Daily Papers

Knowledge distillation compresses large teachers into smaller students, but performance saturates at a loss floor that persists across training methods and objectives. We argue this floor is geometric: neural networks represent far more features than dimensions through superposition, and a student of width d_S can encode at most d_S cdot g(α) features, where g(α) = 1/((1-α)ln1{1-α}) is a sparsity-dependent capacity function. Features beyond this budget are permanently lost, yielding an importance-weighted loss floor. We validate on a toy model (48 configurations, median accuracy >93%) and on Pythia-410M, where sparse autoencoders measure F approx 28{,}700 features at αapprox 0.992 (critical width d_S^* approx 1{,}065). Distillation into five student widths confirms the predicted monotonic floor ordering. The observed floor decomposes into a geometric component and a width-independent architectural baseline (R^2 = 0.993). Linear probing shows coarse concepts survive even 88% feature loss, revealing the floor arises from aggregate loss of fine-grained features in the importance distribution's long tail. Our results connect representation geometry to distillation limits and provide a practical tool for predicting distillation performance from SAE measurements alone.

3D LiDAR sensors are essential for autonomous navigation, environmental monitoring, and precision mapping in remote sensing applications. To efficiently process the massive point clouds generated by these sensors, LiDAR data is often projected into 2D range images that organize points by their angular positions and distances. While these range image representations enable efficient processing, conventional projection methods suffer from fundamental geometric inconsistencies that cause irreversible information loss, compromising high-fidelity applications. We present ALICE-LRI (Automatic LiDAR Intrinsic Calibration Estimation for Lossless Range Images), the first general, sensor-agnostic method that achieves lossless range image generation from spinning LiDAR point clouds without requiring manufacturer metadata or calibration files. Our algorithm automatically reverse-engineers the intrinsic geometry of any spinning LiDAR sensor by inferring critical parameters including laser beam configuration, angular distributions, and per-beam calibration corrections, enabling lossless projection and complete point cloud reconstruction with zero point loss. Comprehensive evaluation across the complete KITTI and DurLAR datasets demonstrates that ALICE-LRI achieves perfect point preservation, with zero points lost across all point clouds. Geometric accuracy is maintained well within sensor precision limits, establishing geometric losslessness with real-time performance. We also present a compression case study that validates substantial downstream benefits, demonstrating significant quality improvements in practical applications. This paradigm shift from approximate to lossless LiDAR projections opens new possibilities for high-precision remote sensing applications requiring complete geometric preservation.

In representation learning, uniformity refers to the uniform feature distribution in the latent space (i.e., unit hypersphere). Previous work has shown that improving uniformity contributes to the learning of under-represented classes. However, most of the previous work focused on classification; the representation space of imbalanced regression remains unexplored. Classification-based methods are not suitable for regression tasks because they cluster features into distinct groups without considering the continuous and ordered nature essential for regression. In a geometric aspect, we uniquely focus on ensuring uniformity in the latent space for imbalanced regression through two key losses: enveloping and homogeneity. The enveloping loss encourages the induced trace to uniformly occupy the surface of a hypersphere, while the homogeneity loss ensures smoothness, with representations evenly spaced at consistent intervals. Our method integrates these geometric principles into the data representations via a Surrogate-driven Representation Learning (SRL) framework. Experiments with real-world regression and operator learning tasks highlight the importance of uniformity in imbalanced regression and validate the efficacy of our geometry-based loss functions.

Recent advances in video generation have enabled the synthesis of high-quality and visually realistic clips using diffusion transformer models. However, most existing approaches operate purely in the 2D pixel space and lack explicit mechanisms for modeling 3D structures, often resulting in temporally inconsistent geometries, implausible motions, and structural artifacts. In this work, we introduce geometric regularization losses into video generation by augmenting latent diffusion models with per-frame depth prediction. We adopted depth as the geometric representation because of the great progress in depth prediction and its compatibility with image-based latent encoders. Specifically, to enforce structural consistency over time, we propose a multi-view geometric loss that aligns the predicted depth maps across frames within a shared 3D coordinate system. Our method bridges the gap between appearance generation and 3D structure modeling, leading to improved spatio-temporal coherence, shape consistency, and physical plausibility. Experiments across multiple datasets show that our approach produces significantly more stable and geometrically consistent results than existing baselines.

Previous works that leverage video models for image-to-3D scene generation often suffer from geometric distortions and blurry content. Using video generation models to implicitly maintain geometric consistency according to a single-frame input is ineffective. In this paper, we present a two-stage method, named GeoWorld, that renovates the image-to-3D scene generation pipeline by providing full-frame geometry features. The first-stage video generation model, followed by a multi-view geometry model, produces full-frame geometry features, which are then used as a mental draft of geometric conditions to aid the second-stage video-generation model. A geometric loss is proposed to impose real-world geometric constraints, and a geometry adaptation module is introduced to ensure the effective utilization of geometry features. Thanks to full-frame geometric modeling, the two smaller video models in our two-stage method can generate higher-fidelity 3D scenes than SOTA methods, while being even faster, e.g. 7.5times faster than Hunyuan-Voyager. Project page: https://peaes.github.io/GeoWorld.

Natural and expressive human motion generation is the holy grail of computer animation. It is a challenging task, due to the diversity of possible motion, human perceptual sensitivity to it, and the difficulty of accurately describing it. Therefore, current generative solutions are either low-quality or limited in expressiveness. Diffusion models, which have already shown remarkable generative capabilities in other domains, are promising candidates for human motion due to their many-to-many nature, but they tend to be resource hungry and hard to control. In this paper, we introduce Motion Diffusion Model (MDM), a carefully adapted classifier-free diffusion-based generative model for the human motion domain. MDM is transformer-based, combining insights from motion generation literature. A notable design-choice is the prediction of the sample, rather than the noise, in each diffusion step. This facilitates the use of established geometric losses on the locations and velocities of the motion, such as the foot contact loss. As we demonstrate, MDM is a generic approach, enabling different modes of conditioning, and different generation tasks. We show that our model is trained with lightweight resources and yet achieves state-of-the-art results on leading benchmarks for text-to-motion and action-to-motion. https://guytevet.github.io/mdm-page/ .

2D-to-3D reconstruction is an ill-posed problem, yet humans are good at solving this problem due to their prior knowledge of the 3D world developed over years. Driven by this observation, we propose NeRDi, a single-view NeRF synthesis framework with general image priors from 2D diffusion models. Formulating single-view reconstruction as an image-conditioned 3D generation problem, we optimize the NeRF representations by minimizing a diffusion loss on its arbitrary view renderings with a pretrained image diffusion model under the input-view constraint. We leverage off-the-shelf vision-language models and introduce a two-section language guidance as conditioning inputs to the diffusion model. This is essentially helpful for improving multiview content coherence as it narrows down the general image prior conditioned on the semantic and visual features of the single-view input image. Additionally, we introduce a geometric loss based on estimated depth maps to regularize the underlying 3D geometry of the NeRF. Experimental results on the DTU MVS dataset show that our method can synthesize novel views with higher quality even compared to existing methods trained on this dataset. We also demonstrate our generalizability in zero-shot NeRF synthesis for in-the-wild images.

Previous deep learning-based line segment detection (LSD) suffers from the immense model size and high computational cost for line prediction. This constrains them from real-time inference on computationally restricted environments. In this paper, we propose a real-time and light-weight line segment detector for resource-constrained environments named Mobile LSD (M-LSD). We design an extremely efficient LSD architecture by minimizing the backbone network and removing the typical multi-module process for line prediction found in previous methods. To maintain competitive performance with a light-weight network, we present novel training schemes: Segments of Line segment (SoL) augmentation, matching and geometric loss. SoL augmentation splits a line segment into multiple subparts, which are used to provide auxiliary line data during the training process. Moreover, the matching and geometric loss allow a model to capture additional geometric cues. Compared with TP-LSD-Lite, previously the best real-time LSD method, our model (M-LSD-tiny) achieves competitive performance with 2.5% of model size and an increase of 130.5% in inference speed on GPU. Furthermore, our model runs at 56.8 FPS and 48.6 FPS on the latest Android and iPhone mobile devices, respectively. To the best of our knowledge, this is the first real-time deep LSD available on mobile devices. Our code is available.

Existing methods for capturing datasets of 3D heads in dense semantic correspondence are slow, and commonly address the problem in two separate steps; multi-view stereo (MVS) reconstruction followed by non-rigid registration. To simplify this process, we introduce TEMPEH (Towards Estimation of 3D Meshes from Performances of Expressive Heads) to directly infer 3D heads in dense correspondence from calibrated multi-view images. Registering datasets of 3D scans typically requires manual parameter tuning to find the right balance between accurately fitting the scans surfaces and being robust to scanning noise and outliers. Instead, we propose to jointly register a 3D head dataset while training TEMPEH. Specifically, during training we minimize a geometric loss commonly used for surface registration, effectively leveraging TEMPEH as a regularizer. Our multi-view head inference builds on a volumetric feature representation that samples and fuses features from each view using camera calibration information. To account for partial occlusions and a large capture volume that enables head movements, we use view- and surface-aware feature fusion, and a spatial transformer-based head localization module, respectively. We use raw MVS scans as supervision during training, but, once trained, TEMPEH directly predicts 3D heads in dense correspondence without requiring scans. Predicting one head takes about 0.3 seconds with a median reconstruction error of 0.26 mm, 64% lower than the current state-of-the-art. This enables the efficient capture of large datasets containing multiple people and diverse facial motions. Code, model, and data are publicly available at https://tempeh.is.tue.mpg.de.

We introduce Efficient Motion Diffusion Model (EMDM) for fast and high-quality human motion generation. Current state-of-the-art generative diffusion models have produced impressive results but struggle to achieve fast generation without sacrificing quality. On the one hand, previous works, like motion latent diffusion, conduct diffusion within a latent space for efficiency, but learning such a latent space can be a non-trivial effort. On the other hand, accelerating generation by naively increasing the sampling step size, e.g., DDIM, often leads to quality degradation as it fails to approximate the complex denoising distribution. To address these issues, we propose EMDM, which captures the complex distribution during multiple sampling steps in the diffusion model, allowing for much fewer sampling steps and significant acceleration in generation. This is achieved by a conditional denoising diffusion GAN to capture multimodal data distributions among arbitrary (and potentially larger) step sizes conditioned on control signals, enabling fewer-step motion sampling with high fidelity and diversity. To minimize undesired motion artifacts, geometric losses are imposed during network learning. As a result, EMDM achieves real-time motion generation and significantly improves the efficiency of motion diffusion models compared to existing methods while achieving high-quality motion generation. Our code will be publicly available upon publication.

There is an emerging effort to combine the two popular 3D frameworks using Multi-View Stereo (MVS) and Neural Implicit Surfaces (NIS) with a specific focus on the few-shot / sparse view setting. In this paper, we introduce a novel integration scheme that combines the multi-view stereo with neural signed distance function representations, which potentially overcomes the limitations of both methods. MVS uses per-view depth estimation and cross-view fusion to generate accurate surfaces, while NIS relies on a common coordinate volume. Based on this strategy, we propose to construct per-view cost frustum for finer geometry estimation, and then fuse cross-view frustums and estimate the implicit signed distance functions to tackle artifacts that are due to noise and holes in the produced surface reconstruction. We further apply a cascade frustum fusion strategy to effectively captures global-local information and structural consistency. Finally, we apply cascade sampling and a pseudo-geometric loss to foster stronger integration between the two architectures. Extensive experiments demonstrate that our method reconstructs robust surfaces and outperforms existing state-of-the-art methods.

Generating the motion of orchestral conductors from a given piece of symphony music is a challenging task since it requires a model to learn semantic music features and capture the underlying distribution of real conducting motion. Prior works have applied Generative Adversarial Networks (GAN) to this task, but the promising diffusion model, which recently showed its advantages in terms of both training stability and output quality, has not been exploited in this context. This paper presents Diffusion-Conductor, a novel DDIM-based approach for music-driven conducting motion generation, which integrates the diffusion model to a two-stage learning framework. We further propose a random masking strategy to improve the feature robustness, and use a pair of geometric loss functions to impose additional regularizations and increase motion diversity. We also design several novel metrics, including Frechet Gesture Distance (FGD) and Beat Consistency Score (BC) for a more comprehensive evaluation of the generated motion. Experimental results demonstrate the advantages of our model.

The process of camera calibration involves estimating the intrinsic and extrinsic parameters, which are essential for accurately performing tasks such as 3D reconstruction, object tracking and augmented reality. In this work, we propose a novel constraints-based loss for measuring the intrinsic (focal length: (f_x, f_y) and principal point: (p_x, p_y)) and extrinsic (baseline: (b), disparity: (d), translation: (t_x, t_y, t_z), and rotation specifically pitch: (theta_p)) camera parameters. Our novel constraints are based on geometric properties inherent in the camera model, including the anatomy of the projection matrix (vanishing points, image of world origin, axis planes) and the orthonormality of the rotation matrix. Thus we proposed a novel Unsupervised Geometric Constraint Loss (UGCL) via a multitask learning framework. Our methodology is a hybrid approach that employs the learning power of a neural network to estimate the desired parameters along with the underlying mathematical properties inherent in the camera projection matrix. This distinctive approach not only enhances the interpretability of the model but also facilitates a more informed learning process. Additionally, we introduce a new CVGL Camera Calibration dataset, featuring over 900 configurations of camera parameters, incorporating 63,600 image pairs that closely mirror real-world conditions. By training and testing on both synthetic and real-world datasets, our proposed approach demonstrates improvements across all parameters when compared to the state-of-the-art (SOTA) benchmarks. The code and the updated dataset can be found here: https://github.com/CVLABLUMS/CVGL-Camera-Calibration

Denoising diffusion models have shown great potential in multiple research areas. Existing diffusion-based generative methods on de novo 3D molecule generation face two major challenges. Since majority heavy atoms in molecules allow connections to multiple atoms through single bonds, solely using pair-wise distance to model molecule geometries is insufficient. Therefore, the first one involves proposing an effective neural network as the denoising kernel that is capable to capture complex multi-body interatomic relationships and learn high-quality features. Due to the discrete nature of graphs, mainstream diffusion-based methods for molecules heavily rely on predefined rules and generate edges in an indirect manner. The second challenge involves accommodating molecule generation to diffusion and accurately predicting the existence of bonds. In our research, we view the iterative way of updating molecule conformations in diffusion process is consistent with molecular dynamics and introduce a novel molecule generation method named Geometric-Facilitated Molecular Diffusion (GFMDiff). For the first challenge, we introduce a Dual-Track Transformer Network (DTN) to fully excevate global spatial relationships and learn high quality representations which contribute to accurate predictions of features and geometries. As for the second challenge, we design Geometric-Facilitated Loss (GFLoss) which intervenes the formation of bonds during the training period, instead of directly embedding edges into the latent space. Comprehensive experiments on current benchmarks demonstrate the superiority of GFMDiff.

Recent advances in single-view 3D scene reconstruction have highlighted the challenges in capturing fine geometric details and ensuring structural consistency, particularly in high-fidelity outdoor scene modeling. This paper presents Niagara, a new single-view 3D scene reconstruction framework that can faithfully reconstruct challenging outdoor scenes from a single input image for the first time. Our approach integrates monocular depth and normal estimation as input, which substantially improves its ability to capture fine details, mitigating common issues like geometric detail loss and deformation. Additionally, we introduce a geometric affine field (GAF) and 3D self-attention as geometry-constraint, which combines the structural properties of explicit geometry with the adaptability of implicit feature fields, striking a balance between efficient rendering and high-fidelity reconstruction. Our framework finally proposes a specialized encoder-decoder architecture, where a depth-based 3D Gaussian decoder is proposed to predict 3D Gaussian parameters, which can be used for novel view synthesis. Extensive results and analyses suggest that our Niagara surpasses prior SoTA approaches such as Flash3D in both single-view and dual-view settings, significantly enhancing the geometric accuracy and visual fidelity, especially in outdoor scenes.

