified evaluation protocol: predictive generation requires inferring outcomes without guidance, while descriptive generation specifies the target outcome for faithful realization. Our experimental results reveal a clear long-tail gap in physical world modeling: performance degrades from Regular to Unconventional and Impossible scenarios, indicating limited generalization beyond common interactions. Failure analysis further shows that models rely on superficial visual patterns: image models fail to realize correct state changes, while video models further suffer from temporal inconsistencies.

This work investigates the use of smooth neural networks for modeling dynamic variations of implicit surfaces under the level set equation (LSE). For this, it extends the representation of neural implicit surfaces to the space-time R^3times R, which opens up mechanisms for continuous geometric transformations. Examples include evolving an initial surface towards general vector fields, smoothing and sharpening using the mean curvature equation, and interpolations of initial conditions. The network training considers two constraints. A data term is responsible for fitting the initial condition to the corresponding time instant, usually R^3 times {0}. Then, a LSE term forces the network to approximate the underlying geometric evolution given by the LSE, without any supervision. The network can also be initialized based on previously trained initial conditions, resulting in faster convergence compared to the standard approach.

We prove stability for a class of heterogeneous catalysis models in the L_p-setting. We consider a setting in a finite three-dimensional pore of cylinder-like geometry, with the lateral walls acting as a catalytic surface. Under a reasonable condition on the involved parameters, we show that given equilibria are normally stable, i.e. solutions are attracted at an exponential rate. The potential incidence of instability is discussed as well.

Understanding the loss landscape is an important problem in machine learning. One key feature of the loss function, common to many neural network architectures, is the presence of exponentially many low lying local minima. Physical systems with similar energy landscapes may provide useful insights. In this work, we point out that black holes naturally give rise to such landscapes, owing to the existence of black hole entropy. For definiteness, we consider 1/8 BPS black holes in N = 8 string theory. These provide an infinite family of potential landscapes arising in the microscopic descriptions of corresponding black holes. The counting of minima amounts to black hole microstate counting. Moreover, the exact numbers of the minima for these landscapes are a priori known from dualities in string theory. Some of the minima are connected by paths of low loss values, resembling mode connectivity. We estimate the number of runs needed to find all the solutions. Initial explorations suggest that Stochastic Gradient Descent can find a significant fraction of the minima.

We propose a method, named DualMesh-UDF, to extract a surface from unsigned distance functions (UDFs), encoded by neural networks, or neural UDFs. Neural UDFs are becoming increasingly popular for surface representation because of their versatility in presenting surfaces with arbitrary topologies, as opposed to the signed distance function that is limited to representing a closed surface. However, the applications of neural UDFs are hindered by the notorious difficulty in extracting the target surfaces they represent. Recent methods for surface extraction from a neural UDF suffer from significant geometric errors or topological artifacts due to two main difficulties: (1) A UDF does not exhibit sign changes; and (2) A neural UDF typically has substantial approximation errors. DualMesh-UDF addresses these two difficulties. Specifically, given a neural UDF encoding a target surface S to be recovered, we first estimate the tangent planes of S at a set of sample points close to S. Next, we organize these sample points into local clusters, and for each local cluster, solve a linear least squares problem to determine a final surface point. These surface points are then connected to create the output mesh surface, which approximates the target surface. The robust estimation of the tangent planes of the target surface and the subsequent minimization problem constitute our core strategy, which contributes to the favorable performance of DualMesh-UDF over other competing methods. To efficiently implement this strategy, we employ an adaptive Octree. Within this framework, we estimate the location of a surface point in each of the octree cells identified as containing part of the target surface. Extensive experiments show that our method outperforms existing methods in terms of surface reconstruction quality while maintaining comparable computational efficiency.

The complexity of learning a concept class under Gaussian marginals in the difficult agnostic model is closely related to its L_1-approximability by low-degree polynomials. For any concept class with Gaussian surface area at most Γ, Klivans et al. (2008) show that degree d = O(Γ^2 / varepsilon^4) suffices to achieve an varepsilon-approximation. This leads to the best-known bounds on the complexity of learning a variety of concept classes. In this note, we improve their analysis by showing that degree d = tilde O (Γ^2 / varepsilon^2) is enough. In light of lower bounds due to Diakonikolas et al. (2021), this yields (near) optimal bounds on the complexity of agnostically learning polynomial threshold functions in the statistical query model. Our proof relies on a direct analogue of a construction of Feldman et al. (2020), who considered L_1-approximation on the Boolean hypercube.

The relationship between balance laws and the Principle of Virtual Work as well as the structure of contact interactions in continua remain foundational issues in Mechanics. In this work, we revisit these issues within the distributional framework emphasized by Paul Germain. We show that while the Principle of Virtual Work implies balance of forces and moments for nth-gradient continua, balance laws alone do not suffice to characterize equilibrium for n geq 2. We then reexamine Noll's classical theorem asserting that surface contact forces depend solely on the unit normal to the surface and identify the precise role of his additional assumptions, namely the absence of edge and wedge contact interactions and the boundedness of the surface contact density on the space of oriented surfaces. We demonstrate that these hypotheses fail for general higher-gradient continua. Consequently, the presence of curvature-dependent surface contact forces in such materials does not conflict with Noll's theorem and refutes his claim of their nonexistence.

This article provides an introduction to surface code quantum computing. We first estimate the size and speed of a surface code quantum computer. We then introduce the concept of the stabilizer, using two qubits, and extend this concept to stabilizers acting on a two-dimensional array of physical qubits, on which we implement the surface code. We next describe how logical qubits are formed in the surface code array and give numerical estimates of their fault-tolerance. We outline how logical qubits are physically moved on the array, how qubit braid transformations are constructed, and how a braid between two logical qubits is equivalent to a controlled-NOT. We then describe the single-qubit Hadamard, S and T operators, completing the set of required gates for a universal quantum computer. We conclude by briefly discussing physical implementations of the surface code. We include a number of appendices in which we provide supplementary information to the main text.

Accurate potential energy surface (PES) descriptions are essential for atomistic simulations of materials. Universal machine learning interatomic potentials (UMLIPs)^{1-3} offer a computationally efficient alternative to density functional theory (DFT)^4 for PES modeling across the periodic table. However, their accuracy today is fundamentally constrained due to a reliance on DFT relaxation data.^{5,6} Here, we introduce MatPES, a foundational PES dataset comprising sim 400,000 structures carefully sampled from 281 million molecular dynamics snapshots that span 16 billion atomic environments. We demonstrate that UMLIPs trained on the modestly sized MatPES dataset can rival, or even outperform, prior models trained on much larger datasets across a broad range of equilibrium, near-equilibrium, and molecular dynamics property benchmarks. We also introduce the first high-fidelity PES dataset based on the revised regularized strongly constrained and appropriately normed (r^2SCAN) functional^7 with greatly improved descriptions of interatomic bonding. The open source MatPES initiative emphasizes the importance of data quality over quantity in materials science and enables broad community-driven advancements toward more reliable, generalizable, and efficient UMLIPs for large-scale materials discovery and design.

3D surface reconstruction from point clouds is a key step in areas such as content creation, archaeology, digital cultural heritage, and engineering. Current approaches either try to optimize a non-data-driven surface representation to fit the points, or learn a data-driven prior over the distribution of commonly occurring surfaces and how they correlate with potentially noisy point clouds. Data-driven methods enable robust handling of noise and typically either focus on a global or a local prior, which trade-off between robustness to noise on the global end and surface detail preservation on the local end. We propose PPSurf as a method that combines a global prior based on point convolutions and a local prior based on processing local point cloud patches. We show that this approach is robust to noise while recovering surface details more accurately than the current state-of-the-art. Our source code, pre-trained model and dataset are available at: https://github.com/cg-tuwien/ppsurf

Recovery of an underlying scene geometry from multiview images stands as a long-time challenge in computer vision research. The recent promise leverages neural implicit surface learning and differentiable volume rendering, and achieves both the recovery of scene geometry and synthesis of novel views, where deep priors of neural models are used as an inductive smoothness bias. While promising for object-level surfaces, these methods suffer when coping with complex scene surfaces. In the meanwhile, traditional multi-view stereo can recover the geometry of scenes with rich textures, by globally optimizing the local, pixel-wise correspondences across multiple views. We are thus motivated to make use of the complementary benefits from the two strategies, and propose a method termed Helix-shaped neural implicit Surface learning or HelixSurf; HelixSurf uses the intermediate prediction from one strategy as the guidance to regularize the learning of the other one, and conducts such intertwined regularization iteratively during the learning process. We also propose an efficient scheme for differentiable volume rendering in HelixSurf. Experiments on surface reconstruction of indoor scenes show that our method compares favorably with existing methods and is orders of magnitude faster, even when some of existing methods are assisted with auxiliary training data. The source code is available at https://github.com/Gorilla-Lab-SCUT/HelixSurf.

We present Orbit-Space Geometric Probability Paths (OGPP), a particle-native flow-matching framework for generative modeling of particle systems. OGPP is motivated by two insights: (i) particles are defined up to permutation symmetries, so anonymous indexing inflates per-index target variance and yields curved, hard-to-learn flows; and (ii) particles live in physical space, so the flow terminal velocity has physical meaning and can encode geometric attributes, e.g., surface normals. OGPP instantiates three key components: (1) orbit-space canonicalization of the probability-path terminal endpoint, (2) particle index embeddings for role specialization, and (3) geometric probability paths with arc-length-aware terminal velocities that generate normals as a byproduct of the flow. We evaluate OGPP on minimal-surface benchmarks, where it reduces metric error by up to two orders of magnitude in a single inference step; on ShapeNet, where it matches the state of the art with 5x fewer steps and reaches airplane EMD comparable to DiT-3D with 26x fewer parameters and 5x fewer steps; and on single-shape encoding, where it produces normals and reconstructions competitive with 6D generators while operating entirely in 3D.

Sphere packing, Hilbert's eighteenth problem, asks for the densest arrangement of congruent spheres in n-dimensional Euclidean space. Although relevant to areas such as cryptography, crystallography, and medical imaging, the problem remains unresolved: beyond a few special dimensions, neither optimal packings nor tight upper bounds are known. Even a major breakthrough in dimension n=8, later recognised with a Fields Medal, underscores its difficulty. A leading technique for upper bounds, the three-point method, reduces the problem to solving large, high-precision semidefinite programs (SDPs). Because each candidate SDP may take days to evaluate, standard data-intensive AI approaches are infeasible. We address this challenge by formulating SDP construction as a sequential decision process, the SDP game, in which a policy assembles SDP formulations from a set of admissible components. Using a sample-efficient model-based framework that combines Bayesian optimisation with Monte Carlo Tree Search, we obtain new state-of-the-art upper bounds in dimensions 4-16, showing that model-based search can advance computational progress in longstanding geometric problems. Together, these results demonstrate that sample-efficient, model-based search can make tangible progress on mathematically rigid, evaluation limited problems, pointing towards a complementary direction for AI-assisted discovery beyond large-scale LLM-driven exploration.

Computational catalysis is playing an increasingly significant role in the design of catalysts across a wide range of applications. A common task for many computational methods is the need to accurately compute the minimum binding energy - the adsorption energy - for an adsorbate and a catalyst surface of interest. Traditionally, the identification of low energy adsorbate-surface configurations relies on heuristic methods and researcher intuition. As the desire to perform high-throughput screening increases, it becomes challenging to use heuristics and intuition alone. In this paper, we demonstrate machine learning potentials can be leveraged to identify low energy adsorbate-surface configurations more accurately and efficiently. Our algorithm provides a spectrum of trade-offs between accuracy and efficiency, with one balanced option finding the lowest energy configuration, within a 0.1 eV threshold, 86.33% of the time, while achieving a 1331x speedup in computation. To standardize benchmarking, we introduce the Open Catalyst Dense dataset containing nearly 1,000 diverse surfaces and 85,658 unique configurations.

Consider a topological surface Σ. We introduce the spectrum of a representation from the fundamental group of Σ to SL(2,R), which is a subset of projective measured lamination on the surface, which captures the directions along which the representation fails to be Fuchsian, and which characterizes the action of the mapping class group on this representation. In the case of the once-punctured torus, we show that the spectrum of a generic representation is a Cantor set, and that it completely describes the dynamics of the familly of locally constant cocycles above interval exchange transformations associated to the representation.

It is shown that in the complex trajectory representation of quantum mechanics, the Born's Psi^{\star}Ψprobability density can be obtained from the imaginary part of the velocity field of particles on the real axis. Extending this probability axiom to the complex plane, we first attempt to find a probability density by solving an appropriate conservation equation. The characteristic curves of this conservation equation are found to be the same as the complex paths of particles in the new representation. The boundary condition in this case is that the extended probability density should agree with the quantum probability rule along the real line. For the simple, time-independent, one-dimensional problems worked out here, we find that a conserved probability density can be derived from the velocity field of particles, except in regions where the trajectories were previously suspected to be nonviable. An alternative method to find this probability density in terms of a trajectory integral, which is easier to implement on a computer and useful for single particle solutions, is also presented. Most importantly, we show, by using the complex extension of Schrodinger equation, that the desired conservation equation can be derived from this definition of probability density.

A common assumption holds that enough observational and interventional data, given to a strong enough predictor, suffices. We report a failure mode that contradicts it. Across hundreds of structural causal models, on identified quantities a strong predictor and a Bayesian baseline both succeed, but on unidentified quantities (the couplings between counterfactual worlds) the predictor collapses to a point, on 28% of models to one no valid model can produce, while the truth is an admissible interval more data never narrows. The gap is structural: prediction cannot represent uncertainty over counterfactual couplings. We cast a world model as a single positive semidefinite coupling kernel K(T,T') over admissible worlds, whose diagonal is the ordinary posterior (what a predictor recovers) and whose off-diagonal is the cross-world coupling it cannot, which every counterfactual reads. The paper is the theory of that off-diagonal. It is real: two states with identical posteriors differ on a cross-world query, and the off-diagonal is the coupling that fixes counterfactuals. It can be bounded: positive semidefiniteness is partial-identifying information the marginals lack, and enforcing it bounds counterfactuals in polynomial time where the exact response-type program is intractable. Logical structure sharpens it: ontology axioms tighten the bound by up to a third, propagating to couplings they never touch. It can be acquired: targeted scars, constraints learned from encountered infeasibilities, close the gap several times faster than untargeted ones. Its full reconstruction is approximate counting of the admissible worlds, tractable below the Sly-Sun threshold and inapproximable above; we do not claim to beat the worst case.

Representing 3D surfaces as level sets of continuous functions over R^3 is the common denominator of neural implicit representations, which recently enabled remarkable progress in geometric deep learning and computer vision tasks. In order to represent 3D motion within this framework, it is often assumed (either explicitly or implicitly) that the transformations which a surface may undergo are homeomorphic: this is not necessarily true, for instance, in the case of fluid dynamics. In order to represent more general classes of deformations, we propose to apply this theoretical framework as regularizers for the optimization of simple 4D implicit functions (such as signed distance fields). We show that our representation is capable of capturing both homeomorphic and topology-changing deformations, while also defining correspondences over the continuously-reconstructed surfaces.

We prove a new existence theorem for proper solutions of Huisken and Ilmanen's weak inverse mean curvature flow, assuming certain non-degeneracy conditions on the isoperimetric profile. In particular, no curvature assumption is imposed in our existence theorem.

The aim of this paper is to present an elementary computable theory of probability, random variables and stochastic processes. The probability theory is baed on existing approaches using valuations and lower integrals. Various approaches to random variables are discussed, including the approach based on completions in a Polish space. We apply the theory to the study of stochastic dynamical systems in discrete-time, and give a brief exposition of the Wiener process as a foundation for stochastic differential equations. The theory is based within the framework of type-two effectivity, so has an explicit direct link with Turing computation, and is expressed in a system of computable types and operations, so has a clean mathematical description.

The Omega numbers-the halting probabilities of universal prefix-free machines-are known to be exactly the Martin-L{\"o}f random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-L{\"o}f random left-c.e. real alpha, a universal prefix-free machine U whose halting probability is alpha. We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real alpha, one cannot uniformly produce a left-c.e. real beta such that alpha -- beta is neither left-c.e. nor right-c.e.

We present an algorithm for efficiently exploring inequivalent Calabi-Yau threefold hypersurfaces in toric varieties. A direct enumeration of fine, regular, star triangulations (FRSTs) of polytopes in the Kreuzer-Skarke database is foreseeably impossible due to the large count of distinct FRSTs. Moreover, such an enumeration is needlessly redundant because many such triangulations have the same restrictions to 2-faces and hence, by Wall's theorem, lead to equivalent Calabi-Yau threefolds. We show that this redundancy can be circumvented by finding a height vector in the strict interior of the intersection of the secondary cones associated with each 2-face triangulation. We demonstrate that such triangulations are generated with orders of magnitude fewer operations than the naive approach of generating all FRSTs and selecting only those differing on 2-faces. Similar methods are also presented to directly generate (the support of) the secondary subfan of all fine triangulations, relevant for random sampling of FRSTs.

A key step in any scanning-based asset creation workflow is to convert unordered point clouds to a surface. Classical methods (e.g., Poisson reconstruction) start to degrade in the presence of noisy and partial scans. Hence, deep learning based methods have recently been proposed to produce complete surfaces, even from partial scans. However, such data-driven methods struggle to generalize to new shapes with large geometric and topological variations. We present Points2Surf, a novel patch-based learning framework that produces accurate surfaces directly from raw scans without normals. Learning a prior over a combination of detailed local patches and coarse global information improves generalization performance and reconstruction accuracy. Our extensive comparison on both synthetic and real data demonstrates a clear advantage of our method over state-of-the-art alternatives on previously unseen classes (on average, Points2Surf brings down reconstruction error by 30\% over SPR and by 270\%+ over deep learning based SotA methods) at the cost of longer computation times and a slight increase in small-scale topological noise in some cases. Our source code, pre-trained model, and dataset are available on: https://github.com/ErlerPhilipp/points2surf

This article concerns the computational complexity of a fundamental problem in number theory: counting points on curves and surfaces over finite fields. There is no subexponential-time algorithm known and it is unclear if it can be NP-hard. Given a curve, we present the first efficient Arthur-Merlin protocol to certify its point-count, its Jacobian group structure, and its Hasse-Weil zeta function. We extend this result to a smooth projective surface to certify the factor P_{1}(T), corresponding to the first Betti number, of the zeta function; by using the counting oracle. We give the first algorithm to compute P_{1}(T) that is poly(log q)-time if the degree D of the input surface is fixed; and in quantum poly(Dlog q)-time in general. Our technique in the curve case, is to sample hash functions using the Weil and Riemann-Roch bounds, to certify the group order of its Jacobian. For higher dimension varieties, we first reduce to the case of a surface, which is fibred as a Lefschetz pencil of hyperplane sections over P^{1}. The formalism of vanishing cycles, and the inherent big monodromy, enable us to prove an effective version of Deligne's `theoreme du pgcd' using the hard-Lefschetz theorem and an equidistribution result due to Katz. These reduce our investigations to that of computing the zeta function of a curve, defined over a finite field extension F_{Q}/F_{q} of poly-bounded degree. This explicitization of the theory yields the first nontrivial upper bounds on the computational complexity.

3D reconstruction of indoor and urban environments is a prominent research topic with various downstream applications. However, existing geometric priors for addressing low-texture regions in indoor and urban settings often lack global consistency. Moreover, Gaussian Splatting and implicit SDF fields often suffer from discontinuities or exhibit computational inefficiencies, resulting in a loss of detail. To address these issues, we propose an Atlanta-world guided implicit-structured Gaussian Splatting that achieves smooth indoor and urban scene reconstruction while preserving high-frequency details and rendering efficiency. By leveraging the Atlanta-world model, we ensure the accurate surface reconstruction for low-texture regions, while the proposed novel implicit-structured GS representations provide smoothness without sacrificing efficiency and high-frequency details. Specifically, we propose a semantic GS representation to predict the probability of all semantic regions and deploy a structure plane regularization with learnable plane indicators for global accurate surface reconstruction. Extensive experiments demonstrate that our method outperforms state-of-the-art approaches in both indoor and urban scenes, delivering superior surface reconstruction quality.