In this paper, we introduce Geometry-Inverse-Meet-Pixel-Insert, short for GEO, an exceptionally versatile image editing technique designed to cater to customized user requirements at both local and global scales. Our approach seamlessly integrates text prompts and image prompts to yield diverse and precise editing outcomes. Notably, our method operates without the need for training and is driven by two key contributions: (i) a novel geometric accumulation loss that enhances DDIM inversion to faithfully preserve pixel space geometry and layout, and (ii) an innovative boosted image prompt technique that combines pixel-level editing for text-only inversion with latent space geometry guidance for standard classifier-free reversion. Leveraging the publicly available Stable Diffusion model, our approach undergoes extensive evaluation across various image types and challenging prompt editing scenarios, consistently delivering high-fidelity editing results for real images.

While text-3D editing has made significant strides in leveraging score distillation sampling, emerging approaches still fall short in delivering separable, precise and consistent outcomes that are vital to content creation. In response, we introduce FocalDreamer, a framework that merges base shape with editable parts according to text prompts for fine-grained editing within desired regions. Specifically, equipped with geometry union and dual-path rendering, FocalDreamer assembles independent 3D parts into a complete object, tailored for convenient instance reuse and part-wise control. We propose geometric focal loss and style consistency regularization, which encourage focal fusion and congruent overall appearance. Furthermore, FocalDreamer generates high-fidelity geometry and PBR textures which are compatible with widely-used graphics engines. Extensive experiments have highlighted the superior editing capabilities of FocalDreamer in both quantitative and qualitative evaluations.

Object pose estimation, which plays a vital role in robotics, augmented reality, and autonomous driving, has been of great interest in computer vision. Existing studies either require multi-stage pose regression or rely on 2D-3D feature matching. Though these approaches have shown promising results, they rely heavily on appearance information, requiring complex input (i.e., multi-view reference input, depth, or CAD models) and intricate pipeline (i.e., feature extraction-SfM-2D to 3D matching-PnP). We propose AxisPose, a model-free, matching-free, single-shot solution for robust 6D pose estimation, which fundamentally diverges from the existing paradigm. Unlike existing methods that rely on 2D-3D or 2D-2D matching using 3D techniques, such as SfM and PnP, AxisPose directly infers a robust 6D pose from a single view by leveraging a diffusion model to learn the latent axis distribution of objects without reference views. Specifically, AxisPose constructs an Axis Generation Module (AGM) to capture the latent geometric distribution of object axes through a diffusion model. The diffusion process is guided by injecting the gradient of geometric consistency loss into the noise estimation to maintain the geometric consistency of the generated tri-axis. With the generated tri-axis projection, AxisPose further adopts a Triaxial Back-projection Module (TBM) to recover the 6D pose from the object tri-axis. The proposed AxisPose achieves robust performance at the cross-instance level (i.e., one model for N instances) using only a single view as input without reference images, with great potential for generalization to unseen-object level.

Accurate monocular metric depth estimation (MMDE) is crucial to solving downstream tasks in 3D perception and modeling. However, the remarkable accuracy of recent MMDE methods is confined to their training domains. These methods fail to generalize to unseen domains even in the presence of moderate domain gaps, which hinders their practical applicability. We propose a new model, UniDepth, capable of reconstructing metric 3D scenes from solely single images across domains. Departing from the existing MMDE methods, UniDepth directly predicts metric 3D points from the input image at inference time without any additional information, striving for a universal and flexible MMDE solution. In particular, UniDepth implements a self-promptable camera module predicting dense camera representation to condition depth features. Our model exploits a pseudo-spherical output representation, which disentangles camera and depth representations. In addition, we propose a geometric invariance loss that promotes the invariance of camera-prompted depth features. Thorough evaluations on ten datasets in a zero-shot regime consistently demonstrate the superior performance of UniDepth, even when compared with methods directly trained on the testing domains. Code and models are available at: https://github.com/lpiccinelli-eth/unidepth

Accurate monocular metric depth estimation (MMDE) is crucial to solving downstream tasks in 3D perception and modeling. However, the remarkable accuracy of recent MMDE methods is confined to their training domains. These methods fail to generalize to unseen domains even in the presence of moderate domain gaps, which hinders their practical applicability. We propose a new model, UniDepthV2, capable of reconstructing metric 3D scenes from solely single images across domains. Departing from the existing MMDE paradigm, UniDepthV2 directly predicts metric 3D points from the input image at inference time without any additional information, striving for a universal and flexible MMDE solution. In particular, UniDepthV2 implements a self-promptable camera module predicting a dense camera representation to condition depth features. Our model exploits a pseudo-spherical output representation, which disentangles the camera and depth representations. In addition, we propose a geometric invariance loss that promotes the invariance of camera-prompted depth features. UniDepthV2 improves its predecessor UniDepth model via a new edge-guided loss which enhances the localization and sharpness of edges in the metric depth outputs, a revisited, simplified and more efficient architectural design, and an additional uncertainty-level output which enables downstream tasks requiring confidence. Thorough evaluations on ten depth datasets in a zero-shot regime consistently demonstrate the superior performance and generalization of UniDepthV2. Code and models are available at https://github.com/lpiccinelli-eth/UniDepth

Driving scenes are extremely diverse and complicated that it is impossible to collect all cases with human effort alone. While data augmentation is an effective technique to enrich the training data, existing methods for camera data in autonomous driving applications are confined to the 2D image plane, which may not optimally increase data diversity in 3D real-world scenarios. To this end, we propose a 3D data augmentation approach termed Drive-3DAug, aiming at augmenting the driving scenes on camera in the 3D space. We first utilize Neural Radiance Field (NeRF) to reconstruct the 3D models of background and foreground objects. Then, augmented driving scenes can be obtained by placing the 3D objects with adapted location and orientation at the pre-defined valid region of backgrounds. As such, the training database could be effectively scaled up. However, the 3D object modeling is constrained to the image quality and the limited viewpoints. To overcome these problems, we modify the original NeRF by introducing a geometric rectified loss and a symmetric-aware training strategy. We evaluate our method for the camera-only monocular 3D detection task on the Waymo and nuScences datasets. The proposed data augmentation approach contributes to a gain of 1.7% and 1.4% in terms of detection accuracy, on Waymo and nuScences respectively. Furthermore, the constructed 3D models serve as digital driving assets and could be recycled for different detectors or other 3D perception tasks.

Recent progress in Sign Language Translation (SLT) has focussed primarily on improving the representational capacity of large language models to incorporate Sign Language features. This work explores an alternative direction: enhancing the geometric properties of skeletal representations themselves. We propose Geo-Sign, a method that leverages the properties of hyperbolic geometry to model the hierarchical structure inherent in sign language kinematics. By projecting skeletal features derived from Spatio-Temporal Graph Convolutional Networks (ST-GCNs) into the Poincar\'e ball model, we aim to create more discriminative embeddings, particularly for fine-grained motions like finger articulations. We introduce a hyperbolic projection layer, a weighted Fr\'echet mean aggregation scheme, and a geometric contrastive loss operating directly in hyperbolic space. These components are integrated into an end-to-end translation framework as a regularisation function, to enhance the representations within the language model. This work demonstrates the potential of hyperbolic geometry to improve skeletal representations for Sign Language Translation, improving on SOTA RGB methods while preserving privacy and improving computational efficiency. Code available here: https://github.com/ed-fish/geo-sign.

Recent 3D content generation pipelines often leverage Variational Autoencoders (VAEs) to encode shapes into compact latent representations, facilitating diffusion-based generation. Efficiently compressing 3D shapes while preserving intricate geometric details remains a key challenge. Existing 3D shape VAEs often employ uniform point sampling and 1D/2D latent representations, such as vector sets or triplanes, leading to significant geometric detail loss due to inadequate surface coverage and the absence of explicit 3D representations in the latent space. Although recent work explores 3D latent representations, their large scale hinders high-resolution encoding and efficient training. Given these challenges, we introduce Hyper3D, which enhances VAE reconstruction through efficient 3D representation that integrates hybrid triplane and octree features. First, we adopt an octree-based feature representation to embed mesh information into the network, mitigating the limitations of uniform point sampling in capturing geometric distributions along the mesh surface. Furthermore, we propose a hybrid latent space representation that integrates a high-resolution triplane with a low-resolution 3D grid. This design not only compensates for the lack of explicit 3D representations but also leverages a triplane to preserve high-resolution details. Experimental results demonstrate that Hyper3D outperforms traditional representations by reconstructing 3D shapes with higher fidelity and finer details, making it well-suited for 3D generation pipelines.

Recently, image-to-3D approaches have achieved significant results with a natural image as input. However, it is not always possible to access these enriched color input samples in practical applications, where only sketches are available. Existing sketch-to-3D researches suffer from limitations in broad applications due to the challenges of lacking color information and multi-view content. To overcome them, this paper proposes a novel generation paradigm Sketch3D to generate realistic 3D assets with shape aligned with the input sketch and color matching the textual description. Concretely, Sketch3D first instantiates the given sketch in the reference image through the shape-preserving generation process. Second, the reference image is leveraged to deduce a coarse 3D Gaussian prior, and multi-view style-consistent guidance images are generated based on the renderings of the 3D Gaussians. Finally, three strategies are designed to optimize 3D Gaussians, i.e., structural optimization via a distribution transfer mechanism, color optimization with a straightforward MSE loss and sketch similarity optimization with a CLIP-based geometric similarity loss. Extensive visual comparisons and quantitative analysis illustrate the advantage of our Sketch3D in generating realistic 3D assets while preserving consistency with the input.

The loss functions of deep neural networks are complex and their geometric properties are not well understood. We show that the optima of these complex loss functions are in fact connected by simple curves over which training and test accuracy are nearly constant. We introduce a training procedure to discover these high-accuracy pathways between modes. Inspired by this new geometric insight, we also propose a new ensembling method entitled Fast Geometric Ensembling (FGE). Using FGE we can train high-performing ensembles in the time required to train a single model. We achieve improved performance compared to the recent state-of-the-art Snapshot Ensembles, on CIFAR-10, CIFAR-100, and ImageNet.

There has been a recent surge of interest in learning to perceive depth from monocular videos in an unsupervised fashion. A key challenge in this field is achieving robust and accurate depth estimation in challenging scenarios, particularly in regions with weak textures or where dynamic objects are present. This study makes three major contributions by delving deeply into dense correspondence priors to provide existing frameworks with explicit geometric constraints. The first novelty is a contextual-geometric depth consistency loss, which employs depth maps triangulated from dense correspondences based on estimated ego-motion to guide the learning of depth perception from contextual information, since explicitly triangulated depth maps capture accurate relative distances among pixels. The second novelty arises from the observation that there exists an explicit, deducible relationship between optical flow divergence and depth gradient. A differential property correlation loss is, therefore, designed to refine depth estimation with a specific emphasis on local variations. The third novelty is a bidirectional stream co-adjustment strategy that enhances the interaction between rigid and optical flows, encouraging the former towards more accurate correspondence and making the latter more adaptable across various scenarios under the static scene hypotheses. DCPI-Depth, a framework that incorporates all these innovative components and couples two bidirectional and collaborative streams, achieves state-of-the-art performance and generalizability across multiple public datasets, outperforming all existing prior arts. Specifically, it demonstrates accurate depth estimation in texture-less and dynamic regions, and shows more reasonable smoothness. Our source code will be publicly available at mias.group/DCPI-Depth upon publication.

Text-guided 3D face synthesis has achieved remarkable results by leveraging text-to-image (T2I) diffusion models. However, most existing works focus solely on the direct generation, ignoring the editing, restricting them from synthesizing customized 3D faces through iterative adjustments. In this paper, we propose a unified text-guided framework from face generation to editing. In the generation stage, we propose a geometry-texture decoupled generation to mitigate the loss of geometric details caused by coupling. Besides, decoupling enables us to utilize the generated geometry as a condition for texture generation, yielding highly geometry-texture aligned results. We further employ a fine-tuned texture diffusion model to enhance texture quality in both RGB and YUV space. In the editing stage, we first employ a pre-trained diffusion model to update facial geometry or texture based on the texts. To enable sequential editing, we introduce a UV domain consistency preservation regularization, preventing unintentional changes to irrelevant facial attributes. Besides, we propose a self-guided consistency weight strategy to improve editing efficacy while preserving consistency. Through comprehensive experiments, we showcase our method's superiority in face synthesis. Project page: https://faceg2e.github.io/.

Multi-organ medical segmentation is a crucial component of medical image processing, essential for doctors to make accurate diagnoses and develop effective treatment plans. Despite significant progress in this field, current multi-organ segmentation models often suffer from inaccurate details, dependence on geometric prompts and loss of spatial information. Addressing these challenges, we introduce a novel model named CRISP-SAM2 with CRoss-modal Interaction and Semantic Prompting based on SAM2. This model represents a promising approach to multi-organ medical segmentation guided by textual descriptions of organs. Our method begins by converting visual and textual inputs into cross-modal contextualized semantics using a progressive cross-attention interaction mechanism. These semantics are then injected into the image encoder to enhance the detailed understanding of visual information. To eliminate reliance on geometric prompts, we use a semantic prompting strategy, replacing the original prompt encoder to sharpen the perception of challenging targets. In addition, a similarity-sorting self-updating strategy for memory and a mask-refining process is applied to further adapt to medical imaging and enhance localized details. Comparative experiments conducted on seven public datasets indicate that CRISP-SAM2 outperforms existing models. Extensive analysis also demonstrates the effectiveness of our method, thereby confirming its superior performance, especially in addressing the limitations mentioned earlier. Our code is available at: https://github.com/YU-deep/CRISP\_SAM2.git.

We study the task of panoptic symbol spotting, which involves identifying both individual instances of countable things and the semantic regions of uncountable stuff in computer-aided design (CAD) drawings composed of vector graphical primitives. Existing methods typically rely on image rasterization, graph construction, or point-based representation, but these approaches often suffer from high computational costs, limited generality, and loss of geometric structural information. In this paper, we propose VecFormer, a novel method that addresses these challenges through line-based representation of primitives. This design preserves the geometric continuity of the original primitive, enabling more accurate shape representation while maintaining a computation-friendly structure, making it well-suited for vector graphic understanding tasks. To further enhance prediction reliability, we introduce a Branch Fusion Refinement module that effectively integrates instance and semantic predictions, resolving their inconsistencies for more coherent panoptic outputs. Extensive experiments demonstrate that our method establishes a new state-of-the-art, achieving 91.1 PQ, with Stuff-PQ improved by 9.6 and 21.2 points over the second-best results under settings with and without prior information, respectively, highlighting the strong potential of line-based representation as a foundation for vector graphic understanding.