The relationship between material properties and independent variables such as temperature, external field or time, is usually represented by a curve or surface in a multi-dimensional space. Determining such a curve or surface requires a series of experiments or calculations which are often time and cost consuming. A general strategy uses an appropriate utility function to sample the space to recommend the next optimal experiment or calculation within an active learning loop. However, knowing what the optimal sampling strategy to use to minimize the number of experiments is an outstanding problem. We compare a number of strategies based on directed exploration on several materials problems of varying complexity using a Kriging based model. These include one dimensional curves such as the fatigue life curve for 304L stainless steel and the Liquidus line of the Fe-C phase diagram, surfaces such as the Hartmann 3 function in 3D space and the fitted intermolecular potential for Ar-SH, and a four dimensional data set of experimental measurements for BaTiO3 based ceramics. We also consider the effects of experimental noise on the Hartmann 3 function. We find that directed exploration guided by maximum variance provides better performance overall, converging faster across several data sets. However, for certain problems, the trade-off methods incorporating exploitation can perform at least as well, if not better than maximum variance. Thus, we discuss how the choice of the utility function depends on the distribution of the data, the model performance and uncertainties, additive noise as well as the budget.

3D Gaussian Splatting techniques have enabled efficient photo-realistic rendering of static scenes. Recent works have extended these approaches to support surface reconstruction and tracking. However, tracking dynamic surfaces with 3D Gaussians remains challenging due to complex topology changes, such as surfaces appearing, disappearing, or splitting. To address these challenges, we propose GSTAR, a novel method that achieves photo-realistic rendering, accurate surface reconstruction, and reliable 3D tracking for general dynamic scenes with changing topology. Given multi-view captures as input, GSTAR binds Gaussians to mesh faces to represent dynamic objects. For surfaces with consistent topology, GSTAR maintains the mesh topology and tracks the meshes using Gaussians. In regions where topology changes, GSTAR adaptively unbinds Gaussians from the mesh, enabling accurate registration and the generation of new surfaces based on these optimized Gaussians. Additionally, we introduce a surface-based scene flow method that provides robust initialization for tracking between frames. Experiments demonstrate that our method effectively tracks and reconstructs dynamic surfaces, enabling a range of applications. Our project page with the code release is available at https://eth-ait.github.io/GSTAR/.

Recently, 3D Gaussian splatting (3D-GS) has achieved great success in reconstructing and rendering real-world scenes. To transfer the high rendering quality to generation tasks, a series of research works attempt to generate 3D-Gaussian assets from text. However, the generated assets have not achieved the same quality as those in reconstruction tasks. We observe that Gaussians tend to grow without control as the generation process may cause indeterminacy. Aiming at highly enhancing the generation quality, we propose a novel framework named GaussianDreamerPro. The main idea is to bind Gaussians to reasonable geometry, which evolves over the whole generation process. Along different stages of our framework, both the geometry and appearance can be enriched progressively. The final output asset is constructed with 3D Gaussians bound to mesh, which shows significantly enhanced details and quality compared with previous methods. Notably, the generated asset can also be seamlessly integrated into downstream manipulation pipelines, e.g. animation, composition, and simulation etc., greatly promoting its potential in wide applications. Demos are available at https://taoranyi.com/gaussiandreamerpro/.

The paper considers a new quantitative-qualitative proximity measure for the features of information objects, where data enters a common information resource from several sources independently. The goal is to determine the possibility of their relation to the same physical object (observation object). The proposed measure accounts for the possibility of differences in individual feature values - both quantitative and qualitative - caused by existing determination errors. To analyze the proximity of quantitative feature values, the author employs a probabilistic measure; for qualitative features, a measure of possibility is used. The paper demonstrates the feasibility of the proposed measure by checking its compliance with the axioms required of any measure. Unlike many known measures, the proposed approach does not require feature value transformation to ensure comparability. The work also proposes several variants of measures to determine the proximity of information objects (IO) based on a group of diverse features.

Can a micron sized sack of interacting molecules understand, and adapt to a constantly-fluctuating environment? Cellular life provides an existence proof in the affirmative, but the principles that allow for life's existence are far from being proven. One challenge in engineering and understanding biochemical computation is the intrinsic noise due to chemical fluctuations. In this paper, we draw insights from machine learning theory, chemical reaction network theory, and statistical physics to show that the broad and biologically relevant class of detailed balanced chemical reaction networks is capable of representing and conditioning complex distributions. These results illustrate how a biochemical computer can use intrinsic chemical noise to perform complex computations. Furthermore, we use our explicit physical model to derive thermodynamic costs of inference.

The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability --- via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.

Words of estimative probability (WEP) are expressions of a statement's plausibility (probably, maybe, likely, doubt, likely, unlikely, impossible...). Multiple surveys demonstrate the agreement of human evaluators when assigning numerical probability levels to WEP. For example, highly likely corresponds to a median chance of 0.90+-0.08 in Fagen-Ulmschneider (2015)'s survey. In this work, we measure the ability of neural language processing models to capture the consensual probability level associated to each WEP. Firstly, we use the UNLI dataset (Chen et al., 2020) which associates premises and hypotheses with their perceived joint probability p, to construct prompts, e.g. "[PREMISE]. [WEP], [HYPOTHESIS]." and assess whether language models can predict whether the WEP consensual probability level is close to p. Secondly, we construct a dataset of WEP-based probabilistic reasoning, to test whether language models can reason with WEP compositions. When prompted "[EVENTA] is likely. [EVENTB] is impossible.", a causal language model should not express that [EVENTA&B] is likely. We show that both tasks are unsolved by off-the-shelf English language models, but that fine-tuning leads to transferable improvement.

Real-world phenomena do not generate arbitrary variability: their signals concentrate on compact, low-variability subsets of functional space, enabling rapid generalization from few examples. A small child can recognize a dog after extremely limited exposure because the perceptual manifold of "dog" is compact, structured, and low-dimensional. We formalize this principle through a deterministic functional-topological framework in which the set of valid realizations produced by a physical process forms a compact subset of a Banach space, endowed with stable invariants, a finite Hausdorff radius, and an induced continuous perceptual functional. This geometry provides explicit limits on knowledge, conditions for identifiability, and guarantees for generalization from sparse evidence -- properties fundamental to both natural and artificial intelligence. Across electromechanical, electrochemical, and physiological domains, we show that real-world processes consistently generate compact perceptual manifolds with the same geometric characteristics. Their boundaries can be discovered in a fully self-supervised manner as the empirical radius saturates with increasing sampling, even when the governing equations are unknown. These results demonstrate that deterministic functional topology offers a unified mathematical foundation for perception, representation, and world-model construction. It provides a geometric explanation for why biological learners and self-supervised AI systems can generalize from few observations, and establishes compact perceptual manifolds as a fundamental building block for future AI architectures. Finally, this work unifies biological perception and modern self-supervised models under a single geometric principle: both derive their generalization ability from the compactness and invariants of real-world perceptual manifolds.

Despite advances in depth estimation, flying points remain a persistent failure mode: near object boundaries, depth estimators often predict spurious 3D points in the empty space between foreground and background surfaces. We trace this artifact to a standard modeling choice: assigning each pixel a single depth hypothesis. At boundaries, a pixel can straddle a foreground and a background surface, so its true depth is ambiguous between the two. A model that predicts a single depth cannot keep both possibilities, so training instead pulls the prediction toward an intermediate depth that lies on neither surface. We address this with MDA, a mixture-density representation that lets the model predict multiple depth hypotheses and their associated probabilities for each pixel. Near boundaries, different hypotheses can align with different surfaces, and the decoded depth is selected from one of these hypotheses rather than placed in the empty space between them. Across different backbones, MDA substantially improves boundary reconstruction and largely removes flying-point artifacts even under severe input blur, while adding negligible runtime overhead. The same mixture-density framework naturally extends to transparent objects, where it predicts multiple depth layers at transparent pixels, and to sky regions, where a dedicated component separates the unbounded sky from finite-depth regions, producing flying-point-free skylines. Project Page: https://biansy000.github.io/mda-site/.

World models for embodied AI must be physically viable: constructed to answer intervention queries by representing the physical structure governing action outcomes, rather than merely predicting future observations. Existing observation-predictive world models can produce visually plausible but physically wrong rollouts. This failure is structural; distinct physical systems can look identical yet diverge under intervention. We expose this problem with controlled benchmarks that fix the visible scene while varying latent physics. We show that such models may recommend infeasible actions, mispredict interaction outcomes, or certify unsafe behavior. We argue that embodied AI requires world models that identify the simplest physical abstraction sufficient to answer an intervention query. Such a model comprises modular components, including environment representation, latent state and parameter estimation, action specification, interventional dynamics, and query-level response. An autonomous orchestrator should identify the relevant abstraction and compose compatible learned and structured components per query. When closed-form physics is unavailable, uncertain, or costly, the transition model may be analytic, simulated, learned, or hybrid, but it must preserve the structure that determines interventional outcomes. This decomposition makes the model interpretable, its components verifiable, and its outputs auditable against the query. It also provides a design principle for new world models and a feasibility test for existing ones: the right abstraction is not the most detailed model of the world, but the simplest model that preserves the distinctions relevant to the query. We demonstrate this approach on queries that existing systems fail to answer correctly, and outline how an orchestrator can dynamically assemble and adapt physically viable models for planning, control, and verification.

Mesh deformation plays a pivotal role in many 3D vision tasks including dynamic simulations, rendering, and reconstruction. However, defining an efficient discrepancy between predicted and target meshes remains an open problem. A prevalent approach in current deep learning is the set-based approach which measures the discrepancy between two surfaces by comparing two randomly sampled point-clouds from the two meshes with Chamfer pseudo-distance. Nevertheless, the set-based approach still has limitations such as lacking a theoretical guarantee for choosing the number of points in sampled point-clouds, and the pseudo-metricity and the quadratic complexity of the Chamfer divergence. To address these issues, we propose a novel metric for learning mesh deformation. The metric is defined by sliced Wasserstein distance on meshes represented as probability measures that generalize the set-based approach. By leveraging probability measure space, we gain flexibility in encoding meshes using diverse forms of probability measures, such as continuous, empirical, and discrete measures via varifold representation. After having encoded probability measures, we can compare meshes by using the sliced Wasserstein distance which is an effective optimal transport distance with linear computational complexity and can provide a fast statistical rate for approximating the surface of meshes. To the end, we employ a neural ordinary differential equation (ODE) to deform the input surface into the target shape by modeling the trajectories of the points on the surface. Our experiments on cortical surface reconstruction demonstrate that our approach surpasses other competing methods in multiple datasets and metrics.

Graphical User Interface (GUI) grounding aims to translate natural language instructions into executable screen coordinates, enabling automated GUI interaction. Nevertheless, incorrect grounding can result in costly, hard-to-reverse actions (e.g., erroneous payment approvals), raising concerns about model reliability. In this paper, we introduce SafeGround, an uncertainty-aware framework for GUI grounding models that enables risk-aware predictions through calibrations before testing. SafeGround leverages a distribution-aware uncertainty quantification method to capture the spatial dispersion of stochastic samples from outputs of any given model. Then, through the calibration process, SafeGround derives a test-time decision threshold with statistically guaranteed false discovery rate (FDR) control. We apply SafeGround on multiple GUI grounding models for the challenging ScreenSpot-Pro benchmark. Experimental results show that our uncertainty measure consistently outperforms existing baselines in distinguishing correct from incorrect predictions, while the calibrated threshold reliably enables rigorous risk control and potentials of substantial system-level accuracy improvements. Across multiple GUI grounding models, SafeGround improves system-level accuracy by up to 5.38% percentage points over Gemini-only inference.

Recent advances in deep generative models have led to immense progress in 3D shape synthesis. While existing models are able to synthesize shapes represented as voxels, point-clouds, or implicit functions, these methods only indirectly enforce the plausibility of the final 3D shape surface. Here we present a 3D shape synthesis framework (SurfGen) that directly applies adversarial training to the object surface. Our approach uses a differentiable spherical projection layer to capture and represent the explicit zero isosurface of an implicit 3D generator as functions defined on the unit sphere. By processing the spherical representation of 3D object surfaces with a spherical CNN in an adversarial setting, our generator can better learn the statistics of natural shape surfaces. We evaluate our model on large-scale shape datasets, and demonstrate that the end-to-end trained model is capable of generating high fidelity 3D shapes with diverse topology.

The increasing population of objects in geostationary orbit has raised concerns about the potential risks posed by debris clouds resulting from fragmentation. The short-term evolution and associated hazards of debris generated by collisions in the geostationary region is investigated in this study. The initial distribution of two debris clouds is modeled using a single probability density function.The combined distribution of the evolved clouds is determined by solving boundary value problems.The risks associated with these debris clouds are evaluated by calculating the instantaneous impact rate and cumulative collision probability.The probability of collisions with millimeter-sized fragments may increase to 1% within 36 hours, while the probability of collisions with fragments 5 cm or larger is approximately 10^{-5}.These findings underscore the vulnerability of the geostationary region to space traffic accidents.

The theories of offline and online reinforcement learning, despite having evolved in parallel, have begun to show signs of the possibility for a unification, with algorithms and analysis techniques for one setting often having natural counterparts in the other. However, the notion of density ratio modeling, an emerging paradigm in offline RL, has been largely absent from online RL, perhaps for good reason: the very existence and boundedness of density ratios relies on access to an exploratory dataset with good coverage, but the core challenge in online RL is to collect such a dataset without having one to start. In this work we show -- perhaps surprisingly -- that density ratio-based algorithms have online counterparts. Assuming only the existence of an exploratory distribution with good coverage, a structural condition known as coverability (Xie et al., 2023), we give a new algorithm (GLOW) that uses density ratio realizability and value function realizability to perform sample-efficient online exploration. GLOW addresses unbounded density ratios via careful use of truncation, and combines this with optimism to guide exploration. GLOW is computationally inefficient; we complement it with a more efficient counterpart, HyGLOW, for the Hybrid RL setting (Song et al., 2022) wherein online RL is augmented with additional offline data. HyGLOW is derived as a special case of a more general meta-algorithm that provides a provable black-box reduction from hybrid RL to offline RL, which may be of independent interest.

Much research has been carried out on large language models (LLMs) and LLM-powered agentic workflows. However, many works within the field state emergence of, ascribe to, or assume, generalised anthropomorphic attributes to them (e.g., morality or understanding of natural language). Our goal is not to argue in favour or against the existence of these attributes, but to point out that these conclusions could be incorrect. For this we build and train a simple neural network on the videogame Age of Empires II, and note that any entity in a sufficiently-powerful substrate, such as LEGO or the Greater Boston Area, could also present such attributes. Hence, the purported anthropomorphic attributes of LLMs are empirically non-unique: although some properties (e.g., responses to prompts) could remain invariant, others, such as the interpretation of their perceived behaviour, might change with the substrate. Thus, any empirically-grounded discussion on these attributes requires explicit measurement criteria; otherwise the interpretation is left to the representation. We then show that assuming that these attributes exist or not in a system, independent of the substrate and in a generalised way, leads to either circular or uninformative conclusions. This is regardless of the experimenter's viewpoint on the subject, or whether the outcome shows existence or non-existence. Finally we propose a 'null' assumption, where one assumes LLM non-uniqueness instead of assuming anthropomorphic attributes to set up an experiment, along with examples of it. We also discuss potential objections to our work, briefly survey the field, and prove that Age of Empires II is functionally- and Turing-complete.

Building world models that accurately and comprehensively represent the real world is the utmost aspiration for conditional image generative models as it would enable their use as world simulators. For these models to be successful world models, they should not only excel at image quality and prompt-image consistency but also ensure high representation diversity. However, current research in generative models mostly focuses on creative applications that are predominantly concerned with human preferences of image quality and aesthetics. We note that generative models have inference time mechanisms - or knobs - that allow the control of generation consistency, quality, and diversity. In this paper, we use state-of-the-art text-to-image and image-and-text-to-image models and their knobs to draw consistency-diversity-realism Pareto fronts that provide a holistic view on consistency-diversity-realism multi-objective. Our experiments suggest that realism and consistency can both be improved simultaneously; however there exists a clear tradeoff between realism/consistency and diversity. By looking at Pareto optimal points, we note that earlier models are better at representation diversity and worse in consistency/realism, and more recent models excel in consistency/realism while decreasing significantly the representation diversity. By computing Pareto fronts on a geodiverse dataset, we find that the first version of latent diffusion models tends to perform better than more recent models in all axes of evaluation, and there exist pronounced consistency-diversity-realism disparities between geographical regions. Overall, our analysis clearly shows that there is no best model and the choice of model should be determined by the downstream application. With this analysis, we invite the research community to consider Pareto fronts as an analytical tool to measure progress towards world models.

The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere.

We present a probabilistic model for point cloud generation, which is fundamental for various 3D vision tasks such as shape completion, upsampling, synthesis and data augmentation. Inspired by the diffusion process in non-equilibrium thermodynamics, we view points in point clouds as particles in a thermodynamic system in contact with a heat bath, which diffuse from the original distribution to a noise distribution. Point cloud generation thus amounts to learning the reverse diffusion process that transforms the noise distribution to the distribution of a desired shape. Specifically, we propose to model the reverse diffusion process for point clouds as a Markov chain conditioned on certain shape latent. We derive the variational bound in closed form for training and provide implementations of the model. Experimental results demonstrate that our model achieves competitive performance in point cloud generation and auto-encoding. The code is available at https://github.com/luost26/diffusion-point-cloud.

We address the problem of optimizing a Brownian motion. We consider a (random) realization W of a Brownian motion with input space in [0,1]. Given W, our goal is to return an ε-approximation of its maximum using the smallest possible number of function evaluations, the sample complexity of the algorithm. We provide an algorithm with sample complexity of order log^2(1/ε). This improves over previous results of Al-Mharmah and Calvin (1996) and Calvin et al. (2017) which provided only polynomial rates. Our algorithm is adaptive---each query depends on previous values---and is an instance of the optimism-in-the-face-of-uncertainty principle.

In this note we explore the distribution of vacuum expectation values of the superpotential W_0 in explicit Type IIB flux compactifications. We show that the distribution can be approximated, universally across geometries, by a two-dimensional Gaussian with a model dependent standard deviation. We identify this behaviour in 20 Calabi-Yau orientifold compactifications with between two and five complex structure moduli by constructing a total of O(10^7) flux vacua. We observe a characteristic scaling behaviour of the width σ of our distributions with respect to the D3-charge contributions N_{flux} from fluxes which can be approximated by σsim N_{text{flux}}. This W_0 distribution implies that locating small values of |W_0| as a preferred regime associated with classes of string theory solutions typically featuring hierarchies, simplifies to the basic statement of finding small Euclidean norms of normally distributed values. We do also identify small modifications to this Gaussian behaviour in our samples which might be seen as indications for the breakdown of the continuous flux approximation commonly used in the context of statistical analyses of the flux landscape.

We introduce a theoretical framework for differentiable surface evolution that allows discrete topology changes through the use of topological derivatives for variational optimization of image functionals. While prior methods for inverse rendering of geometry rely on silhouette gradients for topology changes, such signals are sparse. In contrast, our theory derives topological derivatives that relate the introduction of vanishing holes and phases to changes in image intensity. As a result, we enable differentiable shape perturbations in the form of hole or phase nucleation. We validate the proposed theory with optimization of closed curves in 2D and surfaces in 3D to lend insights into limitations of current methods and enable improved applications such as image vectorization, vector-graphics generation from text prompts, single-image reconstruction of shape ambigrams and multi-view 3D reconstruction.

While surface-based view synthesis algorithms are appealing due to their low computational requirements, they often struggle to reproduce thin structures. In contrast, more expensive methods that model the scene's geometry as a volumetric density field (e.g. NeRF) excel at reconstructing fine geometric detail. However, density fields often represent geometry in a "fuzzy" manner, which hinders exact localization of the surface. In this work, we modify density fields to encourage them to converge towards surfaces, without compromising their ability to reconstruct thin structures. First, we employ a discrete opacity grid representation instead of a continuous density field, which allows opacity values to discontinuously transition from zero to one at the surface. Second, we anti-alias by casting multiple rays per pixel, which allows occlusion boundaries and subpixel structures to be modelled without using semi-transparent voxels. Third, we minimize the binary entropy of the opacity values, which facilitates the extraction of surface geometry by encouraging opacity values to binarize towards the end of training. Lastly, we develop a fusion-based meshing strategy followed by mesh simplification and appearance model fitting. The compact meshes produced by our model can be rendered in real-time on mobile devices and achieve significantly higher view synthesis quality compared to existing mesh-based approaches.