Recent 3D content generation pipelines commonly employ Variational Autoencoders (VAEs) to encode shapes into compact latent representations for diffusion-based generation. However, the widely adopted uniform point sampling strategy in Shape VAE training often leads to a significant loss of geometric details, limiting the quality of shape reconstruction and downstream generation tasks. We present Dora-VAE, a novel approach that enhances VAE reconstruction through our proposed sharp edge sampling strategy and a dual cross-attention mechanism. By identifying and prioritizing regions with high geometric complexity during training, our method significantly improves the preservation of fine-grained shape features. Such sampling strategy and the dual attention mechanism enable the VAE to focus on crucial geometric details that are typically missed by uniform sampling approaches. To systematically evaluate VAE reconstruction quality, we additionally propose Dora-bench, a benchmark that quantifies shape complexity through the density of sharp edges, introducing a new metric focused on reconstruction accuracy at these salient geometric features. Extensive experiments on the Dora-bench demonstrate that Dora-VAE achieves comparable reconstruction quality to the state-of-the-art dense XCube-VAE while requiring a latent space at least 8times smaller (1,280 vs. > 10,000 codes).

Large language models (LLMs) achieve remarkable performance through ever-increasing parameter counts, but scaling incurs steep computational costs. To better understand LLM scaling, we study representational differences between LLMs and their smaller counterparts, with the goal of replicating the representational qualities of larger models in the smaller models. We observe a geometric phenomenon which we term embedding condensation, where token embeddings collapse into a narrow cone-like subspace in some language models. Through systematic analyses across multiple Transformer families, we show that small models such as GPT2 and Qwen3-0.6B exhibit severe condensation, whereas the larger models such as GPT2-xl and Qwen3-32B are more resistant to this phenomenon. Additional observations show that embedding condensation is not reliably mitigated by knowledge distillation from larger models. To fight against it, we formulate a dispersion loss that explicitly encourages embedding dispersion during training. Experiments demonstrate that it mitigates condensation, recovers dispersion patterns seen in larger models, and yields performance gains across 10 benchmarks. We believe this work offers a principled path toward improving smaller Transformers without additional parameters.

Diffusion models for single image novel view synthesis (NVS) can generate highly realistic and plausible images, but they are limited in the geometric consistency to the given relative poses. The generated images often show significant errors with respect to the epipolar constraints that should be fulfilled, as given by the target pose. In this paper we address this issue by proposing a methodology to improve the geometric correctness of images generated by a diffusion model for single image NVS. We formulate a loss function based on image matching and epipolar constraints, and optimize the starting noise in a diffusion sampling process such that the generated image should both be a realistic image and fulfill geometric constraints derived from the given target pose. Our method does not require training data or fine-tuning of the diffusion models, and we show that we can apply it to multiple state-of-the-art models for single image NVS. The method is evaluated on the MegaScenes dataset and we show that geometric consistency is improved compared to the baseline models while retaining the quality of the generated images.

Latent variable collaborative filtering methods have been a standard approach to modelling user-click interactions due to their simplicity and effectiveness. However, there is limited work on analyzing the mathematical properties of these methods in particular on preventing the overfitting towards the identity, and such methods typically utilize loss functions that overlook the geometry between items. In this work, we introduce a notion of generalization gap in collaborative filtering and analyze this with respect to latent collaborative filtering models. We present a geometric upper bound that gives rise to loss functions, and a way to meaningfully utilize the geometry of item-metadata to improve recommendations. We show how these losses can be minimized and gives the recipe to a new latent collaborative filtering algorithm, which we refer to as GeoCF, due to the geometric nature of our results. We then show experimentally that our proposed GeoCF algorithm can outperform other all existing methods on the Movielens20M and Netflix datasets, as well as two large-scale internal datasets. In summary, our work proposes a theoretically sound method which paves a way to better understand generalization of collaborative filtering at large.

Camera-controllable image editing aims to synthesize novel views of a given scene under varying camera poses while strictly preserving cross-view geometric consistency. However, existing methods typically rely on fragmented geometric guidance, such as only injecting point clouds at the representation level despite models containing multiple levels, and are mainly based on image diffusion models that operate on discrete view mappings. These two limitations jointly lead to geometric drift and structural degradation under continuous camera motion. We observe that while leveraging video models provides continuous viewpoint priors for camera-controllable image editing, they still struggle to form stable geometric understanding if geometric guidance remains fragmented. To systematically address this, we inject unified geometric guidance across three levels that jointly determine the generative output: representation, architecture, and loss function. To this end, we propose UniGeo, a novel camera-controllable editing framework. Specifically, at the representation level, UniGeo incorporates a frame-decoupled geometric reference injection mechanism to provide robust cross-view geometry context. At the architecture level, it introduces geometric anchor attention to align multi-view features. At the loss function level, it proposes a trajectory-endpoint geometric supervision strategy to explicitly reinforce the structural fidelity of target views. Comprehensive experiments across multiple public benchmarks, encompassing both extensive and limited camera motion settings, demonstrate that UniGeo significantly outperforms existing methods in both visual quality and geometric consistency.

Machine unlearning, the removal of a training subset's influence from a deployed model, is critical for privacy preservation and model reliability, yet gradient ascent on forget samples often harms retained knowledge. Existing approaches face a persistent tradeoff between effective forgetting and preservation on the retain set. While previous methods provide useful heuristics, they often lack a formal analysis on how exactly forgetting updates harm retained knowledge, and whether the side effects can be removed with theoretical guarantees. To explore a theoretically sound and simple solution, we start from the first principle on how performance on the retain set is actually affected: a first-order analysis of the local change of the retain loss under small parameter updates during model training. We start from a crisp equivalence: the retain loss is unchanged to first order iff the update direction is orthogonal to the subspace spanned by retain gradients ("retain-invariant"). This identifies the entangled component as the tangential part of forget update within the retain-gradient subspace, and characterizes disentanglement as orthogonality. Guided by this, we propose the Geometric-disentanglement Unlearning (GU) that decomposes any candidate forget gradient update into tangential and normal components to retain space and executes only the normal component. Under a standard trust-region budget, the projected direction aligned with the raw forget gradient is optimal among all first-order retain-invariant moves, and we also derive the optimal projected direction for joint forget-retain updating objectives. Our method is plug-and-play and can be attached to existing gradient-based unlearning procedures to mitigate side effects. GU achieves consistent improvement on various methods across three benchmarks TOFU, MUSE, and WMDP.

Pretraining is the cornerstone of Large Language Models (LLMs), dominating the vast majority of computational budget and data to serve as the primary engine for their capabilities. During pretraining, LLMs acquire foundational knowledge from an unprecedentedly massive and diverse data sources, encompassing a vast array of domains such as general language, mathematics, code, and complex reasoning. In this work, we investigate an interesting geometric question regarding the converged state of pretraining: Does the model converge to a common minimizer across all data sources (e.g., fig:cwa_illustration:close), or merely a minimizer of the summed loss (e.g., fig:cwa_illustration:distant)? We hypothesize that the geometric "closeness" of task-specific minima is intrinsically linked to downstream generalization. We reveal that standard optimizers (e.g., AdamW) often converge to points where task-specific minima are distant from each other. To address this, we propose the Nexus optimizer, which encourages the closeness of these minima by maximizing gradient similarity during optimization. Experiments across models ranging from 130M to 3B parameters, various data mixtures and hyperparameter schedules, show that Nexus significantly boosts downstream performance, despite achieving the same pretraining loss (see fig:demo:benchmark). Notably, on the 3B model, Nexus reduces the out-of-distribution loss by 0.012 and yields up to a 15.0\% accuracy improvement on complex reasoning tasks (e.g., GSM8k). This finding challenges the reliance on pretraining loss as the sole proxy for model evaluation and demonstrates the importance of implicit biases in unlocking downstream generalization.

3D Morphable Models (3DMMs) provide promising 3D face reconstructions in various applications. However, existing methods struggle to reconstruct faces with extreme expressions due to deficiencies in supervisory signals, such as sparse or inaccurate landmarks. Segmentation information contains effective geometric contexts for face reconstruction. Certain attempts intuitively depend on differentiable renderers to compare the rendered silhouettes of reconstruction with segmentation, which is prone to issues like local optima and gradient instability. In this paper, we fully utilize the facial part segmentation geometry by introducing Part Re-projection Distance Loss (PRDL). Specifically, PRDL transforms facial part segmentation into 2D points and re-projects the reconstruction onto the image plane. Subsequently, by introducing grid anchors and computing different statistical distances from these anchors to the point sets, PRDL establishes geometry descriptors to optimize the distribution of the point sets for face reconstruction. PRDL exhibits a clear gradient compared to the renderer-based methods and presents state-of-the-art reconstruction performance in extensive quantitative and qualitative experiments. Our project is available at https://github.com/wang-zidu/3DDFA-V3 .

Text and image conditioned 3D models now generate convincing assets, but they still offer little direct control over the space an object should occupy or avoid. In authoring, this spatial intent is often known before generation starts. A chair should fit a seating envelope, a prop should leave clearance for motion, or a part should expose a contact surface. Prompts and image views are poor carriers for such constraints, requiring the need for an explicit control interface. We present Arbor, a trainable attachment for text conditioned latent 3D generation. Arbor introduces constraint meshes as a native 3D control interface. The interface uses hull regions where geometry should exist, avoidance regions that should remain empty, and touch regions the object should contact. Unlike completion or whole object scaffold control, these meshes are not target evidence. They are local typed requirements and can include regions where no surface should appear. Arbor keeps this signal as geometry by converting constraint meshes into tokens and learning a routed attachment inside a frozen denoiser. Each latent region can therefore receive the part of the constraint that matters for its spatial location. We evaluate Arbor on automatic and artist curated control benchmarks with hull, avoidance, and touch constraints, and compare the metric trends to a user preference study. Even without dedicated compliance losses, Arbor improves constraint obedience while preserving object quality and variation under fixed constraints.

Improving the quality of hyperspectral images (HSIs), such as through super-resolution, is a crucial research area. However, generative modeling for HSIs presents several challenges. Due to their high spectral dimensionality, HSIs are too memory-intensive for direct input into conventional diffusion models. Furthermore, general generative models lack an understanding of the topological and geometric structures of ground objects in remote sensing imagery. In addition, most diffusion models optimize loss functions at the noise level, leading to a non-intuitive convergence behavior and suboptimal generation quality for complex data. To address these challenges, we propose a Geometric Enhanced Wavelet-based Diffusion Model (GEWDiff), a novel framework for reconstructing hyperspectral images at 4-times super-resolution. A wavelet-based encoder-decoder is introduced that efficiently compresses HSIs into a latent space while preserving spectral-spatial information. To avoid distortion during generation, we incorporate a geometry-enhanced diffusion process that preserves the geometric features. Furthermore, a multi-level loss function was designed to guide the diffusion process, promoting stable convergence and improved reconstruction fidelity. Our model demonstrated state-of-the-art results across multiple dimensions, including fidelity, spectral accuracy, visual realism, and clarity.

Sparse Mixture-of-Experts (SMoE) models enable scaling language models efficiently, but training them remains challenging, as routing can collapse onto few experts and auxiliary load-balancing losses can reduce specialization. Motivated by these hurdles, we study how routing decisions in SMoEs are formed mechanistically. First, we reveal a geometric coupling between routers and their corresponding experts. For a given token, the router weights for the selected expert and the expert weights processing it receive gradients along the same input direction, differing only in scalar coefficients. Thus, matched router--expert directions accumulate the same routed token history. This theoretical coupling also appears empirically in routing dynamics. In a 1B SMoE trained from scratch, higher router scores predict stronger expert neuron activations, showing that routing decisions are mirrored inside the selected expert. Next, we analyze the effects of auxiliary load balancing on the router--expert geometric coupling, showing that such losses break this structure by spreading input-directed gradients across router weights, making distinct router directions nearly three times more similar to each other. Last, we demonstrate the centrality of geometric coupling for effective routing with a parameter-free online K-Means router, in which each expert maintains a running average of the hidden states routed to it and tokens are assigned based on cosine similarity. Compared with auxiliary-loss and loss-free balancing, this router achieves the lowest load imbalance with only a modest perplexity increase, indicating that geometric coupling captures a substantial part of what the router learns. Overall, our results explain how routers form assignment geometry that supports an effective division of labor.

Prior ReLoc3R achieves breakthrough performance with fast 25ms inference and state-of-the-art regression accuracy, yet our analysis reveals subtle geometric inconsistencies in its internal representations that prevent reaching the precision ceiling of correspondence-based methods like MASt3R (which require 300ms per pair). In this work, we present GeLoc3r, a novel approach to relative camera pose estimation that enhances pose regression methods through Geometric Consistency Regularization (GCR). GeLoc3r overcomes the speed-accuracy dilemma by training regression networks to produce geometrically consistent poses without inference-time geometric computation. During training, GeLoc3r leverages ground-truth depth to generate dense 3D-2D correspondences, weights them using a FusionTransformer that learns correspondence importance, and computes geometrically-consistent poses via weighted RANSAC. This creates a consistency loss that transfers geometric knowledge into the regression network. Unlike FAR method which requires both regression and geometric solving at inference, GeLoc3r only uses the enhanced regression head at test time, maintaining ReLoc3R's fast speed and approaching MASt3R's high accuracy. On challenging benchmarks, GeLoc3r consistently outperforms ReLoc3R, achieving significant improvements including 40.45% vs. 34.85% AUC@5° on the CO3Dv2 dataset (16% relative improvement), 68.66% vs. 66.70% AUC@5° on RealEstate10K, and 50.45% vs. 49.60% on MegaDepth1500. By teaching geometric consistency during training rather than enforcing it at inference, GeLoc3r represents a paradigm shift in how neural networks learn camera geometry, achieving both the speed of regression and the geometric understanding of correspondence methods.

Accurate segmentation of topological tubular structures, such as blood vessels and roads, is crucial in various fields, ensuring accuracy and efficiency in downstream tasks. However, many factors complicate the task, including thin local structures and variable global morphologies. In this work, we note the specificity of tubular structures and use this knowledge to guide our DSCNet to simultaneously enhance perception in three stages: feature extraction, feature fusion, and loss constraint. First, we propose a dynamic snake convolution to accurately capture the features of tubular structures by adaptively focusing on slender and tortuous local structures. Subsequently, we propose a multi-view feature fusion strategy to complement the attention to features from multiple perspectives during feature fusion, ensuring the retention of important information from different global morphologies. Finally, a continuity constraint loss function, based on persistent homology, is proposed to constrain the topological continuity of the segmentation better. Experiments on 2D and 3D datasets show that our DSCNet provides better accuracy and continuity on the tubular structure segmentation task compared with several methods. Our codes will be publicly available.

The recent integration of deep learning and pairwise similarity annotation-based constrained clustering -- i.e., deep constrained clustering (DCC) -- has proven effective for incorporating weak supervision into massive data clustering: Less than 1% of pair similarity annotations can often substantially enhance the clustering accuracy. However, beyond empirical successes, there is a lack of understanding of DCC. In addition, many DCC paradigms are sensitive to annotation noise, but performance-guaranteed noisy DCC methods have been largely elusive. This work first takes a deep look into a recently emerged logistic loss function of DCC, and characterizes its theoretical properties. Our result shows that the logistic DCC loss ensures the identifiability of data membership under reasonable conditions, which may shed light on its effectiveness in practice. Building upon this understanding, a new loss function based on geometric factor analysis is proposed to fend against noisy annotations. It is shown that even under unknown annotation confusions, the data membership can still be provably identified under our proposed learning criterion. The proposed approach is tested over multiple datasets to validate our claims.

PGD adversarial training, the standard robustness method, can reduce Jacobian Frobenius norm yet worsen clean-input geometry (e.g., TDI 1.336 vs. ERM 1.093). We show this is not an implementation artifact but a theorem-level consequence of supervised learning. We prove that any encoder minimizing supervised loss must retain non-zero sensitivity along directions correlated with training labels, including directions that are nuisance at test time. This holds across proper scoring rules, architectures, and dataset sizes. We call this the geometric blind spot of supervised learning. This theorem unifies four empirical phenomena often treated separately: non-robust features, texture bias, corruption fragility, and the robustness-accuracy tradeoff. It also explains why suppressing sensitivity in one adversarial direction can redistribute sensitivity elsewhere. We introduce Trajectory Deviation Index (TDI), a diagnostic of geometric isotropy. Unlike CKA, intrinsic dimension, or Jacobian Frobenius norm alone, TDI captures the failure mode above. In our experiments, PGD attains low Frobenius norm but high TDI, while PMH attains the lowest TDI with one additional training term and no architectural changes. Across seven tasks, BERT/SST-2, and ImageNet ViT-B/16 (backbone family underlying CLIP/DINO/SAM), the blind spot is measurable and repairable. It appears at foundation-model scale, worsens with model scale and task-specific fine-tuning, and is substantially reduced by PMH. PMH also leads on non-Gaussian corruption types (blur/brightness/contrast) without corruption-specific training.

Multimodal Large Language Models (MLLMs) have achieved remarkable progress in 2D visual tasks but still exhibit limited physical spatial awareness when processing real-world visual streams. Recently, feed-forward geometric foundation models, which implicitly extract geometric priors, have provided a new pathway to address this issue. However, existing geometry-aware MLLMs are predominantly constrained by the paradigm of single deep-layer extraction and input-level fusion. This flattened fusion leads to the loss of local geometric details and causes semantic mismatches in the early layers. To break this bottleneck, we propose GUIDE (Geometric Unrolling Inside MLLM Early-layers), a progressive geometric priors injection framework. GUIDE performs multi-level sampling within the geometric encoder, comprehensively capturing multi-granularity features ranging from local edges to global topologies. Subsequently, we rigorously align and fuse these multi-level geometric priors step-by-step with the early layers of the MLLM. Building upon the injection of multi-granularity geometric information, this design guides the model to progressively learn the 2D-to-3D transitional process. Furthermore, we introduce a context-aware gating that enables the model to fetch requisite spatial cues based on current semantics, thereby maximizing the utilization efficiency of spatial priors and effectively suppressing redundant geometric noise. Extensive experiments demonstrate that GUIDE significantly outperforms existing baselines on multiple complex spatial reasoning and perception tasks, establishing a novel paradigm for integrating 3D geometric priors into large models.

We present a class of novel optimisers for training neural networks that makes use of the Riemannian metric naturally induced when the loss landscape is embedded in higher-dimensional space. This is the same metric that underlies common visualisations of loss landscapes. By taking this geometric perspective literally and using the induced metric, we develop a new optimiser and compare it to existing methods, namely: SGD, Adam, AdamW, and Muon, across a range of tasks and architectures. Empirically, we conclude that this new class of optimisers is highly effective in low dimensional examples, and provides slight improvement over state-of-the-art methods for training neural networks. These new optimisers have theoretically desirable properties. In particular, the effective learning rate is automatically decreased in regions of high curvature acting as a smoothed out form of gradient clipping. Similarly, one variant of these optimisers can also be viewed as inducing an effective scheduled learning rate and decoupled weight decay is the natural choice from our geometric perspective. The basic method can be used to modify any existing preconditioning method. The new optimiser has a computational complexity comparable to that of Adam.

While static word embedding models are known to represent linguistic analogies as parallel lines in high-dimensional space, the underlying mechanism as to why they result in such geometric structures remains obscure. We find that an elementary contrastive-style method employed over distributional information performs competitively with popular word embedding models on analogy recovery tasks, while achieving dramatic speedups in training time. Further, we demonstrate that a contrastive loss is sufficient to create these parallel structures in word embeddings, and establish a precise relationship between the co-occurrence statistics and the geometric structure of the resulting word embeddings.

Bounding box regression is the crucial step in object detection. In existing methods, while ell_n-norm loss is widely adopted for bounding box regression, it is not tailored to the evaluation metric, i.e., Intersection over Union (IoU). Recently, IoU loss and generalized IoU (GIoU) loss have been proposed to benefit the IoU metric, but still suffer from the problems of slow convergence and inaccurate regression. In this paper, we propose a Distance-IoU (DIoU) loss by incorporating the normalized distance between the predicted box and the target box, which converges much faster in training than IoU and GIoU losses. Furthermore, this paper summarizes three geometric factors in bounding box regression, \ie, overlap area, central point distance and aspect ratio, based on which a Complete IoU (CIoU) loss is proposed, thereby leading to faster convergence and better performance. By incorporating DIoU and CIoU losses into state-of-the-art object detection algorithms, e.g., YOLO v3, SSD and Faster RCNN, we achieve notable performance gains in terms of not only IoU metric but also GIoU metric. Moreover, DIoU can be easily adopted into non-maximum suppression (NMS) to act as the criterion, further boosting performance improvement. The source code and trained models are available at https://github.com/Zzh-tju/DIoU.

Learning discriminative shape representations is a crucial issue for large-scale 3D shape retrieval. In this paper, we propose the Collaborative Inner Product Loss (CIP Loss) to obtain ideal shape embedding that discriminative among different categories and clustered within the same class. Utilizing simple inner product operation, CIP loss explicitly enforces the features of the same class to be clustered in a linear subspace, while inter-class subspaces are constrained to be at least orthogonal. Compared to previous metric loss functions, CIP loss could provide more clear geometric interpretation for the embedding than Euclidean margin, and is easy to implement without normalization operation referring to cosine margin. Moreover, our proposed loss term can combine with other commonly used loss functions and can be easily plugged into existing off-the-shelf architectures. Extensive experiments conducted on the two public 3D object retrieval datasets, ModelNet and ShapeNetCore 55, demonstrate the effectiveness of our proposal, and our method has achieved state-of-the-art results on both datasets.

Affine correspondences have received significant attention due to their benefits in tasks like image matching and pose estimation. Existing methods for extracting affine correspondences still have many limitations in terms of performance; thus, exploring a new paradigm is crucial. In this paper, we present a new pipeline designed for extracting accurate affine correspondences by integrating dense matching and geometric constraints. Specifically, a novel extraction framework is introduced, with the aid of dense matching and a novel keypoint scale and orientation estimator. For this purpose, we propose loss functions based on geometric constraints, which can effectively improve accuracy by supervising neural networks to learn feature geometry. The experimental show that the accuracy and robustness of our method outperform the existing ones in image matching tasks. To further demonstrate the effectiveness of the proposed method, we applied it to relative pose estimation. Affine correspondences extracted by our method lead to more accurate poses than the baselines on a range of real-world datasets. The code is available at https://github.com/stilcrad/DenseAffine.

Deep learning approaches have provided state-of-the-art performance in many applications by relying on large and overparameterized neural networks. However, such networks have been shown to be very brittle and are difficult to deploy on resource-limited platforms. Model pruning, i.e., reducing the size of the network, is a widely adopted strategy that can lead to a more robust and compact model. Many heuristics exist for model pruning, but empirical studies show that some heuristics improve performance whereas others can make models more brittle or have other side effects. This work aims to shed light on how different pruning methods alter the network's internal feature representation and the corresponding impact on model performance. To facilitate a comprehensive comparison and characterization of the high-dimensional model feature space, we introduce a visual geometric analysis of feature representations. We decomposed and evaluated a set of critical geometric concepts from the common adopted classification loss, and used them to design a visualization system to compare and highlight the impact of pruning on model performance and feature representation. The proposed tool provides an environment for in-depth comparison of pruning methods and a comprehensive understanding of how model response to common data corruption. By leveraging the proposed visualization, machine learning researchers can reveal the similarities between pruning methods and redundant in robustness evaluation benchmarks, obtain geometric insights about the differences between pruned models that achieve superior robustness performance, and identify samples that are robust or fragile to model pruning and common data corruption to model pruning and data corruption but also obtain insights and explanations on how some pruned models achieve superior robustness performance.

Monocular 3D object detection is an essential task in autonomous driving. However, most current methods consider each 3D object in the scene as an independent training sample, while ignoring their inherent geometric relations, thus inevitably resulting in a lack of leveraging spatial constraints. In this paper, we propose a novel method that takes all the objects into consideration and explores their mutual relationships to help better estimate the 3D boxes. Moreover, since 2D detection is more reliable currently, we also investigate how to use the detected 2D boxes as guidance to globally constrain the optimization of the corresponding predicted 3D boxes. To this end, a differentiable loss function, termed as Homography Loss, is proposed to achieve the goal, which exploits both 2D and 3D information, aiming at balancing the positional relationships between different objects by global constraints, so as to obtain more accurately predicted 3D boxes. Thanks to the concise design, our loss function is universal and can be plugged into any mature monocular 3D detector, while significantly boosting the performance over their baseline. Experiments demonstrate that our method yields the best performance (Nov. 2021) compared with the other state-of-the-arts by a large margin on KITTI 3D datasets.

LiDAR point clouds are fundamental to various applications, yet the extreme sparsity of high-precision geometric details hinders efficient context modeling, thereby limiting the compression speed and performance of existing methods. To address this challenge, we propose a compact representation for efficient predictive lossless coding. Our framework comprises two lightweight modules. First, the Geometry Re-Densification Module iteratively densifies encoded sparse geometry, extracts features at a dense scale, and then sparsifies the features for predictive coding. This module avoids costly computation on highly sparse details while maintaining a lightweight prediction head. Second, the Cross-scale Feature Propagation Module leverages occupancy cues from multiple resolution levels to guide hierarchical feature propagation, enabling information sharing across scales and reducing redundant feature extraction. Additionally, we introduce an integer-only inference pipeline to enable bit-exact cross-platform consistency, which avoids the entropy-coding collapse observed in existing neural compression methods and further accelerates coding. Experiments demonstrate competitive compression performance at real-time speed. Code will be released upon acceptance. Code is available at https://github.com/pengpeng-yu/FastPCC.

Accurate and efficient voxelized representations of 3D meshes are the foundation of 3D reconstruction and generation. However, existing representations based on iso-surface heavily rely on water-tightening or rendering optimization, which inevitably compromise geometric fidelity. We propose Faithful Contouring, a sparse voxelized representation that supports 2048+ resolutions for arbitrary meshes, requiring neither converting meshes to field functions nor extracting the isosurface during remeshing. It achieves near-lossless fidelity by preserving sharpness and internal structures, even for challenging cases with complex geometry and topology. The proposed method also shows flexibility for texturing, manipulation, and editing. Beyond representation, we design a dual-mode autoencoder for Faithful Contouring, enabling scalable and detail-preserving shape reconstruction. Extensive experiments show that Faithful Contouring surpasses existing methods in accuracy and efficiency for both representation and reconstruction. For direct representation, it achieves distance errors at the 10^{-5} level; for mesh reconstruction, it yields a 93\% reduction in Chamfer Distance and a 35\% improvement in F-score over strong baselines, confirming superior fidelity as a representation for 3D learning tasks.

Large Language Models frequently hallucinate in precision-critical domains such as technical diagramming and mechanical design, where outputs must satisfy strict geometric constraints. We study open-ended geometric synthesis from natural language: translating free-form descriptions into precise constructions whose entities must simultaneously satisfy dozens of interacting constraints. To make this tractable, we release PyGeoX, a programmable geometric DSL that compiles declarative constraints into a differentiable loss, and PyGeoX-Bench, a stratified suite of 300 problems with per-constraint verifiable rewards. Using PyGeoX as a verifier, we identify a failure mode we call Outlier Gradient Masking: under global-norm rewards (any scheme that aggregates residuals through a single norm, for example, exp(-MSE)), a single outlier constraint can nullify the learning signal across all others. To address this, we propose Saturating Additive Rewards (SAR), which decompose the reward into bounded per-constraint terms, preserving partial progress and ensuring consistent gradients even under severe violations. Against MSE-based rewards, the natural baseline for geometry solvers, SAR improves the hard-tier solving rate by 2.3times, and the resulting 8B model is competitive with much larger frontier systems on this benchmark. We release the engine, benchmark, and data at https://github.com/Huawei-AI4Math/PyGeoX.

A key challenge in machine learning is to explain how learning dynamics select among the many solutions that achieve identical loss values in overparameterized models - a phenomenon known as implicit bias. Controlling this bias provides a direct mechanism on learned representations, which are central to interpretability, robustness, and reasoning in modern AI systems. Yet, despite its importance, existing explanations remain largely ad hoc and lack a unifying mechanism. We develop a theoretical and constructive framework in which implicit bias emerges as a geometric correction induced by the interplay between gradient noise and continuous symmetries of the loss. We compute the induced bias across a range of architectures, predicting new behaviors and explaining known ones. The approach also enables inverse design: by engineering predictor - preserving parameterizations, it is possible to shape the bias, with sparsity and spectral sparsity emerging as canonical instances. Numerical experiments support the theory and validate the inverse - design framework in controlled settings.

Learning discriminative face features plays a major role in building high-performing face recognition models. The recent state-of-the-art face recognition solutions proposed to incorporate a fixed penalty margin on commonly used classification loss function, softmax loss, in the normalized hypersphere to increase the discriminative power of face recognition models, by minimizing the intra-class variation and maximizing the inter-class variation. Marginal penalty softmax losses, such as ArcFace and CosFace, assume that the geodesic distance between and within the different identities can be equally learned using a fixed penalty margin. However, such a learning objective is not realistic for real data with inconsistent inter-and intra-class variation, which might limit the discriminative and generalizability of the face recognition model. In this paper, we relax the fixed penalty margin constrain by proposing elastic penalty margin loss (ElasticFace) that allows flexibility in the push for class separability. The main idea is to utilize random margin values drawn from a normal distribution in each training iteration. This aims at giving the decision boundary chances to extract and retract to allow space for flexible class separability learning. We demonstrate the superiority of our ElasticFace loss over ArcFace and CosFace losses, using the same geometric transformation, on a large set of mainstream benchmarks. From a wider perspective, our ElasticFace has advanced the state-of-the-art face recognition performance on seven out of nine mainstream benchmarks.

Knowledge distillation (KD) methods can transfer knowledge of a parameter-heavy teacher model to a light-weight student model. The status quo for feature KD methods is to utilize loss functions based on logits (i.e., pre-softmax class scores) and intermediate layer features (i.e., latent representations). Unlike previous approaches, we propose a feature KD framework for training the student's backbone using feature-based losses exclusively (i.e., without logit-based losses such as cross entropy). Leveraging recent discoveries about the geometry of latent representations, we introduce a knowledge quality metric for identifying which teacher layers provide the most effective knowledge for distillation. Experiments on three image classification datasets with four diverse student-teacher pairs, spanning convolutional neural networks and vision transformers, demonstrate our KD method achieves state-of-the-art performance, delivering top-1 accuracy boosts of up to 15% over standard approaches. We publically share our code to facilitate future work at https://github.com/Thegolfingocto/KD_wo_CE.

Challenges remain in ego-centric 3D scene generation due to limited view overlap and the dominant influence of individual perspectives on scene interpretation. These factors hinder the creation of viewpoint-consistent and semantically aligned visual content, as well as the construction of accurate geometric structures. In this paper, we propose CGGS, a text-to-3D framework aiming to enhance 3D-content-awareness and address geometric distortions in ego-centric scene generation. Firstly, the Ego-centric Generator is proposed by fine-tuning a Multi-View Latent Diffusion Model with consistency-augmented loss to generate consistent, high-fidelity 2D content aligned with textual descriptions. Then, Layout Decorator leverages optical flow and point-track correspondence to estimate depth, therefore producing dense point clouds as coarse layouts from the ego-centric 2D priors. Building on this initialization, Geometric Refiner is proposed to enhance 3D Gaussian reconstruction via an entropy-based Mutual Information Depth Loss (MID) combined with a hierarchical optimization scheme for improving visual quality and geometric structure. Comprehensive experiments demonstrate that softred{CGGS} outperforms previous methods in generating coherent and accurate text-driven 3D scenes. Project page: https://cggs-26.github.io/cggs26/.