The presence of point defects such as vacancies plays an important role in material design. Here, we demonstrate that a graph neural network (GNN) model trained only on perfect materials can also be used to predict vacancy formation energies (E_{vac}) of defect structures without the need for additional training data. Such GNN-based predictions are considerably faster than density functional theory (DFT) calculations with reasonable accuracy and show the potential that GNNs are able to capture a functional form for energy predictions. To test this strategy, we developed a DFT dataset of 508 E_{vac} consisting of 3D elemental solids, alloys, oxides, nitrides, and 2D monolayer materials. We analyzed and discussed the applicability of such direct and fast predictions. We applied the model to predict 192494 E_{vac} for 55723 materials in the JARVIS-DFT database.

World modeling requires forecasting uncertain futures while preserving information useful for downstream perception. Existing visual world models often struggle to satisfy both goals: VAE-based stochastic models operate in low-dimensional reconstruction latents, which can limit perception performance, while deterministic predictors using strong pretrained features collapse multimodal futures into a single blurry mean. In this work, we propose FlowWM, a stochastic world model that performs flow matching directly within pretrained feature space (e.g., DINOv3). This is challenging because pretrained features are substantially high-dimensional, making standard diffusion recipes suboptimal. To address this, we investigate the design choices needed for feature-space flow matching and introduce a differentiable one-step projection mechanism that enables efficient training with temporal consistency and task-driven objectives. We evaluate FlowWM on two benchmarks: a synthetic benchmark for systematic evaluation of accuracy and diversity, and a real-world benchmark FuturePerception. FlowWM improves perception performance, mode coverage, and horizon robustness, validating our proposed design for stochastic world modeling in high-dimensional feature spaces.

In this paper we present SurfaceNet, an approach for estimating spatially-varying bidirectional reflectance distribution function (SVBRDF) material properties from a single image. We pose the problem as an image translation task and propose a novel patch-based generative adversarial network (GAN) that is able to produce high-quality, high-resolution surface reflectance maps. The employment of the GAN paradigm has a twofold objective: 1) allowing the model to recover finer details than standard translation models; 2) reducing the domain shift between synthetic and real data distributions in an unsupervised way. An extensive evaluation, carried out on a public benchmark of synthetic and real images under different illumination conditions, shows that SurfaceNet largely outperforms existing SVBRDF reconstruction methods, both quantitatively and qualitatively. Furthermore, SurfaceNet exhibits a remarkable ability in generating high-quality maps from real samples without any supervision at training time.

Agentic theorem provers combine a reasoning model, retrieval, search, and a proof assistant verifier, yet it remains unclear which components actually improve finite-budget proof success and why they help on real mathematical workloads. We study this question through statistical provability: the probability of reaching a verified proof within a budget on a specified stream of theorem instances. We model formal proof search as a finite-horizon reachability MDP with deterministic verifier dynamics, and show that under a faithful state abstraction the optimal success probability coincides with ordinary syntactic provability. We then analyze a simple but practically important pipeline: depth-wise offline action-value regression followed by greedy test-time proving. Our main theorem bounds the provability gap between the learned prover and the optimal prover by an occupancy-weighted sum of uniform action-value errors; in the common uniform-error reading, the leading complexity multiplier is the learned prover's average truncated proof length. The error decomposes into approximation error, geometric coverage of the training distribution, and Monte Carlo label noise, and improves to a fast rate under an action-gap margin condition. The result gives a component-sensitive account of why verifier feedback, retrieval, representation geometry, and proof-shortening mechanisms help on biased theorem workloads, without contradicting classical worst-case hardness.

This paper has two aims, first one is to introduce special kind of proximal contractions guaranteeing a finite number of best proximity points, and second one is to derive best proximity point results for perimetric contractions. To meet these two aims, we introduce two new proximal contractions: perimetric proximal contractions of the first and the second kind, and derive best proximity point results for these mappings. We establish that for these particular mappings, best proximity points are not necessarily unique; however, we provide an upper bound, proving that at most two such points can exist. To establish the validity of our results, we provide illustrative examples demonstrating that these newly defined mappings can possess unique or exactly two best proximity points.

Given a black-box AI system and a task, at what confidence level can a practitioner trust the system's output? We answer with a reliability level -- a single number per system-task pair, derived from self-consistency sampling and conformal calibration, that serves as a black-box deployment gate with exact, finite-sample, distribution-free guarantees. Self-consistency sampling reduces uncertainty exponentially; conformal calibration guarantees correctness within 1/(n+1) of the target level, regardless of the system's errors -- made transparently visible through larger answer sets for harder questions. Weaker models earn lower reliability levels (not accuracy -- see Definition 2.4): GPT-4.1 earns 94.6% on GSM8K and 96.8% on TruthfulQA, while GPT-4.1-nano earns 89.8% on GSM8K and 66.5% on MMLU. We validate across five benchmarks, five models from three families, and both synthetic and real data. Conditional coverage on solvable items exceeds 0.93 across all configurations; sequential stopping reduces API costs by around 50%.

We formalize a transfinite Phi process that treats all possibility embeddings as operators on structured state spaces including complete lattices, Banach and Hilbert spaces, and orthomodular lattices. We prove a determinization lemma showing that lifting to sets or distributions yields a deterministic global dynamic, an ordinal stabilization theorem sending operator transforms to the fixed subspace by stage omega under normal spectral contraction, and a product of Riesz projections theorem for commuting layers. We establish a compositionality law for lifted maps, show closure of Phi packings, and present a quantitative application to sensory depletion that models tissue removal as a projection and derives strict decreases in the attainable fixed point under minimal monotonicity and positivity assumptions. We also state measurable conditions for probabilistic lifts, give explicit non normal and non commuting counterexamples, and provide finite dimensional and stochastic witnesses together with per theorem scope tables and a small reproducible code appendix.

Large-scale land enclosure for speculative mega-development constitutes a non-equilibrium spatial process whose velocity, topology, and irreversibility remain poorly quantified. We study the Pantai Indah Kapuk 2 (PIK2) coastal mega-development north of Jakarta, Indonesia, using eight years (2017--2024) of Sentinel-2 land-use/land-cover (LULC) data at 10-meter resolution. The landscape is projected onto a Marxian probability simplex partitioning terrestrial pixels into Commons, Agrarian, and Capital fractions. Fisher-Rao (FR) geodesic distances on this simplex identify a transformation pulse of 0.405~rad/yr during 2019--2020, coinciding with major construction activity. Absorbing Markov chain analysis yields expected absorption times into the built environment of 46.0~years for cropland and 38.1~years for tree cover, with a pooled built-area self-retention rate of 96.4%. Percolation analysis reveals that a giant connected component containing 89--95% of all built pixels persists at occupation probabilities p in [0.096, 0.162], far below the random percolation threshold p_c approx 0.593, indicating planned rather than stochastic spatial growth. The box-counting fractal dimension of the urban boundary increases from d_f = 1.316 to 1.397, consistent with increasingly irregular frontier expansion. These results suggest that information-geometric and statistical-mechanical tools can characterize the kinematic and topological signatures of capitalist spatial accumulation with quantitative precision.

In this paper, we deal with the gluing of two surfaces, where the gluing locus is assumed to be a curve. We consider a moving frame along the gluing locus, and define developable surfaces with respect to the frame. Considering geometric properties of these developable surfaces, we study the geometry of gluing two surfaces.

Accurate aerodynamic modeling of satellites in very low Earth orbit (VLEO) requires gas-surface interaction (GSI) models that capture the full velocity spectrum from thermal to orbital speeds. Atmospheric particles initially strike spacecraft surfaces at hypersonic velocities of 6 000 - 10 000 m/s. Due to surface roughness and complex geometries, especially within air-breathing electric propulsion (ABEP) intake systems, multiple collisions occur, progressively reducing the particle velocities. A recent machine learning framework for deriving scattering kernels from molecular dynamics (MD) simulations has shown promise, but remains limited to high-velocity single impacts and possibly violates fundamental equilibrium principles such as detailed balance. This work extends this machine learning based scattering kernel to cover the complete velocity range using conditional normalizing flows trained with physics-informed constraints, enabling accurate modeling of multi-bounce scenarios in realistic VLEO applications. We train a conditional Real-valued Non-Volume Preserving (cRealNVP) model on expanded molecular dynamics simulations covering velocities from thermal to hypersonic speeds, incorporating a detailed balance loss term. The resulting model demonstrates improved accuracy compared to previous approaches even in the original high-velocity regime, while successfully capturing thermal-velocity scattering. Quantitative assessment shows that thermalization is approximated within acceptable tolerances. This framework provides essential capabilities for accurate ABEP intake optimization and VLEO mission planning while offering a general methodology applicable to broader rarefied gas dynamics problems requiring thermodynamic consistency.

Using a long-standing conjecture from combinatorial group theory, we explore, from multiple perspectives, the challenges of finding rare instances carrying disproportionately high rewards. Based on lessons learned in the context defined by the Andrews-Curtis conjecture, we propose algorithmic enhancements and a topological hardness measure with implications for a broad class of search problems. As part of our study, we also address several open mathematical questions. Notably, we demonstrate the length reducibility of all but two presentations in the Akbulut-Kirby series (1981), and resolve various potential counterexamples in the Miller-Schupp series (1991), including three infinite subfamilies.

Stochastic rounding (SR) is a probabilistic method used to round numbers to floating-point and fixed-point representations. In length n summation, the worst-case error of SR grows as n with high probability, unlike for standard modes, like round-to-nearest (RN), which grows as n. For this reason, the former is increasingly employed in large-scale, low-precision computations as an RN alternative. Additionally, SR alleviates stagnation, whereby relatively small summands are completely rounded off and do not contribute to the sum. We provide an update to [Croci et al., Roy. Soc. Open Sci. 9.3 (2022), pp. 1-25], a survey which discusses the development and use of SR between 1949 and 2022, citing over 100 references. Since then, there has been a surge of new research, and this update covers almost four years of further progress in applying, analysing, and implementing SR. Our main focus is limited-precision stochastic rounding, a new variant that fixes the precision of the random numbers used. We provide insights into industrial and numerical analysis activities surrounding SR, highlighting the next possible steps in making this rounding mode more widely available in hardware.