Vision-Language Models (VLMs) often struggle with robust 3D spatial reasoning. Prevailing methods that rely on fine-tuning with 3D visual question-answering (VQA) datasets may overfit dataset-specific biases, while integrating specialized 3D visual encoders is often inflexible and cumbersome. In this paper, we argue that genuine spatial understanding should emerge from learning fundamental geometric priors, not only from high-level VQA supervision. We propose GASP (Geometric-Aware Spatial Priors), a framework that injects these priors directly into the LLM's transformer layers. GASP employs a small correspondence head, applied as a deep supervision signal across all layers, and is trained with a dual objective leveraging ground-truth geometry from large-scale video scenes: a contrastive loss on ground-truth point correspondences enforces 2D view-invariance, while a depth consistency supervision resolves 3D geometric ambiguities. Our analysis first provides a diagnostic showing that standard VLMs' internal correspondence matching accuracy is very low (often below 5%). We then demonstrate that our training substantially improves this behavior, boosting peak layer-wise correspondence to over 70% and maintaining over 85% temporal robustness while baselines remain below 5%. These internal improvements translate to significant gains on downstream spatial benchmarks including +18.2% on All-Angles Bench and +29.0% on VSI-Bench, all without training on any 3D VQA data. Our findings indicate that learning from fundamental geometric priors is a promising and generalizable pathway towards VLMs with more reliable 3D spatial reasoning.

We present a novel framework to regularize Neural Radiance Field (NeRF) in a few-shot setting with a geometry-aware consistency regularization. The proposed approach leverages a rendered depth map at unobserved viewpoint to warp sparse input images to the unobserved viewpoint and impose them as pseudo ground truths to facilitate learning of NeRF. By encouraging such geometry-aware consistency at a feature-level instead of using pixel-level reconstruction loss, we regularize the NeRF at semantic and structural levels while allowing for modeling view dependent radiance to account for color variations across viewpoints. We also propose an effective method to filter out erroneous warped solutions, along with training strategies to stabilize training during optimization. We show that our model achieves competitive results compared to state-of-the-art few-shot NeRF models. Project page is available at https://ku-cvlab.github.io/GeCoNeRF/.

Recently, a popular line of research in face recognition is adopting margins in the well-established softmax loss function to maximize class separability. In this paper, we first introduce an Additive Angular Margin Loss (ArcFace), which not only has a clear geometric interpretation but also significantly enhances the discriminative power. Since ArcFace is susceptible to the massive label noise, we further propose sub-center ArcFace, in which each class contains K sub-centers and training samples only need to be close to any of the K positive sub-centers. Sub-center ArcFace encourages one dominant sub-class that contains the majority of clean faces and non-dominant sub-classes that include hard or noisy faces. Based on this self-propelled isolation, we boost the performance through automatically purifying raw web faces under massive real-world noise. Besides discriminative feature embedding, we also explore the inverse problem, mapping feature vectors to face images. Without training any additional generator or discriminator, the pre-trained ArcFace model can generate identity-preserved face images for both subjects inside and outside the training data only by using the network gradient and Batch Normalization (BN) priors. Extensive experiments demonstrate that ArcFace can enhance the discriminative feature embedding as well as strengthen the generative face synthesis.

Novel View Synthesis (NVS) has traditionally relied on models with explicit 3D inductive biases combined with known camera parameters from Structure-from-Motion (SfM) beforehand. Recent vision foundation models like VGGT take an orthogonal approach -- 3D knowledge is gained implicitly through training data and loss objectives, enabling feed-forward prediction of both camera parameters and 3D representations directly from a set of uncalibrated images. While flexible, VGGT features lack explicit multi-view geometric consistency, and we find that improving such 3D feature consistency benefits both NVS and pose estimation tasks. We introduce Selfi, a self-improving 3D reconstruction pipeline via feature alignment, transforming a VGGT backbone into a high-fidelity 3D reconstruction engine by leveraging its own outputs as pseudo-ground-truth. Specifically, we train a lightweight feature adapter using a reprojection-based consistency loss, which distills VGGT outputs into a new geometrically-aligned feature space that captures spatial proximity in 3D. This enables state-of-the-art performance in both NVS and camera pose estimation, demonstrating that feature alignment is a highly beneficial step for downstream 3D reasoning.

Relative representations are an established approach to zero-shot model stitching, consisting of a non-trainable transformation of the latent space of a deep neural network. Based on insights of topological and geometric nature, we propose two improvements to relative representations. First, we introduce a normalization procedure in the relative transformation, resulting in invariance to non-isotropic rescalings and permutations. The latter coincides with the symmetries in parameter space induced by common activation functions. Second, we propose to deploy topological densification when fine-tuning relative representations, a topological regularization loss encouraging clustering within classes. We provide an empirical investigation on a natural language task, where both the proposed variations yield improved performance on zero-shot model stitching.

Functional magnetic resonance imaging techniques benefit from echo-planar imaging's fast image acquisition but are susceptible to inhomogeneities in the main magnetic field, resulting in geometric distortion and signal loss artifacts in the images. Traditional methods leverage a field map or voxel displacement map for distortion correction. However, voxel displacement map estimation requires additional sequence acquisitions, and the accuracy of the estimation influences correction performance. This work implements a novel approach called GDCNet, which estimates a geometric distortion map by non-linear registration to T1-weighted anatomical images and applies it for distortion correction. GDCNet demonstrated fast distortion correction of functional images in retrospectively and prospectively acquired datasets. Among the compared models, the 2D self-supervised configuration resulted in a statistically significant improvement to normalized mutual information between distortion-corrected functional and T1-weighted images compared to the benchmark methods FUGUE and TOPUP. Furthermore, GDCNet models achieved processing speeds 14 times faster than TOPUP in the prospective dataset.

Stochastic Gradient Descent (SGD) stands as a cornerstone optimization algorithm with proven real-world empirical successes but relatively limited theoretical understanding. Recent research has illuminated a key factor contributing to its practical efficacy: the implicit regularization it instigates. Several studies have investigated the linear stability property of SGD in the vicinity of a stationary point as a predictive proxy for sharpness and generalization error in overparameterized neural networks (Wu et al., 2022; Jastrzebski et al., 2019; Cohen et al., 2021). In this paper, we delve deeper into the relationship between linear stability and sharpness. More specifically, we meticulously delineate the necessary and sufficient conditions for linear stability, contingent on hyperparameters of SGD and the sharpness at the optimum. Towards this end, we introduce a novel coherence measure of the loss Hessian that encapsulates pertinent geometric properties of the loss function that are relevant to the linear stability of SGD. It enables us to provide a simplified sufficient condition for identifying linear instability at an optimum. Notably, compared to previous works, our analysis relies on significantly milder assumptions and is applicable for a broader class of loss functions than known before, encompassing not only mean-squared error but also cross-entropy loss.

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/.

FP4 training promises substantial reductions in memory and computation cost for LLM pretraining, yet current FP4 hardware paths and recipes, including NVIDIA Blackwell/Rubin-class systems and AMD MI350-series GPUs, remain centered on E2M1 data elements. In this study, we identify a fundamental limitation of that choice: non-uniform formats such as E2M1 inherently suffer from Shrinkage Bias, a systematic negative rounding error caused by the geometric asymmetry of their representable bins. We show that this bias accumulates multiplicatively across layers and is amplified by the Random Hadamard Transform (RHT), providing a unified explanation for the training instability observed in existing E2M1-based FP4 recipes. In contrast, uniform grids (E1M2/INT4) bypass this grid-geometry error and better convert the improved bucket utilization from RHT into higher quantization quality. Based on this finding, we propose UFP4, a uniform 4-bit training recipe that applies RHT to all three training GEMMs while restricting stochastic rounding to dY alone. On Dense 1.5B, MoE 7.9B, and MoE 124B long-run pretraining, UFP4 consistently achieves lower BF16-relative loss degradation than strong E2M1-based baselines, supported by scaling-law analysis and ablation studies. Our results suggest that future accelerators should support E1M2/INT4-style uniform 4-bit grids as first-class training primitives alongside E2M1.

Recent advances in 3D Gaussian Splatting (3DGS) have enabled RGB-only SLAM systems to achieve high-fidelity scene representation. However, the heavy reliance of existing systems on photometric rendering loss for camera tracking undermines their robustness, especially in unbounded outdoor environments with severe viewpoint and illumination changes. To address these challenges, we propose TVG-SLAM, a robust RGB-only 3DGS SLAM system that leverages a novel tri-view geometry paradigm to ensure consistent tracking and high-quality mapping. We introduce a dense tri-view matching module that aggregates reliable pairwise correspondences into consistent tri-view matches, forming robust geometric constraints across frames. For tracking, we propose Hybrid Geometric Constraints, which leverage tri-view matches to construct complementary geometric cues alongside photometric loss, ensuring accurate and stable pose estimation even under drastic viewpoint shifts and lighting variations. For mapping, we propose a new probabilistic initialization strategy that encodes geometric uncertainty from tri-view correspondences into newly initialized Gaussians. Additionally, we design a Dynamic Attenuation of Rendering Trust mechanism to mitigate tracking drift caused by mapping latency. Experiments on multiple public outdoor datasets show that our TVG-SLAM outperforms prior RGB-only 3DGS-based SLAM systems. Notably, in the most challenging dataset, our method improves tracking robustness, reducing the average Absolute Trajectory Error (ATE) by 69.0\% while achieving state-of-the-art rendering quality. The implementation of our method will be released as open-source.

Machine unlearning (MU) for large language models has become critical for AI safety, yet existing methods fail to generalize to Mixture-of-Experts (MoE) architectures. We identify that traditional unlearning methods exploit MoE's architectural vulnerability: they manipulate routers to redirect queries away from knowledgeable experts rather than erasing knowledge, causing a loss of model utility and superficial forgetting. We propose Geometric Routing Invariance Preservation (GRIP), an algorithm-agnostic framework for unlearning for MoE. Our core contribution is a geometric constraint, implemented by projecting router gradient updates into an expert-specific null-space. Crucially, this decouples routing stability from parameter rigidity: while discrete expert selections remain stable for retained knowledge, the continuous router parameters remain plastic within the null space, allowing the model to undergo necessary internal reconfiguration to satisfy unlearning objectives. This forces the unlearning optimization to erase knowledge directly from expert parameters rather than exploiting the superficial router manipulation shortcut. GRIP functions as an adapter, constraining router parameter updates without modifying the underlying unlearning algorithm. Extensive experiments on large-scale MoE models demonstrate that our adapter eliminates expert selection shift (achieving over 95% routing stability) across all tested unlearning methods while preserving their utility. By preventing existing algorithms from exploiting MoE model's router vulnerability, GRIP adapts existing unlearning research from dense architectures to MoEs.

We study zero-shot 3D alignment of two given meshes, using a text prompt describing their spatial relation -- an essential capability for content creation and scene assembly. Earlier approaches primarily rely on geometric alignment procedures, while recent work leverages pretrained 2D diffusion models to model language-conditioned object-object spatial relationships. In contrast, we directly optimize the relative pose at test time, updating translation, rotation, and isotropic scale with CLIP-driven gradients via a differentiable renderer, without training a new model. Our framework augments language supervision with geometry-aware objectives: a variant of soft-Iterative Closest Point (ICP) term to encourage surface attachment and a penetration loss to discourage interpenetration. A phased schedule strengthens contact constraints over time, and camera control concentrates the optimization on the interaction region. To enable evaluation, we curate a benchmark containing diverse categories and relations, and compare against baselines. Our method outperforms all alternatives, yielding semantically faithful and physically plausible alignments.

3D human pose estimation from sketches has broad applications in computer animation and film production. Unlike traditional human pose estimation, this task presents unique challenges due to the abstract and disproportionate nature of sketches. Previous sketch-to-pose methods, constrained by the lack of large-scale sketch-3D pose annotations, primarily relied on optimization with heuristic rules-an approach that is both time-consuming and limited in generalizability. To address these challenges, we propose a novel approach leveraging a "learn from synthesis" strategy. First, a diffusion model is trained to synthesize sketch images from 2D poses projected from 3D human poses, mimicking disproportionate human structures in sketches. This process enables the creation of a synthetic dataset, SKEP-120K, consisting of 120k accurate sketch-3D pose annotation pairs across various sketch styles. Building on this synthetic dataset, we introduce an end-to-end data-driven framework for estimating human poses and shapes from diverse sketch styles. Our framework combines existing 2D pose detectors and generative diffusion priors for sketch feature extraction with a feed-forward neural network for efficient 2D pose estimation. Multiple heuristic loss functions are incorporated to guarantee geometric coherence between the derived 3D poses and the detected 2D poses while preserving accurate self-contacts. Qualitative, quantitative, and subjective evaluations collectively show that our model substantially surpasses previous ones in both estimation accuracy and speed for sketch-to-pose tasks.

To accelerate diffusion model inference, numerical solvers perform poorly at extremely small steps, while distillation techniques often introduce complexity and instability. This work presents an intermediate strategy, balancing performance and cost, by learning ODE integration using loss functions derived from the derivative-integral relationship, inspired by Monte Carlo integration and Picard iteration. From a geometric perspective, the losses operate by gradually extending the tangent to the secant, thus are named as secant losses. The target of secant losses is the same as that of diffusion models, or the diffusion model itself, leading to great training stability. By fine-tuning or distillation, the secant version of EDM achieves a 10-step FID of 2.14 on CIFAR-10, while the secant version of SiT-XL/2 attains a 4-step FID of 2.27 and an 8-step FID of 1.96 on ImageNet-256times256. Code is available at https://github.com/poppuppy/secant-expectation.

Maps are a key component in image-based camera localization and visual SLAM systems: they are used to establish geometric constraints between images, correct drift in relative pose estimation, and relocalize cameras after lost tracking. The exact definitions of maps, however, are often application-specific and hand-crafted for different scenarios (e.g. 3D landmarks, lines, planes, bags of visual words). We propose to represent maps as a deep neural net called MapNet, which enables learning a data-driven map representation. Unlike prior work on learning maps, MapNet exploits cheap and ubiquitous sensory inputs like visual odometry and GPS in addition to images and fuses them together for camera localization. Geometric constraints expressed by these inputs, which have traditionally been used in bundle adjustment or pose-graph optimization, are formulated as loss terms in MapNet training and also used during inference. In addition to directly improving localization accuracy, this allows us to update the MapNet (i.e., maps) in a self-supervised manner using additional unlabeled video sequences from the scene. We also propose a novel parameterization for camera rotation which is better suited for deep-learning based camera pose regression. Experimental results on both the indoor 7-Scenes dataset and the outdoor Oxford RobotCar dataset show significant performance improvement over prior work. The MapNet project webpage is https://goo.gl/mRB3Au.