Slippery road conditions pose significant challenges for autonomous driving. Beyond predicting road grip, it is crucial to estimate its uncertainty reliably to ensure safe vehicle control. In this work, we benchmark several uncertainty prediction methods to assess their effectiveness for grip uncertainty estimation. Additionally, we propose a novel approach that leverages road surface state segmentation to predict grip uncertainty. Our method estimates a pixel-wise grip probability distribution based on inferred road surface conditions. Experimental results indicate that the proposed approach enhances the robustness of grip uncertainty prediction.

Every generative model for crystalline materials harbors a critical structure size beyond which its outputs quietly become unreliable -- we call this the extrapolation frontier. Despite its direct consequences for nanomaterial design, this frontier has never been systematically measured. We introduce RADII, a radius-resolved benchmark of {sim}75,000 nanoparticle structures (55-11,298 atoms) that treats radius as a continuous scaling knob to trace generation quality from in-distribution to out-of-distribution regimes under leakage-free splits. RADII provides frontier-specific diagnostics: per-radius error profiles pinpoint each architecture's scaling ceiling, surface-interior decomposition tests whether failures originate at boundaries or in bulk, and cross-metric failure sequencing reveals which aspect of structural fidelity breaks first. Benchmarking five state-of-the-art architectures, we find that: (i) all models degrade by {sim}13% in global positional error beyond training radii, yet local bond fidelity diverges wildly across architectures -- from near-zero to over 2times collapse; (ii) no two architectures share the same failure sequence, revealing the frontier as a multi-dimensional surface shaped by model family; and (iii) well-behaved models obey a power-law scaling exponent αapprox 1/3 whose in-distribution fit accurately predicts out-of-distribution error, making their frontiers quantitatively forecastable. These findings establish output scale as a first-class evaluation axis for geometric generative models. The dataset and code are available at https://github.com/KurbanIntelligenceLab/RADII.

Let S be a rational projective surface given by means of a projective rational parametrization whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides three rational maps f,g,h:A^2 --to Ssubset P^n such that the union of the three images covers S. As a consequence, we present a second algorithm that generates two rational maps f,g:A^2 --to S, such that the union of their images covers the affine surface Scap A^n. In the affine case, the number of rational maps involved in the cover is in general optimal.

In physics and engineering literature, the distribution of the excursion-above-zero time distribution (exceedance distribution) for a stationary Gaussian process has been approximated by a stationary switching process with independently distributed switching times. The approach matched the covariance of the clipped Gaussian process with the one for the stationary switching process and the distribution of the latter was used as the so-called independent interval approximation (IIA). The approach successfully assessed the persistency exponent for many physically important processes but left an unanswered question when such an approach leads to a mathematically meaningful and proper exceedance distribution. Here we address this question by proposing an alternative matching of the expected values of the clipped Slepian process and the corresponding switched process initiated at the origin. The method has allowed resolving the mathematical correctness of the matching method for a large subclass of the Gaussian processes with monotonic covariance, for which we provide a sufficient condition for the validity of the IIA. Within this class, the IIA produces a valid distribution for the excursion time and is represented in an explicit stochastic form that connects directly to the covariance of the underlying Gaussian process. We compare the excursion level distributions as well as the corresponding persistency exponents obtained through the IIA method with numerically computed exact distributions, and the simulated distribution for several important Gaussian models. We also argue that for stationary Gaussian processes with a non-monotonic covariance, the IIA fails and should not be used.

Despite the growing demand for accurate surface normal estimation models, existing methods use general-purpose dense prediction models, adopting the same inductive biases as other tasks. In this paper, we discuss the inductive biases needed for surface normal estimation and propose to (1) utilize the per-pixel ray direction and (2) encode the relationship between neighboring surface normals by learning their relative rotation. The proposed method can generate crisp - yet, piecewise smooth - predictions for challenging in-the-wild images of arbitrary resolution and aspect ratio. Compared to a recent ViT-based state-of-the-art model, our method shows a stronger generalization ability, despite being trained on an orders of magnitude smaller dataset. The code is available at https://github.com/baegwangbin/DSINE.

We analyze surface patches with a corner that is rounded in the sense that the partial derivatives at that point are antiparallel. Sufficient conditions for G^1 smoothness are given, which, up to a certain degenerate case, are also necessary. Further, we investigate curvature integrability and present examples

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

Recovering pixel-wise geometric properties from a single image is fundamentally ill-posed due to appearance ambiguity and non-injective mappings between 2D observations and 3D structures. While discriminative regression models achieve strong performance through large-scale supervision, their success is bounded by the scale, quality and diversity of available data and limited physical reasoning. Recent diffusion models exhibit powerful world priors that encode geometry and semantics learned from massive image-text data, yet directly reusing their stochastic generative formulation is suboptimal for deterministic geometric inference: the former is optimized for diverse and high-fidelity image generation, whereas the latter requires stable and accurate predictions. In this work, we propose Lotus-2, a two-stage deterministic framework for stable, accurate and fine-grained geometric dense prediction, aiming to provide an optimal adaption protocol to fully exploit the pre-trained generative priors. Specifically, in the first stage, the core predictor employs a single-step deterministic formulation with a clean-data objective and a lightweight local continuity module (LCM) to generate globally coherent structures without grid artifacts. In the second stage, the detail sharpener performs a constrained multi-step rectified-flow refinement within the manifold defined by the core predictor, enhancing fine-grained geometry through noise-free deterministic flow matching. Using only 59K training samples, less than 1% of existing large-scale datasets, Lotus-2 establishes new state-of-the-art results in monocular depth estimation and highly competitive surface normal prediction. These results demonstrate that diffusion models can serve as deterministic world priors, enabling high-quality geometric reasoning beyond traditional discriminative and generative paradigms.

Estimating the pose of an object from a monocular image is an inverse problem fundamental in computer vision. The ill-posed nature of this problem requires incorporating deformation priors to solve it. In practice, many materials do not perceptibly shrink or extend when manipulated, constituting a powerful and well-known prior. Mathematically, this translates to the preservation of the Riemannian metric. Neural networks offer the perfect playground to solve the surface reconstruction problem as they can approximate surfaces with arbitrary precision and allow the computation of differential geometry quantities. This paper presents an approach to inferring continuous deformable surfaces from a sequence of images, which is benchmarked against several techniques and obtains state-of-the-art performance without the need for offline training.

We study a version of adversarial classification where an adversary is empowered to corrupt data inputs up to some distance varepsilon, using tools from variational analysis. In particular, we describe necessary conditions associated with the optimal classifier subject to such an adversary. Using the necessary conditions, we derive a geometric evolution equation which can be used to track the change in classification boundaries as varepsilon varies. This evolution equation may be described as an uncoupled system of differential equations in one dimension, or as a mean curvature type equation in higher dimension. In one dimension, and under mild assumptions on the data distribution, we rigorously prove that one can use the initial value problem starting from varepsilon=0, which is simply the Bayes classifier, in order to solve for the global minimizer of the adversarial problem for small values of varepsilon. In higher dimensions we provide a similar result, albeit conditional to the existence of regular solutions of the initial value problem. In the process of proving our main results we obtain a result of independent interest connecting the original adversarial problem with an optimal transport problem under no assumptions on whether classes are balanced or not. Numerical examples illustrating these ideas are also presented.

Effective decision-making in Large Language Models (LLMs) is essential for handling intricate tasks. However, existing approaches prioritize performance but often overlook the balance between effectiveness and computational cost. To address this, we first introduce the 3E Criteria to systematically assess the cost-effectiveness of search strategies, revealing that existing methods often trade significant efficiency for marginal performance gains. To improve LLM decision-making while maintaining efficiency, we propose the Speculative Reward Model (SRM), a plug-and-play framework that seamlessly integrates with existing search strategies. Specifically, SRM employs an external reward assigner to predict optimal actions, reducing reliance on LLMs' internal self-evaluation. And a speculative verification mechanism is used to prune suboptimal choices and guide the search toward more promising steps. We evaluate SRM on several complex decision-making tasks including mathematical reasoning, planning and numerical reasoning in specialized domains. Experimental results show that SRM reduces costs to 1/10 of the original search framework on average while maintaining effectiveness.

Inference-time steering enables pretrained diffusion/flow models to be adapted to new tasks without retraining. A widely used approach is the ratio-of-densities method, which defines a time-indexed target path by reweighting probability-density trajectories from multiple models with positive, or in some cases, negative exponents. This construction, however, harbors a critical and previously unformalized failure mode: Marginal Path Collapse, where intermediate densities become non-normalizable even though endpoints remain valid. Collapse arises systematically when composing heterogeneous models trained on different noise schedules or datasets, including a common setting in molecular design where de-novo, conformer, and pocket-conditioned models must be combined for tasks such as flexible-pose scaffold decoration. We provide a novel and complete solution for the problem. First, we derive a simple path existence criterion that predicts exactly when collapse occurs from noise schedules and exponents alone. Second, we introduce Adaptive path Correction with Exponents (ACE), which extends Feynman-Kac steering to time-varying exponents and guarantees a valid probability path. On a synthetic 2D benchmark and on flexible-pose scaffold decoration, ACE eliminates collapse and enables high-guidance compositional generation, improving distributional and docking metrics over constant-exponent baselines and even specialized task-specific scaffold decoration models. Our work turns ratio-of-densities steering with heterogeneous experts from an unstable heuristic into a reliable tool for controllable generation.

One of the main goals and challenges of materials discovery is to find the best candidates for each interest property or application. Machine learning rises in this context to efficiently optimize this search, exploring the immense materials space, consisting of simultaneously the atomic, compositional, and structural spaces. Topological insulators, presenting symmetry-protected metallic edge states, are a promising class of materials for different applications. However, further, development is limited by the scarcity of viable candidates. Here we present and discuss machine learning-accelerated strategies for searching the materials space for two-dimensional topological materials. We show the importance of detailed investigations of each machine learning component, leading to different results. Using recently created databases containing thousands of ab initio calculations of 2D materials, we train machine learning models capable of determining the electronic topology of materials, with an accuracy of over 90%. We can then generate and screen thousands of novel materials, efficiently predicting their topological character without the need for a priori structural knowledge. We discover 56 non-trivial materials, of which 17 novel insulating candidates for further investigation, for which we corroborate their topological properties with density functional theory calculations. This strategy is 10times more efficient than the trial-and-error approach while few orders of magnitude faster and is a proof of concept for guiding improved materials discovery search strategies.

This work aims to develop a method based on a structurally reliable ice model and a statistically and physico-chemically robust approach for BE distribution inference, with the aim to be applicable to various relevant interstellar species. A multiscale computational approach is presented, with a Molecular Dynamics (MD) Heat & Quench protocol for the amorphous water ice model, and an ONIOM(B3LYP-D3(BJ)/6-311+G**:GFN2-xtb) scheme for the BE inference, with a prime emphasis onto the BE/real system size convergence. The sampling of the binding configurations is twofold, exploring both regularly spaced binding sites, as well as various adsorbate-to-substrate orientations on each locally distinct site. This second source of BE diversity accounts for the local roughness of the potential energy landscape of the substrate. Three different adsorbate test cases are considered, i.e. NH3, CO and CH4, owing to their significance in dust icy mantles, and their distinct binding behavior with water ices. The BE distributions for NH3, CO and CH4 have been inferred, with converged statistics. The distribution for NH3 is better represented by a double Gaussian component profile. Three starting adsorbate orientations per site are required to reach convergence for both Gaussian components of NH3, while 2 orientations are sufficient for CO, and one unique for CH4 (symmetric). Further geometrical and molecular surrounding insights have been provided. These results encompass previously reported results.

Image-based 3D reconstruction has increasingly stunning results over the past few years with the latest improvements in computer vision and graphics. Geometry and topology are two fundamental concepts when dealing with 3D mesh structures. But the latest often remains a side issue in the 3D mesh-based reconstruction literature. Indeed, performing per-vertex elementary displacements over a 3D sphere mesh only impacts its geometry and leaves the topological structure unchanged and fixed. Whereas few attempts propose to update the geometry and the topology, all need to lean on costly 3D ground-truth to determine the faces/edges to prune. We present in this work a method that aims to refine the topology of any 3D mesh through a face-pruning strategy that extensively relies upon 2D alpha masks and camera pose information. Our solution leverages a differentiable renderer that renders each face as a 2D soft map. Its pixel intensity reflects the probability of being covered during the rendering process by such a face. Based on the 2D soft-masks available, our method is thus able to quickly highlight all the incorrectly rendered faces for a given viewpoint. Because our module is agnostic to the network that produces the 3D mesh, it can be easily plugged into any self-supervised image-based (either synthetic or natural) 3D reconstruction pipeline to get complex meshes with a non-spherical topology.

Industrial visual sim-to-real is often described as transferring from synthetic images to real images, but industrial deployment usually involves a broader mismatch between available evidence and required decisions. A system may be built from CAD renderings, simulated RGB-D observations, normal reference images, synthetic defects, pretrained feature spaces, or language prompts, yet deployed under different sensors, lighting, materials, fixtures, calibration, production variation, and rare defect modes. This review reframes industrial visual sim-to-real as a domain-gap problem organized by prior availability. We distinguish CAD-available settings, where explicit object geometry can support rendering, calibration, pose estimation, segmentation, and test-time geometric verification; CAD-unavailable settings, where geometry is replaced by normal-reference appearance, feature distributions, teacher-student residuals, synthetic anomaly assumptions, foundation features, or vision-language priors; and boundary-prior settings, where approximate models, templates, reference views, or semantic correspondences preserve only part of the CAD role. This framing connects CAD-based detection and 6D pose-estimation literature with industrial anomaly and surface-inspection literature that is usually reviewed separately. To make the taxonomy concrete, we use empirical anchors on T-LESS/BOP, MVTec AD, and VisA. The anchors show that CAD render count alone does not close transfer; source-distribution design, detector capacity, and small real calibration can matter more. They also show that CAD at test time creates a distinct verification channel through mask, pose, and depth consistency, whereas CAD-unavailable inspection relies on calibrated normality and feature deviation. The review therefore argues against a single cross-task leaderboard and instead asks what prior grounds the deployment decision.

As the most common representation for 3D shapes, mesh is often stored discretely with arrays of vertices and faces. However, 3D shapes in the real world are presented continuously. In this paper, we propose to learn a continuous representation for meshes with fixed topology, a common and practical setting in many faces-, hand-, and body-related applications. First, we split the template into multiple closed manifold genus-0 meshes so that each genus-0 mesh can be parameterized onto the unit sphere. Then we learn spherical implicit surface (SIS), which takes a spherical coordinate and a global feature or a set of local features around the coordinate as inputs, predicting the vertex corresponding to the coordinate as an output. Since the spherical coordinates are continuous, SIS can depict a mesh in an arbitrary resolution. SIS representation builds a bridge between discrete and continuous representation in 3D shapes. Specifically, we train SIS networks in a self-supervised manner for two tasks: a reconstruction task and a super-resolution task. Experiments show that our SIS representation is comparable with state-of-the-art methods that are specifically designed for meshes with a fixed resolution and significantly outperforms methods that work in arbitrary resolutions.

Though Gaussian splatting has achieved impressive results in novel view synthesis, it requires millions of primitives to model highly textured scenes, even when the geometry of the scene is simple. We propose a representation that goes beyond point-based rendering and decouples geometry and appearance in order to achieve a compact representation. We use surfels for geometry and a combination of a global neural field and per-primitive colours for appearance. The neural field textures a fixed number of primitives for each pixel, ensuring that the added compute is low. Our representation matches the perceptual quality of 3D Gaussian splatting while using 9.7times fewer primitives and 5.5times less memory on outdoor scenes and using 31times fewer primitives and 3.7times less memory on indoor scenes. Our representation also renders twice as fast as existing textured primitives while improving upon their visual quality.

We present dla-ideal-solver, a high-performance framework for simulating two-dimensional Diffusion-Limited Aggregation (DLA) using Numba-accelerated Python. By leveraging just-in-time (JIT) compilation, we achieve computational throughput comparable to legacy static implementations while retaining high-level flexibility. We investigate the Laplacian growth instability across varying injection geometries and walker concentrations. Our analysis confirms the robustness of the standard fractal dimension D_f approx 1.71 for dilute regimes, consistent with the Witten-Sander universality class. However, we report a distinct crossover to Eden-like compact growth (D_f approx 1.87) in high-density environments, attributed to the saturation of the screening length. Beyond standard mass-radius scaling, we employ generalized Rényi dimensions and lacunarity metrics to quantify the monofractal character and spatial heterogeneity of the aggregates. This work establishes a reproducible, open-source testbed for exploring phase transitions in non-equilibrium statistical mechanics.

We study the Stochastic Boolean Function Certification (SBFC) problem, where we are given n Bernoulli random variables {X_e: e in U} on a ground set U of n elements with joint distribution p, a Boolean function f: 2^U to {0, 1}, and an (unknown) scenario S = {e in U: X_e = 1} of active elements sampled from p. We seek to probe the elements one-at-a-time to reveal if they are active until we can certify f(S) = 1, while minimizing the expected number of probes. Unlike most previous results that assume independence, we study correlated distributions p and give approximation algorithms for several classes of functions f. When f(S) is the indicator function for whether S is the spanning set of a given matroid, our problem reduces to finding a basis of active elements of a matroid by probing elements. We give a non-adaptive O(log n)-approximation algorithm for arbitrary distributions p, and show that this is tight up to constants unless P = NP, even for partition matroids. For uniform matroids, we give constant factor 4.642-approximation ([BBFT20]) that can be further improved to a 2-approximation if additionally the random variables are negatively correlated for the case of 1-uniform matroid. We also give an adaptive O(log k)-approximation algorithm for SBFC for k-uniform matroids for the Graph Probing problem, where we seek to probe the edges of a graph one-at-a-time until we find k active edges. The underlying distribution on edges arises from (hidden) independent vertex random variables, with an edge being active if at least one of its endpoints is active. This significantly improves over the information-theoretic lower bound on Ω(poly(n)) ([JGM19]) for adaptive algorithms for k-uniform matroids with arbitrary distributions.

Neural surface representation has demonstrated remarkable success in the areas of novel view synthesis and 3D reconstruction. However, assessing the geometric quality of 3D reconstructions in the absence of ground truth mesh remains a significant challenge, due to its rendering-based optimization process and entangled learning of appearance and geometry with photometric losses. In this paper, we present a novel framework, i.e, GURecon, which establishes a geometric uncertainty field for the neural surface based on geometric consistency. Different from existing methods that rely on rendering-based measurement, GURecon models a continuous 3D uncertainty field for the reconstructed surface, and is learned by an online distillation approach without introducing real geometric information for supervision. Moreover, in order to mitigate the interference of illumination on geometric consistency, a decoupled field is learned and exploited to finetune the uncertainty field. Experiments on various datasets demonstrate the superiority of GURecon in modeling 3D geometric uncertainty, as well as its plug-and-play extension to various neural surface representations and improvement on downstream tasks such as incremental reconstruction. The code and supplementary material are available on the project website: https://zju3dv.github.io/GURecon/.

A desired closure property in Bayesian probability is that an updated posterior distribution be in the same class of distributions --- say Gaussians --- as the prior distribution. When the updating takes place via a statistical model, one calls the class of prior distributions the `conjugate priors' of the model. This paper gives (1) an abstract formulation of this notion of conjugate prior, using channels, in a graphical language, (2) a simple abstract proof that such conjugate priors yield Bayesian inversions, and (3) a logical description of conjugate priors that highlights the required closure of the priors under updating. The theory is illustrated with several standard examples, also covering multiple updating.

Recent works have extended notions of feature importance to semantic concepts that are inherently interpretable to the users interacting with a black-box predictive model. Yet, precise statistical guarantees, such as false positive rate control, are needed to communicate findings transparently and to avoid unintended consequences in real-world scenarios. In this paper, we formalize the global (i.e., over a population) and local (i.e., for a sample) statistical importance of semantic concepts for the predictions of opaque models, by means of conditional independence, which allows for rigorous testing. We use recent ideas of sequential kernelized testing (SKIT) to induce a rank of importance across concepts, and showcase the effectiveness and flexibility of our framework on synthetic datasets as well as on image classification tasks using vision-language models such as CLIP.