3D Gaussian Splatting (3DGS) has exhibited remarkable efficacy in novel view synthesis (NVS). However, it suffers from a significant drawback: achieving high-fidelity rendering typically necessitates a large number of 3D Gaussians, resulting in substantial memory consumption and storage requirements. To address this challenge, we propose the first knowledge distillation framework for 3DGS, featuring various teacher models, including vanilla 3DGS, noise-augmented variants, and dropout-regularized versions. The outputs of these teachers are aggregated to guide the optimization of a lightweight student model. To distill the hidden geometric structure, we propose a structural similarity loss to boost the consistency of spatial geometric distributions between the student and teacher model. Through comprehensive quantitative and qualitative evaluations across diverse datasets, the proposed Distilled-3DGS, a simple yet effective framework without bells and whistles, achieves promising rendering results in both rendering quality and storage efficiency compared to state-of-the-art methods. Project page: https://distilled3dgs.github.io . Code: https://github.com/lt-xiang/Distilled-3DGS .

Existing single-image 3D human avatar methods primarily rely on rigid joint transformations, limiting their ability to model realistic cloth dynamics. We present DynaAvatar, a zero-shot framework that reconstructs animatable 3D human avatars with motion-dependent cloth dynamics from a single image. Trained on large-scale multi-person motion datasets, DynaAvatar employs a Transformer-based feed-forward architecture that directly predicts dynamic 3D Gaussian deformations without subject-specific optimization. To overcome the scarcity of dynamic captures, we introduce a static-to-dynamic knowledge transfer strategy: a Transformer pretrained on large-scale static captures provides strong geometric and appearance priors, which are efficiently adapted to motion-dependent deformations through lightweight LoRA fine-tuning on dynamic captures. We further propose the DynaFlow loss, an optical flow-guided objective that provides reliable motion-direction geometric cues for cloth dynamics in rendered space. Finally, we reannotate the missing or noisy SMPL-X fittings in existing dynamic capture datasets, as most public dynamic capture datasets contain incomplete or unreliable fittings that are unsuitable for training high-quality 3D avatar reconstruction models. Experiments demonstrate that DynaAvatar produces visually rich and generalizable animations, outperforming prior methods.

Despite recent progress in 3D self-supervised learning, collecting large-scale 3D scene scans remains expensive and labor-intensive. In this work, we investigate whether 3D representations can be learned from unlabeled videos recorded without any real 3D sensors. We present Laplacian-Aware Multi-level 3D Clustering with Sinkhorn-Knopp (LAM3C), a self-supervised framework that learns from video-generated point clouds from unlabeled videos. We first introduce RoomTours, a video-generated point cloud dataset constructed by collecting room-walkthrough videos from the web (e.g., real-estate tours) and generating 49,219 scenes using an off-the-shelf feed-forward reconstruction model. We also propose a noise-regularized loss that stabilizes representation learning by enforcing local geometric smoothness and ensuring feature stability under noisy point clouds. Remarkably, without using any real 3D scans, LAM3C achieves higher performance than the previous self-supervised methods on indoor semantic and instance segmentation. These results suggest that unlabeled videos represent an abundant source of data for 3D self-supervised learning.

Mirror reflections are common in everyday environments and can provide stereo information within a single capture, as the real and reflected virtual views are visible simultaneously. We exploit this property by treating the reflection as an auxiliary view and designing a transformation that constructs a physically valid virtual camera, allowing direct pixel-domain generation of the virtual view while adhering to the real-world imaging process. This enables a multi-view stereo setup from a single image, simplifying the imaging process, making it compatible with powerful feed-forward reconstruction models for generalizable and robust 3D reconstruction. To further exploit the geometric symmetry introduced by mirrors, we propose a symmetric-aware loss to refine pose estimation. Our framework also naturally extends to dynamic scenes, where each frame contains a mirror reflection, enabling efficient per-frame geometry recovery. For quantitative evaluation, we provide a fully customizable synthetic dataset of 16 Blender scenes, each with ground-truth point clouds and camera poses. Extensive experiments on real-world data and synthetic data are conducted to illustrate the effectiveness of our method.

Gaze estimation methods estimate gaze from facial appearance with a single camera. However, due to the limited view of a single camera, the captured facial appearance cannot provide complete facial information and thus complicate the gaze estimation problem. Recently, camera devices are rapidly updated. Dual cameras are affordable for users and have been integrated in many devices. This development suggests that we can further improve gaze estimation performance with dual-view gaze estimation. In this paper, we propose a dual-view gaze estimation network (DV-Gaze). DV-Gaze estimates dual-view gaze directions from a pair of images. We first propose a dual-view interactive convolution (DIC) block in DV-Gaze. DIC blocks exchange dual-view information during convolution in multiple feature scales. It fuses dual-view features along epipolar lines and compensates for the original feature with the fused feature. We further propose a dual-view transformer to estimate gaze from dual-view features. Camera poses are encoded to indicate the position information in the transformer. We also consider the geometric relation between dual-view gaze directions and propose a dual-view gaze consistency loss for DV-Gaze. DV-Gaze achieves state-of-the-art performance on ETH-XGaze and EVE datasets. Our experiments also prove the potential of dual-view gaze estimation. We release codes in https://github.com/yihuacheng/DVGaze.

Understanding the 3D geometry and semantics of driving scenes is critical for safe autonomous driving. Recent advances in 3D occupancy prediction have improved scene representation but often suffer from spatial inconsistencies, leading to floating artifacts and poor surface localization. Existing voxel-wise losses (e.g., cross-entropy) fail to enforce geometric coherence. In this paper, we propose GaussRender, a module that improves 3D occupancy learning by enforcing projective consistency. Our key idea is to project both predicted and ground-truth 3D occupancy into 2D camera views, where we apply supervision. Our method penalizes 3D configurations that produce inconsistent 2D projections, thereby enforcing a more coherent 3D structure. To achieve this efficiently, we leverage differentiable rendering with Gaussian splatting. GaussRender seamlessly integrates with existing architectures while maintaining efficiency and requiring no inference-time modifications. Extensive evaluations on multiple benchmarks (SurroundOcc-nuScenes, Occ3D-nuScenes, SSCBench-KITTI360) demonstrate that GaussRender significantly improves geometric fidelity across various 3D occupancy models (TPVFormer, SurroundOcc, Symphonies), achieving state-of-the-art results, particularly on surface-sensitive metrics. The code is open-sourced at https://github.com/valeoai/GaussRender.

We investigate the relationship between the geometry of token embeddings and their role in the next token prediction within transformer models. An important aspect of this connection uses the notion of empirical measure, which encodes the distribution of token point clouds across transformer layers and drives the evolution of token representations in the mean-field interacting picture. We use metrics such as intrinsic dimension, neighborhood overlap, and cosine similarity to observationally probe these empirical measures across layers. To validate our approach, we compare these metrics to a dataset where the tokens are shuffled, which disrupts the syntactic and semantic structure. Our findings reveal a correlation between the geometric properties of token embeddings and the cross-entropy loss of next token predictions, implying that prompts with higher loss values have tokens represented in higher-dimensional spaces.

Vision foundation models trained via multi-teacher distillation offer a promising path toward unified visual representations, yet the learning dynamics and data efficiency of such approaches remain underexplored. In this paper, we systematically study multi-teacher distillation for vision foundation models and identify key factors that enable training at lower computational cost. We introduce Agglomerative Mixture-of-Experts Vision Foundation Models (AMoE), which distill knowledge from SigLIP2 and DINOv3 simultaneously into a Mixture-of-Experts student. We show that (1) our Asymmetric Relation-Knowledge Distillation loss preserves the geometric properties of each teacher while enabling effective knowledge transfer, (2) token-balanced batching that packs varying-resolution images into sequences with uniform token budgets stabilizes representation learning across resolutions without sacrificing performance, and (3) hierarchical clustering and sampling of training data--typically reserved for self-supervised learning--substantially improves sample efficiency over random sampling for multi-teacher distillation. By combining these findings, we curate OpenLVD200M, a 200M-image corpus that demonstrates superior efficiency for multi-teacher distillation. Instantiated in a Mixture-of-Experts. We release OpenLVD200M and distilled models.

Recent advances in language and vision have demonstrated that scaling up model capacity consistently improves performance across diverse tasks. In 3D visual geometry reconstruction, large-scale training has likewise proven effective for learning versatile representations. However, further scaling of 3D models is challenging due to the complexity of geometric supervision and the diversity of 3D data. To overcome these limitations, we propose MoRE, a dense 3D visual foundation model based on a Mixture-of-Experts (MoE) architecture that dynamically routes features to task-specific experts, allowing them to specialize in complementary data aspects and enhance both scalability and adaptability. Aiming to improve robustness under real-world conditions, MoRE incorporates a confidence-based depth refinement module that stabilizes and refines geometric estimation. In addition, it integrates dense semantic features with globally aligned 3D backbone representations for high-fidelity surface normal prediction. MoRE is further optimized with tailored loss functions to ensure robust learning across diverse inputs and multiple geometric tasks. Extensive experiments demonstrate that MoRE achieves state-of-the-art performance across multiple benchmarks and supports effective downstream applications without extra computation.

We introduce SPFSplat, an efficient framework for 3D Gaussian splatting from sparse multi-view images, requiring no ground-truth poses during training or inference. It employs a shared feature extraction backbone, enabling simultaneous prediction of 3D Gaussian primitives and camera poses in a canonical space from unposed inputs within a single feed-forward step. Alongside the rendering loss based on estimated novel-view poses, a reprojection loss is integrated to enforce the learning of pixel-aligned Gaussian primitives for enhanced geometric constraints. This pose-free training paradigm and efficient one-step feed-forward design make SPFSplat well-suited for practical applications. Remarkably, despite the absence of pose supervision, SPFSplat achieves state-of-the-art performance in novel view synthesis even under significant viewpoint changes and limited image overlap. It also surpasses recent methods trained with geometry priors in relative pose estimation. Code and trained models are available on our project page: https://ranrhuang.github.io/spfsplat/.

Real-valued functions on geometric data -- such as node attributes on a graph -- can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in the loss function. When optimizing a single real-valued function (the one-parameter setting), there is a canonical choice of descriptor for persistent homology: the barcode. The operation mapping a real-valued function to its barcode is differentiable almost everywhere, and the convergence of gradient descent for losses using barcodes is relatively well understood. When optimizing a vector-valued function (the multiparameter setting), there is no unique choice of descriptor for multiparameter persistent homology, and many distinct descriptors have been proposed. This calls for the development of a general framework for differentiability and optimization that applies to a wide range of multiparameter homological descriptors. In this article, we develop such a framework and show that it encompasses well-known descriptors of different flavors, such as signed barcodes and the multiparameter persistence landscape. We complement the theory with numerical experiments supporting the idea that optimizing multiparameter homological descriptors can lead to improved performances compared to optimizing one-parameter descriptors, even when using the simplest and most efficiently computable multiparameter descriptors.

Multimodal LLMs are powerful but prone to object hallucinations, which describe non-existent entities and harm reliability. While recent unlearning methods attempt to mitigate this, we identify a critical flaw: structural fragility. We empirically demonstrate that standard erasure achieves only superficial suppression, trapping the model in sharp minima where hallucinations catastrophically resurge after lightweight relearning. To ensure geometric stability, we propose SARE, which casts unlearning as a targeted min-max optimization problem and uses a Targeted-SAM mechanism to explicitly flatten the loss landscape around hallucinated concepts. By suppressing hallucinations under simulated worst-case parameter perturbations, our framework ensures robust removal stable against weight shifts. Extensive experiments demonstrate that SARE significantly outperforms baselines in erasure efficacy while preserving general generation quality. Crucially, it maintains persistent hallucination suppression against relearning and parameter updates, validating the effectiveness of geometric stabilization.

Direct alignment algorithms such as Direct Preference Optimization (DPO) fine-tune models based on preference data, using only supervised learning instead of two-stage reinforcement learning with human feedback (RLHF). We show that DPO encodes a statistical estimation problem over reward functions induced by a parametric policy class. When the true reward function that generates preferences cannot be realized via the policy class, DPO becomes misspecified, resulting in failure modes such as preference order reversal, worsening of policy reward, and high sensitivity to the input preference data distribution. On the other hand, we study the local behavior of two-stage RLHF for a parametric class and relate it to a natural gradient step in policy space. Our fine-grained geometric characterization allows us to propose AuxDPO, which introduces additional auxiliary variables in the DPO loss function to help move towards the RLHF solution in a principled manner and mitigate the misspecification in DPO. We empirically demonstrate the superior performance of AuxDPO on didactic bandit settings as well as LLM alignment tasks.

Popular representation learning methods encourage feature invariance under transformations applied at the input. However, in 3D perception tasks like object localization and segmentation, outputs are naturally equivariant to some transformations, such as rotation. Using pre-training loss functions that encourage equivariance of features under certain transformations provides a strong self-supervision signal while also retaining information of geometric relationships between transformed feature representations. This can enable improved performance in downstream tasks that are equivariant to such transformations. In this paper, we propose a spatio-temporal equivariant learning framework by considering both spatial and temporal augmentations jointly. Our experiments show that the best performance arises with a pre-training approach that encourages equivariance to translation, scaling, and flip, rotation and scene flow. For spatial augmentations, we find that depending on the transformation, either a contrastive objective or an equivariance-by-classification objective yields best results. To leverage real-world object deformations and motion, we consider sequential LiDAR scene pairs and develop a novel 3D scene flow-based equivariance objective that leads to improved performance overall. We show our pre-training method for 3D object detection which outperforms existing equivariant and invariant approaches in many settings.

Learning signed distance functions (SDFs) from 3D point clouds is an important task in 3D computer vision. However, without ground truth signed distances, point normals or clean point clouds, current methods still struggle from learning SDFs from noisy point clouds. To overcome this challenge, we propose to learn SDFs via a noise to noise mapping, which does not require any clean point cloud or ground truth supervision for training. Our novelty lies in the noise to noise mapping which can infer a highly accurate SDF of a single object or scene from its multiple or even single noisy point cloud observations. Our novel learning manner is supported by modern Lidar systems which capture multiple noisy observations per second. We achieve this by a novel loss which enables statistical reasoning on point clouds and maintains geometric consistency although point clouds are irregular, unordered and have no point correspondence among noisy observations. Our evaluation under the widely used benchmarks demonstrates our superiority over the state-of-the-art methods in surface reconstruction, point cloud denoising and upsampling. Our code, data, and pre-trained models are available at https://github.com/mabaorui/Noise2NoiseMapping/

The computing cost and memory demand of deep learning pipelines have grown fast in recent years and thus a variety of pruning techniques have been developed to reduce model parameters. The majority of these techniques focus on reducing inference costs by pruning the network after a pass of full training. A smaller number of methods address the reduction of training costs, mostly based on compressing the network via low-rank layer factorizations. Despite their efficiency for linear layers, these methods fail to effectively handle convolutional filters. In this work, we propose a low-parametric training method that factorizes the convolutions into tensor Tucker format and adaptively prunes the Tucker ranks of the convolutional kernel during training. Leveraging fundamental results from geometric integration theory of differential equations on tensor manifolds, we obtain a robust training algorithm that provably approximates the full baseline performance and guarantees loss descent. A variety of experiments against the full model and alternative low-rank baselines are implemented, showing that the proposed method drastically reduces the training costs, while achieving high performance, comparable to or better than the full baseline, and consistently outperforms competing low-rank approaches.

Text-conditioned image generation has made significant progress in recent years with generative adversarial networks and more recently, diffusion models. While diffusion models conditioned on text prompts have produced impressive and high-quality images, accurately representing complex text prompts such as the number of instances of a specific object remains challenging. To address this limitation, we propose a novel guidance approach for the sampling process in the diffusion model that leverages bounding box and segmentation map information at inference time without additional training data. Through a novel loss in the sampling process, our approach guides the model with semantic features from CLIP embeddings and enforces geometric constraints, leading to high-resolution images that accurately represent the scene. To obtain bounding box and segmentation map information, we structure the text prompt as a scene graph and enrich the nodes with CLIP embeddings. Our proposed model achieves state-of-the-art performance on two public benchmarks for image generation from scene graphs, surpassing both scene graph to image and text-based diffusion models in various metrics. Our results demonstrate the effectiveness of incorporating bounding box and segmentation map guidance in the diffusion model sampling process for more accurate text-to-image generation.

Vision-language model (VLM) agents increasingly rely on memory-augmented reinforcement learning to reuse experience across long-horizon tasks, yet most existing frameworks store memory as text and depend on proprietary teacher models to summarize or refine it. This design is poorly matched to spatial decision making: geometric priors are compressed into lossy language, and sparse interaction is often supervised through delayed textual feedback rather than dense visually grounded signals. We argue that reusable experience for VLM agents should remain visually grounded. Based on this insight, we propose AtlasVA, a teacher-free visual skill memory framework that organizes memory into three complementary layers: spatial heatmaps, visual exemplars, and symbolic text skills. AtlasVA further evolves danger and affinity atlases directly from trajectory statistics and lightweight grid heuristics, and reuses these self-evolving atlases as potential-based shaping rewards for reinforcement learning. This unifies perception, memory, and optimization without external LLM supervision. Experiments on Sokoban, FrozenLake, 3D embodied navigation, and 3D robotic manipulation benchmarks show that AtlasVA consistently outperforms text-centric memory baselines and competitive VLM agents, with especially strong gains on spatially intensive tasks. Homepage: https://wangpan-ustc.github.io/AtlasvaWeb

Depth Anything has achieved remarkable success in monocular depth estimation with strong generalization ability. However, it suffers from temporal inconsistency in videos, hindering its practical applications. Various methods have been proposed to alleviate this issue by leveraging video generation models or introducing priors from optical flow and camera poses. Nonetheless, these methods are only applicable to short videos (< 10 seconds) and require a trade-off between quality and computational efficiency. We propose Video Depth Anything for high-quality, consistent depth estimation in super-long videos (over several minutes) without sacrificing efficiency. We base our model on Depth Anything V2 and replace its head with an efficient spatial-temporal head. We design a straightforward yet effective temporal consistency loss by constraining the temporal depth gradient, eliminating the need for additional geometric priors. The model is trained on a joint dataset of video depth and unlabeled images, similar to Depth Anything V2. Moreover, a novel key-frame-based strategy is developed for long video inference. Experiments show that our model can be applied to arbitrarily long videos without compromising quality, consistency, or generalization ability. Comprehensive evaluations on multiple video benchmarks demonstrate that our approach sets a new state-of-the-art in zero-shot video depth estimation. We offer models of different scales to support a range of scenarios, with our smallest model capable of real-time performance at 30 FPS.

Real-time monocular 3D object detection remains challenging due to severe depth ambiguity, viewpoint shifts, and the high computational cost of 3D reasoning. Existing approaches either rely on LiDAR or geometric priors to compensate for missing depth, or sacrifice efficiency to achieve competitive accuracy. We introduce LeAD-M3D, a monocular 3D detector that achieves state-of-the-art accuracy and real-time inference without extra modalities. Our method is powered by three key components. Asymmetric Augmentation Denoising Distillation (A2D2) transfers geometric knowledge from a clean-image teacher to a mixup-noised student via a quality- and importance-weighted depth-feature loss, enabling stronger depth reasoning without LiDAR supervision. 3D-aware Consistent Matching (CM3D) improves prediction-to-ground truth assignment by integrating 3D MGIoU into the matching score, yielding more stable and precise supervision. Finally, Confidence-Gated 3D Inference (CGI3D) accelerates detection by restricting expensive 3D regression to top-confidence regions. Together, these components set a new Pareto frontier for monocular 3D detection: LeAD-M3D achieves state-of-the-art accuracy on KITTI and Waymo, and the best reported car AP on Rope3D, while running up to 3.6x faster than prior high-accuracy methods. Our results demonstrate that high fidelity and real-time efficiency in monocular 3D detection are simultaneously attainable - without LiDAR, stereo, or geometric assumptions.

We present MatDecompSDF, a novel framework for recovering high-fidelity 3D shapes and decomposing their physically-based material properties from multi-view images. The core challenge of inverse rendering lies in the ill-posed disentanglement of geometry, materials, and illumination from 2D observations. Our method addresses this by jointly optimizing three neural components: a neural Signed Distance Function (SDF) to represent complex geometry, a spatially-varying neural field for predicting PBR material parameters (albedo, roughness, metallic), and an MLP-based model for capturing unknown environmental lighting. The key to our approach is a physically-based differentiable rendering layer that connects these 3D properties to the input images, allowing for end-to-end optimization. We introduce a set of carefully designed physical priors and geometric regularizations, including a material smoothness loss and an Eikonal loss, to effectively constrain the problem and achieve robust decomposition. Extensive experiments on both synthetic and real-world datasets (e.g., DTU) demonstrate that MatDecompSDF surpasses state-of-the-art methods in geometric accuracy, material fidelity, and novel view synthesis. Crucially, our method produces editable and relightable assets that can be seamlessly integrated into standard graphics pipelines, validating its practical utility for digital content creation.

Recent advancements in implicit neural representations have contributed to high-fidelity surface reconstruction and photorealistic novel view synthesis. However, the computational complexity inherent in these methodologies presents a substantial impediment, constraining the attainable frame rates and resolutions in practical applications. In response to this predicament, we propose VQ-NeRF, an effective and efficient pipeline for enhancing implicit neural representations via vector quantization. The essence of our method involves reducing the sampling space of NeRF to a lower resolution and subsequently reinstating it to the original size utilizing a pre-trained VAE decoder, thereby effectively mitigating the sampling time bottleneck encountered during rendering. Although the codebook furnishes representative features, reconstructing fine texture details of the scene remains challenging due to high compression rates. To overcome this constraint, we design an innovative multi-scale NeRF sampling scheme that concurrently optimizes the NeRF model at both compressed and original scales to enhance the network's ability to preserve fine details. Furthermore, we incorporate a semantic loss function to improve the geometric fidelity and semantic coherence of our 3D reconstructions. Extensive experiments demonstrate the effectiveness of our model in achieving the optimal trade-off between rendering quality and efficiency. Evaluation on the DTU, BlendMVS, and H3DS datasets confirms the superior performance of our approach.

LiDAR relocalization has attracted increasing attention as it can deliver accurate 6-DoF pose estimation in complex 3D environments. Recent learning-based regression methods offer efficient solutions by directly predicting global poses without the need for explicit map storage. However, these methods often struggle in challenging scenes due to their equal treatment of all predicted points, which is vulnerable to noise and outliers. In this paper, we propose LEADER, a robust LiDAR-based relocalization framework enhanced by a simple, yet effective geometric encoder. Specifically, a Robust Projection-based Geometric Encoder architecture which captures multi-scale geometric features is first presented to enhance descriptiveness in geometric representation. A Truncated Relative Reliability loss is then formulated to model point-wise ambiguity and mitigate the influence of unreliable predictions. Extensive experiments on the Oxford RobotCar and NCLT datasets demonstrate that LEADER outperforms state-of-the-art methods, achieving 24.1% and 73.9% relative reductions in position error over existing techniques, respectively. The source code is released on https://github.com/JiansW/LEADER.

Industrial quality inspection plays a critical role in modern manufacturing by identifying defective products during production. While single-modality approaches using either 3D point clouds or 2D RGB images suffer from information incompleteness, multimodal anomaly detection offers promise through the complementary fusion of crossmodal data. However, existing methods face challenges in effectively integrating unimodal results and improving discriminative power. To address these limitations, we first reinterpret memory bank-based anomaly scores in single modalities as isotropic Euclidean distances in local feature spaces. Dynamically evolving from Euclidean metrics, we propose a novel Geometry-Guided Score Fusion (G^{2}SF) framework that progressively learns an anisotropic local distance metric as a unified score for the fusion task. Through a geometric encoding operator, a novel Local Scale Prediction Network (LSPN) is proposed to predict direction-aware scaling factors that characterize first-order local feature distributions, thereby enhancing discrimination between normal and anomalous patterns. Additionally, we develop specialized loss functions and score aggregation strategy from geometric priors to ensure both metric generalization and efficacy. Comprehensive evaluations on the MVTec-3D AD and Eyecandies datasets demonstrate the state-of-the-art detection performance of our method, and detailed ablation analysis validates each component's contribution. Our code is available at https://github.com/ctaoaa/G2SF.

We present TreeON, a novel neural-based framework for reconstructing detailed 3D tree point clouds from sparse top-down geodata, using only a single orthophoto and its corresponding Digital Surface Model (DSM). Our method introduces a new training supervision strategy that combines both geometric supervision and differentiable shadow and silhouette losses to learn point cloud representations of trees without requiring species labels, procedural rules, terrestrial reconstruction data, or ground laser scans. To address the lack of ground truth data, we generate a synthetic dataset of point clouds from procedurally modeled trees and train our network on it. Quantitative and qualitative experiments demonstrate better reconstruction quality and coverage compared to existing methods, as well as strong generalization to real-world data, producing visually appealing and structurally plausible tree point cloud representations suitable for integration into interactive digital 3D maps. The codebase, synthetic dataset, and pretrained model are publicly available at https://angelikigram.github.io/treeON/.

We propose Hier-SLAM++, a comprehensive Neuro-Symbolic semantic 3D Gaussian Splatting SLAM method with both RGB-D and monocular input featuring an advanced hierarchical categorical representation, which enables accurate pose estimation as well as global 3D semantic mapping. The parameter usage in semantic SLAM systems increases significantly with the growing complexity of the environment, making scene understanding particularly challenging and costly. To address this problem, we introduce a novel and general hierarchical representation that encodes both semantic and geometric information in a compact form into 3D Gaussian Splatting, leveraging the capabilities of large language models (LLMs) as well as the 3D generative model. By utilizing the proposed hierarchical tree structure, semantic information is symbolically represented and learned in an end-to-end manner. We further introduce a novel semantic loss designed to optimize hierarchical semantic information through both inter-level and cross-level optimization. Additionally, we propose an improved SLAM system to support both RGB-D and monocular inputs using a feed-forward model. To the best of our knowledge, this is the first semantic monocular Gaussian Splatting SLAM system, significantly reducing sensor requirements for 3D semantic understanding and broadening the applicability of semantic Gaussian SLAM system. We conduct experiments on both synthetic and real-world datasets, demonstrating superior or on-par performance with state-of-the-art NeRF-based and Gaussian-based SLAM systems, while significantly reducing storage and training time requirements.

Robust 6DoF pose estimation with mobile devices is the foundation for applications in robotics, augmented reality, and digital twin localization. In this paper, we extensively investigate the robustness of existing RGBD-based 6DoF pose estimation methods against varying levels of depth sensor noise. We highlight that existing 6DoF pose estimation methods suffer significant performance discrepancies due to depth measurement inaccuracies. In response to the robustness issue, we present a simple and effective transformer-based 6DoF pose estimation approach called DTTDNet, featuring a novel geometric feature filtering module and a Chamfer distance loss for training. Moreover, we advance the field of robust 6DoF pose estimation and introduce a new dataset -- Digital Twin Tracking Dataset Mobile (DTTD-Mobile), tailored for digital twin object tracking with noisy depth data from the mobile RGBD sensor suite of the Apple iPhone 14 Pro. Extensive experiments demonstrate that DTTDNet significantly outperforms state-of-the-art methods at least 4.32, up to 60.74 points in ADD metrics on the DTTD-Mobile. More importantly, our approach exhibits superior robustness to varying levels of measurement noise, setting a new benchmark for robustness to measurement noise. The project page is publicly available at https://openark-berkeley.github.io/DTTDNet/.

The relentless scaling of deep learning models has led to unsustainable computational demands, positioning Mixture-of-Experts (MoE) architectures as a promising path towards greater efficiency. However, MoE models are plagued by two fundamental challenges: 1) a load imbalance problem known as the``rich get richer" phenomenon, where a few experts are over-utilized, and 2) an expert homogeneity problem, where experts learn redundant representations, negating their purpose. Current solutions typically employ an auxiliary load-balancing loss that, while mitigating imbalance, often exacerbates homogeneity by enforcing uniform routing at the expense of specialization. To resolve this, we introduce the Eigen-Mixture-of-Experts (EMoE), a novel architecture that leverages a routing mechanism based on a learned orthonormal eigenbasis. EMoE projects input tokens onto this shared eigenbasis and routes them based on their alignment with the principal components of the feature space. This principled, geometric partitioning of data intrinsically promotes both balanced expert utilization and the development of diverse, specialized experts, all without the need for a conflicting auxiliary loss function. Our code is publicly available at https://github.com/Belis0811/EMoE.

3D facial reconstruction from a single in-the-wild image is a crucial task in human-centered computer vision tasks. While existing methods can recover accurate facial shapes, there remains significant space for improvement in fine-grained expression capture. Current approaches struggle with irregular mouth shapes, exaggerated expressions, and asymmetrical facial movements. We present TEASER (Token EnhAnced Spatial modeling for Expressions Reconstruction), which addresses these challenges and enhances 3D facial geometry performance. TEASER tackles two main limitations of existing methods: insufficient photometric loss for self-reconstruction and inaccurate localization of subtle expressions. We introduce a multi-scale tokenizer to extract facial appearance information. Combined with a neural renderer, these tokens provide precise geometric guidance for expression reconstruction. Furthermore, TEASER incorporates a pose-dependent landmark loss to further improve geometric performances. Our approach not only significantly enhances expression reconstruction quality but also offers interpretable tokens suitable for various downstream applications, such as photorealistic facial video driving, expression transfer, and identity swapping. Quantitative and qualitative experimental results across multiple datasets demonstrate that TEASER achieves state-of-the-art performance in precise expression reconstruction.

Perspective projection has been extensively utilized in monocular 3D object detection methods. It introduces geometric priors from 2D bounding boxes and 3D object dimensions to reduce the uncertainty of depth estimation. However, due to depth errors originating from the object's visual surface, the height of the bounding box often fails to represent the actual projected central height, which undermines the effectiveness of geometric depth. Direct prediction for the projected height unavoidably results in a loss of 2D priors, while multi-depth prediction with complex branches does not fully leverage geometric depth. This paper presents a Transformer-based monocular 3D object detection method called MonoDGP, which adopts perspective-invariant geometry errors to modify the projection formula. We also try to systematically discuss and explain the mechanisms and efficacy behind geometry errors, which serve as a simple but effective alternative to multi-depth prediction. Additionally, MonoDGP decouples the depth-guided decoder and constructs a 2D decoder only dependent on visual features, providing 2D priors and initializing object queries without the disturbance of 3D detection. To further optimize and fine-tune input tokens of the transformer decoder, we also introduce a Region Segment Head (RSH) that generates enhanced features and segment embeddings. Our monocular method demonstrates state-of-the-art performance on the KITTI benchmark without extra data. Code is available at https://github.com/PuFanqi23/MonoDGP.

We present a simple, yet effective, approach for self-supervised 3D human pose estimation. Unlike the prior work, we explore the temporal information next to the multi-view self-supervision. During training, we rely on triangulating 2D body pose estimates of a multiple-view camera system. A temporal convolutional neural network is trained with the generated 3D ground-truth and the geometric multi-view consistency loss, imposing geometrical constraints on the predicted 3D body skeleton. During inference, our model receives a sequence of 2D body pose estimates from a single-view to predict the 3D body pose for each of them. An extensive evaluation shows that our method achieves state-of-the-art performance in the Human3.6M and MPI-INF-3DHP benchmarks. Our code and models are publicly available at https://github.com/vru2020/TM_HPE/.