We propose a novel point-based representation, Gaussian surfels, to combine the advantages of the flexible optimization procedure in 3D Gaussian points and the surface alignment property of surfels. This is achieved by directly setting the z-scale of 3D Gaussian points to 0, effectively flattening the original 3D ellipsoid into a 2D ellipse. Such a design provides clear guidance to the optimizer. By treating the local z-axis as the normal direction, it greatly improves optimization stability and surface alignment. While the derivatives to the local z-axis computed from the covariance matrix are zero in this setting, we design a self-supervised normal-depth consistency loss to remedy this issue. Monocular normal priors and foreground masks are incorporated to enhance the quality of the reconstruction, mitigating issues related to highlights and background. We propose a volumetric cutting method to aggregate the information of Gaussian surfels so as to remove erroneous points in depth maps generated by alpha blending. Finally, we apply screened Poisson reconstruction method to the fused depth maps to extract the surface mesh. Experimental results show that our method demonstrates superior performance in surface reconstruction compared to state-of-the-art neural volume rendering and point-based rendering methods.

World models are increasingly regarded as a decisive step toward artificial general intelligence, yet modeling the physical world demands more than rendering convincing frames on demand: it requires an internal world state that keeps evolving over time, decoupled from observation, so that objects endure and events run to their conclusions whether or not a camera is watching, much as the moon holds to its orbit when no one is looking. This requirement is a blind spot of existing benchmarks, which reward surface properties such as fidelity, motion, and camera controllability while never asking whether a generated world keeps evolving once it is unobserved. We introduce WRBench, the first systematic diagnostic benchmark that treats camera motion as an intervention on observability and resolves evaluation into a human-calibrated chain that asks whether the camera executes the requested interaction, whether the scene stays continuous and identifiable while in view, and whether a returning target remains consistent with the event that was set in motion. Across 9{,}600 videos from 23 models spanning four control paradigms, one finding proves stubborn: current systems maintain the observed world as a tracking shot, resuming a returning target in the state at which it was abandoned rather than advancing the event while it went unseen. Because this failure recurs across control paradigms, model families, and increments of scale, robust world-state evolution does not follow from cleaner imagery, tighter control, richer geometric priors, or sheer parameter count We therefore argue that the stability of the physical state kernel and the consistency of worldlines under viewpoint intervention should become first-class objectives of world-model design, so that a world model captures how the world will unfold rather than how the next frame appears.

The interstellar medium (ISM) is all but empty. To date, more than 300 molecules have already been discovered. Because of the extremely low temperature, the gas-phase chemistry is dominated by barrierless exothermic reactions of radicals and ions. However, several abundant molecules and organic molecules cannot be produced efficiently by gas-phase reactions. To explain the existence of such molecules in the ISM, gas-surface interactions between small molecules and dust particles covered with amorphous solid water (ASW) mantles must be considered. In general, surface processes such as adsorption, diffusion, desorption, and chemical reactions can be linked to the binding energy of molecules to the surface. Hence, a lot of studies have been performed to identify the binding energies of interstellar molecules on ASW surfaces. Cosmic radiation and free electrons may induce a negative charge on the dust particles, and the binding energies may be affected by this charge. In this study, we calculate the binding energies of CO, CH4, and NH3, on neutral and charged ASW surfaces using DFT calculations. Our results indicate that CO can interact with the surface charge, increasing its binding energy. In contrast, the binding energy of CH4 remains unchanged in the presence of surface charge, and that of NH3 typically decreases.

In this work we target a learnable output representation that allows continuous, high resolution outputs of arbitrary shape. Recent works represent 3D surfaces implicitly with a Neural Network, thereby breaking previous barriers in resolution, and ability to represent diverse topologies. However, neural implicit representations are limited to closed surfaces, which divide the space into inside and outside. Many real world objects such as walls of a scene scanned by a sensor, clothing, or a car with inner structures are not closed. This constitutes a significant barrier, in terms of data pre-processing (objects need to be artificially closed creating artifacts), and the ability to output open surfaces. In this work, we propose Neural Distance Fields (NDF), a neural network based model which predicts the unsigned distance field for arbitrary 3D shapes given sparse point clouds. NDF represent surfaces at high resolutions as prior implicit models, but do not require closed surface data, and significantly broaden the class of representable shapes in the output. NDF allow to extract the surface as very dense point clouds and as meshes. We also show that NDF allow for surface normal calculation and can be rendered using a slight modification of sphere tracing. We find NDF can be used for multi-target regression (multiple outputs for one input) with techniques that have been exclusively used for rendering in graphics. Experiments on ShapeNet show that NDF, while simple, is the state-of-the art, and allows to reconstruct shapes with inner structures, such as the chairs inside a bus. Notably, we show that NDF are not restricted to 3D shapes, and can approximate more general open surfaces such as curves, manifolds, and functions. Code is available for research at https://virtualhumans.mpi-inf.mpg.de/ndf/.

Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain poorly understood, as universality is traditionally studied for models accepting inputs of a fixed size, defined on a compact subset of their domain. In sharp contrast, any-dimensional models can be viewed as sequences of functions defined on growing-sized inputs, and it is not clear in which sense they can be universal. We develop a systematic approach to establish any-dimensional universality, by identifying any-dimensional functions with a unique function taking inputs in a suitable infinite-dimensional limit space containing inputs of all finite sizes as well as their limits. Using the symmetries of these inputs and relations between inputs of different sizes, we show that this limit space admits a natural topology with rich families of compact sets on which any-dimensional universality can be established. We illustrate our approach by showing that several existing architectures fail to be universal, and we propose simple modifications that restore universality.

This work addresses the challenge of high-quality surface normal estimation from monocular colored inputs (i.e., images and videos), a field which has recently been revolutionized by repurposing diffusion priors. However, previous attempts still struggle with stochastic inference, conflicting with the deterministic nature of the Image2Normal task, and costly ensembling step, which slows down the estimation process. Our method, StableNormal, mitigates the stochasticity of the diffusion process by reducing inference variance, thus producing "Stable-and-Sharp" normal estimates without any additional ensembling process. StableNormal works robustly under challenging imaging conditions, such as extreme lighting, blurring, and low quality. It is also robust against transparent and reflective surfaces, as well as cluttered scenes with numerous objects. Specifically, StableNormal employs a coarse-to-fine strategy, which starts with a one-step normal estimator (YOSO) to derive an initial normal guess, that is relatively coarse but reliable, then followed by a semantic-guided refinement process (SG-DRN) that refines the normals to recover geometric details. The effectiveness of StableNormal is demonstrated through competitive performance in standard datasets such as DIODE-indoor, iBims, ScannetV2 and NYUv2, and also in various downstream tasks, such as surface reconstruction and normal enhancement. These results evidence that StableNormal retains both the "stability" and "sharpness" for accurate normal estimation. StableNormal represents a baby attempt to repurpose diffusion priors for deterministic estimation. To democratize this, code and models have been publicly available in hf.co/Stable-X

Neural implicit surface learning has shown significant progress in multi-view 3D reconstruction, where an object is represented by multilayer perceptrons that provide continuous implicit surface representation and view-dependent radiance. However, current methods often fail to accurately reconstruct reflective surfaces, leading to severe ambiguity. To overcome this issue, we propose Ref-NeuS, which aims to reduce ambiguity by attenuating the effect of reflective surfaces. Specifically, we utilize an anomaly detector to estimate an explicit reflection score with the guidance of multi-view context to localize reflective surfaces. Afterward, we design a reflection-aware photometric loss that adaptively reduces ambiguity by modeling rendered color as a Gaussian distribution, with the reflection score representing the variance. We show that together with a reflection direction-dependent radiance, our model achieves high-quality surface reconstruction on reflective surfaces and outperforms the state-of-the-arts by a large margin. Besides, our model is also comparable on general surfaces.

Persistent memory is a central capability for AI agents, yet the mathematical foundations of memory retrieval, lifecycle management, and consistency remain unexplored. Current systems employ cosine similarity for retrieval, heuristic decay for salience, and provide no formal contradiction detection. We establish information-geometric foundations through three contributions. First, a retrieval metric derived from the Fisher information structure of diagonal Gaussian families, satisfying Riemannian metric axioms, invariant under sufficient statistics, and computable in O(d) time. Second, memory lifecycle formulated as Riemannian Langevin dynamics with proven existence and uniqueness of the stationary distribution via the Fokker-Planck equation, replacing hand-tuned decay with principled convergence guarantees. Third, a cellular sheaf model where non-trivial first cohomology classes correspond precisely to irreconcilable contradictions across memory contexts. On the LoCoMo benchmark, the mathematical layers yield +12.7 percentage points over engineering baselines across six conversations, reaching +19.9 pp on the most challenging dialogues. A four-channel retrieval architecture achieves 75% accuracy without cloud dependency. Cloud-augmented results reach 87.7%. A zero-LLM configuration satisfies EU AI Act data sovereignty requirements by architectural design. To our knowledge, this is the first work establishing information-geometric, sheaf-theoretic, and stochastic-dynamical foundations for AI agent memory systems.

Conventional production workflow of high-precision mesh assets necessitates a cumbersome and laborious process of manual sculpting by specialized 3D artists/modelers. The recent years have witnessed remarkable advances in AI-empowered 3D content creation for generating plausible structures and intricate appearances from images or text prompts. However, synthesizing realistic surface details still poses great challenges, and enhancing the geometry fidelity of existing lower-quality 3D meshes (instead of image/text-to-3D generation) remains an open problem. In this paper, we introduce SuperCarver, a 3D geometry super-resolution pipeline for supplementing texture-consistent surface details onto a given coarse mesh. We start by rendering the original textured mesh into the image domain from multiple viewpoints. To achieve detail boosting, we construct a deterministic prior-guided normal diffusion model, which is fine-tuned on a carefully curated dataset of paired detail-lacking and detail-rich normal map renderings. To update mesh surfaces from potentially imperfect normal map predictions, we design a noise-resistant inverse rendering scheme through deformable distance field. Experiments demonstrate that our SuperCarver is capable of generating realistic and expressive surface details depicted by the actual texture appearance, making it a powerful tool to both upgrade historical low-quality 3D assets and reduce the workload of sculpting high-poly meshes.