Vision Language Models (VLMs) exhibit a fundamental semantic-to-geometric gap in spatial reasoning: they excel at qualitative semantic inference but their reasoning operates within a lossy semantic space, misaligned with high-fidelity geometry. Current paradigms fail to bridge this gap. Training-based methods suffer from an ``oracle paradox,'' learning flawed spatial logic from imperfect oracles. Tool-integrated methods constrain the final computation but critically leave the VLM's planning process unconstrained, resulting in geometrically flawed plans. In this work, we propose Geometrically-Constrained Agent (GCA), a training-free agentic paradigm that resolves this gap by introducing a formal task constraint. Specifically, we strategically decouples the VLM's role into two stages. First, acting as a semantic analyst, the VLM translates the user's ambiguous query into the formal, verifiable task constraint, which defines the reference frame and objective. Second, acting as a task solver, the VLM generates and executes tool calls strictly within the deterministic bounds defined by the constraint. This geometrically-constrained reasoning strategy successfully resolve the semantic-to-geometric gap, yielding a robust and verifiable reasoning pathway for spatial reasoning. Comprehensive experiments demonstrate that GCA achieves SOTA performance on multiple spatial reasoning benchmarks, surpassing existing training-based and tool-integrated methods by ~27%. Please see our homepage at https://gca-spatial-reasoning.github.io.

3D reconstruction and generation are commonly tackled by separate paradigms: pixel-based regression for reconstruction, and latent diffusion for generation. Recent works attempt to unify them in latent space, but with notable drawbacks: the diffusion objective is defined on latent features rather than the underlying 3D representation, and both branches suffer from information loss introduced by latent encoding, while requiring a pretrained Variational Autoencoder (VAE) or Representation Autoencoder (RAE). In this paper, we reformulate these two tasks under a unified pixel-space diffusion paradigm and introduce PixWorld, a single model that jointly addresses 3D reconstruction and generation. By supervising diffusion directly on rendered images, PixWorld removes the above limitations and aligns optimization with 3D scene fidelity. Beyond photometric and perceptual supervision that operates at the 2D image level and lacks 3D geometric awareness, we further introduce a geometry perception loss that aligns rendered views with their ground truth in the geometry-aware feature space of a pretrained 3D foundation model, providing 3D structural supervision. PixWorld consistently outperforms prior latent-space generation methods and matches state-of-the-art reconstruction methods, demonstrating the superiority of a unified pixel-space approach.

Document image rectification aims to eliminate geometric deformation in photographed documents to facilitate text recognition. However, existing methods often neglect the significance of foreground elements, which provide essential geometric references and layout information for document image correction. In this paper, we introduce Foreground-Centric Network (ForCenNet) to eliminate geometric distortions in document images. Specifically, we initially propose a foreground-centric label generation method, which extracts detailed foreground elements from an undistorted image. Then we introduce a foreground-centric mask mechanism to enhance the distinction between readable and background regions. Furthermore, we design a curvature consistency loss to leverage the detailed foreground labels to help the model understand the distorted geometric distribution. Extensive experiments demonstrate that ForCenNet achieves new state-of-the-art on four real-world benchmarks, such as DocUNet, DIR300, WarpDoc, and DocReal. Quantitative analysis shows that the proposed method effectively undistorts layout elements, such as text lines and table borders. The resources for further comparison are provided at https://github.com/caipeng328/ForCenNet.

Developing a unified pipeline that enables users to remove, re-texture, or replace objects in a versatile manner is crucial for text-guided 3D inpainting. However, there are still challenges in performing multiple 3D inpainting tasks within a unified framework: 1) Single reference inpainting methods lack robustness when dealing with views that are far from the reference view. 2) Appearance inconsistency arises when independently inpainting multi-view images with 2D diffusion priors; 3) Geometry inconsistency limits performance when there are significant geometric changes in the inpainting regions. To tackle these challenges, we introduce DiGA3D, a novel and versatile 3D inpainting pipeline that leverages diffusion models to propagate consistent appearance and geometry in a coarse-to-fine manner. First, DiGA3D develops a robust strategy for selecting multiple reference views to reduce errors during propagation. Next, DiGA3D designs an Attention Feature Propagation (AFP) mechanism that propagates attention features from the selected reference views to other views via diffusion models to maintain appearance consistency. Furthermore, DiGA3D introduces a Texture-Geometry Score Distillation Sampling (TG-SDS) loss to further improve the geometric consistency of inpainted 3D scenes. Extensive experiments on multiple 3D inpainting tasks demonstrate the effectiveness of our method. The project page is available at https://rorisis.github.io/DiGA3D/.

The field of neuroscience and the development of artificial neural networks (ANNs) have mutually influenced each other, drawing from and contributing to many concepts initially developed in statistical mechanics. Notably, Hopfield networks and Boltzmann machines are versions of the Ising model, a model extensively studied in statistical mechanics for over a century. In the first part of this chapter, we provide an overview of the principles, models, and applications of ANNs, highlighting their connections to statistical mechanics and statistical learning theory. Artificial neural networks can be seen as high-dimensional mathematical functions, and understanding the geometric properties of their loss landscapes (i.e., the high-dimensional space on which one wishes to find extrema or saddles) can provide valuable insights into their optimization behavior, generalization abilities, and overall performance. Visualizing these functions can help us design better optimization methods and improve their generalization abilities. Thus, the second part of this chapter focuses on quantifying geometric properties and visualizing loss functions associated with deep ANNs.

Existing research has shown that large language models (LLMs) exhibit remarkable performance in language understanding and generation. However, when LLMs are continuously fine-tuned on complex and diverse domain-specific downstream tasks, the inference performance on historical tasks decreases dramatically, which is known as a catastrophic forgetting problem. A trade-off needs to be kept between learning plasticity and memory stability. Plenty of existing works have explored strategies like memory replay, regularization and parameter isolation, but little is known about the geometric connection of various adjacent minima in the continual LLMs fine-tuning scenarios. In this work, we investigate the geometric connections of different minima through the lens of mode connectivity, which means different minima can be connected by a low-loss valley. Through extensive experiments, we uncover the mode connectivity phenomenon in the LLMs continual learning scenario and find that it can strike a balance between plasticity and stability. Building upon these findings, we propose a simple yet effective method called Interpolation-based LoRA (I-LoRA), which constructs a dual-memory experience replay framework based on LoRA parameter interpolations. Extensive experiments and analysis on eight domain-specific CL benchmarks demonstrate that I-LoRA consistently show significant improvement over the previous state-of-the-art approaches with up to 11% performance gains, providing a strong baseline and insights for future research on the large language model continual learning problem. Our code is available at https://github.com/which47/LLMCL.

Modern deep neural networks have achieved impressive performance on tasks from image classification to natural language processing. Surprisingly, these complex systems with massive amounts of parameters exhibit the same structural properties in their last-layer features and classifiers across canonical datasets when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means, and those class-means are the vertices of a simplex Equiangular Tight Frame (ETF). This phenomenon is known as Neural Collapse (NC). Recent papers have theoretically shown that NC emerges in the global minimizers of training problems with the simplified "unconstrained feature model". In this context, we take a step further and prove the NC occurrences in deep linear networks for the popular mean squared error (MSE) and cross entropy (CE) losses, showing that global solutions exhibit NC properties across the linear layers. Furthermore, we extend our study to imbalanced data for MSE loss and present the first geometric analysis of NC under bias-free setting. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of orthogonal vectors, whose lengths depend on the amount of data in their corresponding classes. Finally, we empirically validate our theoretical analyses on synthetic and practical network architectures with both balanced and imbalanced scenarios.

Dynamic radiance fields have emerged as a promising approach for generating novel views from a monocular video. However, previous methods enforce the geometric consistency to dynamic radiance fields only between adjacent input frames, making it difficult to represent the global scene geometry and degenerates at the viewpoint that is spatio-temporally distant from the input camera trajectory. To solve this problem, we introduce point-based dynamic radiance fields (Point-DynRF), a novel framework where the global geometric information and the volume rendering process are trained by neural point clouds and dynamic radiance fields, respectively. Specifically, we reconstruct neural point clouds directly from geometric proxies and optimize both radiance fields and the geometric proxies using our proposed losses, allowing them to complement each other. We validate the effectiveness of our method with experiments on the NVIDIA Dynamic Scenes Dataset and several causally captured monocular video clips.

We extend neural radiance fields (NeRFs) to dynamic large-scale urban scenes. Prior work tends to reconstruct single video clips of short durations (up to 10 seconds). Two reasons are that such methods (a) tend to scale linearly with the number of moving objects and input videos because a separate model is built for each and (b) tend to require supervision via 3D bounding boxes and panoptic labels, obtained manually or via category-specific models. As a step towards truly open-world reconstructions of dynamic cities, we introduce two key innovations: (a) we factorize the scene into three separate hash table data structures to efficiently encode static, dynamic, and far-field radiance fields, and (b) we make use of unlabeled target signals consisting of RGB images, sparse LiDAR, off-the-shelf self-supervised 2D descriptors, and most importantly, 2D optical flow. Operationalizing such inputs via photometric, geometric, and feature-metric reconstruction losses enables SUDS to decompose dynamic scenes into the static background, individual objects, and their motions. When combined with our multi-branch table representation, such reconstructions can be scaled to tens of thousands of objects across 1.2 million frames from 1700 videos spanning geospatial footprints of hundreds of kilometers, (to our knowledge) the largest dynamic NeRF built to date. We present qualitative initial results on a variety of tasks enabled by our representations, including novel-view synthesis of dynamic urban scenes, unsupervised 3D instance segmentation, and unsupervised 3D cuboid detection. To compare to prior work, we also evaluate on KITTI and Virtual KITTI 2, surpassing state-of-the-art methods that rely on ground truth 3D bounding box annotations while being 10x quicker to train.

Diffusion models have been applied to 3D LiDAR scene completion due to their strong training stability and high completion quality. However, the slow sampling speed limits the practical application of diffusion-based scene completion models since autonomous vehicles require an efficient perception of surrounding environments. This paper proposes a novel distillation method tailored for 3D LiDAR scene completion models, dubbed ScoreLiDAR, which achieves efficient yet high-quality scene completion. ScoreLiDAR enables the distilled model to sample in significantly fewer steps after distillation. To improve completion quality, we also introduce a novel Structural Loss, which encourages the distilled model to capture the geometric structure of the 3D LiDAR scene. The loss contains a scene-wise term constraining the holistic structure and a point-wise term constraining the key landmark points and their relative configuration. Extensive experiments demonstrate that ScoreLiDAR significantly accelerates the completion time from 30.55 to 5.37 seconds per frame (>5times) on SemanticKITTI and achieves superior performance compared to state-of-the-art 3D LiDAR scene completion models. Our code is publicly available at https://github.com/happyw1nd/ScoreLiDAR.

We introduce SPFSplatV2, an efficient feed-forward framework for 3D Gaussian splatting from sparse multi-view images, requiring no ground-truth poses during training and inference. It employs a shared feature extraction backbone, enabling simultaneous prediction of 3D Gaussian primitives and camera poses in a canonical space from unposed inputs. A masked attention mechanism is introduced to efficiently estimate target poses during training, while a reprojection loss enforces pixel-aligned Gaussian primitives, providing stronger geometric constraints. We further demonstrate the compatibility of our training framework with different reconstruction architectures, resulting in two model variants. Remarkably, despite the absence of pose supervision, our method achieves state-of-the-art performance in both in-domain and out-of-domain novel view synthesis, even under extreme viewpoint changes and limited image overlap, and surpasses recent methods that rely on geometric supervision for relative pose estimation. By eliminating dependence on ground-truth poses, our method offers the scalability to leverage larger and more diverse datasets. Code and pretrained models will be available on our project page: https://ranrhuang.github.io/spfsplatv2/.

Distributed Denial of Service (DDoS) attacks pose an increasingly substantial cybersecurity threat to organizations across the globe. In this paper, we introduce a new deep learning-based technique for detecting DDoS attacks, a paramount cybersecurity challenge with evolving complexity and scale. Specifically, we propose a new dual-space prototypical network that leverages a unique dual-space loss function to enhance detection accuracy for various attack patterns through geometric and angular similarity measures. This approach capitalizes on the strengths of representation learning within the latent space (a lower-dimensional representation of data that captures complex patterns for machine learning analysis), improving the model's adaptability and sensitivity towards varying DDoS attack vectors. Our comprehensive evaluation spans multiple training environments, including offline training, simulated online training, and prototypical network scenarios, to validate the model's robustness under diverse data abundance and scarcity conditions. The Multilayer Perceptron (MLP) with Attention, trained with our dual-space prototypical design over a reduced training set, achieves an average accuracy of 94.85% and an F1-Score of 94.71% across our tests, showcasing its effectiveness in dynamic and constrained real-world scenarios.

We study the momentum-based minimization of a diffuse perimeter functional on Euclidean spaces and on graphs with applications to semi-supervised classification tasks in machine learning. While the gradient flow in the task at hand is a parabolic partial differential equation, the momentum-method corresponds to a damped hyperbolic PDE, leading to qualitatively and quantitatively different trajectories. Using a convex-concave splitting-based FISTA-type time discretization, we demonstrate empirically that momentum can lead to faster convergence if the time step size is large but not too large. With large time steps, the PDE analysis offers only limited insight into the geometric behavior of solutions and typical hyperbolic phenomena like loss of regularity are not be observed in sample simulations.

The presence of linear paths in parameter space between two different network solutions in certain cases, i.e., linear mode connectivity (LMC), has garnered interest from both theoretical and practical fronts. There has been significant research that either practically designs algorithms catered for connecting networks by adjusting for the permutation symmetries as well as some others that more theoretically construct paths through which networks can be connected. Yet, the core reasons for the occurrence of LMC, when in fact it does occur, in the highly non-convex loss landscapes of neural networks are far from clear. In this work, we take a step towards understanding it by providing a model of how the loss landscape needs to behave topographically for LMC (or the lack thereof) to manifest. Concretely, we present a `mountainside and ridge' perspective that helps to neatly tie together different geometric features that can be spotted in the loss landscape along the training runs. We also complement this perspective by providing a theoretical analysis of the barrier height, for which we provide empirical support, and which additionally extends as a faithful predictor of layer-wise LMC. We close with a toy example that provides further intuition on how barriers arise in the first place, all in all, showcasing the larger aim of the work -- to provide a working model of the landscape and its topography for the occurrence of LMC.

Previous methods solve feature matching and pose estimation using a two-stage process by first finding matches and then estimating the pose. As they ignore the geometric relationships between the two tasks, they focus on either improving the quality of matches or filtering potential outliers, leading to limited efficiency or accuracy. In contrast, we propose an iterative matching and pose estimation framework (IMP) leveraging the geometric connections between the two tasks: a few good matches are enough for a roughly accurate pose estimation; a roughly accurate pose can be used to guide the matching by providing geometric constraints. To this end, we implement a geometry-aware recurrent attention-based module which jointly outputs sparse matches and camera poses. Specifically, for each iteration, we first implicitly embed geometric information into the module via a pose-consistency loss, allowing it to predict geometry-aware matches progressively. Second, we introduce an efficient IMP, called EIMP, to dynamically discard keypoints without potential matches, avoiding redundant updating and significantly reducing the quadratic time complexity of attention computation in transformers. Experiments on YFCC100m, Scannet, and Aachen Day-Night datasets demonstrate that the proposed method outperforms previous approaches in terms of accuracy and efficiency.