Daily Papers

We study quantum state tomography when the unknown state lies in a trace-distance ball of radius R around a known reference state ρ. For arbitrary collective measurements, we determine the minimax expected risk and confidence radius up to universal constants, uniformly over the dimension, center spectrum, radius, and sample size. Write Γ=d(Trsqrtρ)^2+d^2R. For dge2, 0<Rle1, and nge1 copies, the expected risk is Θ(min{R,Γ/n}). At failure probability 0<δle1/3, the minimax confidence radius is \[ Θ\!\left(\min\left\{R,\frac{Γ+\log(1/δ){n}}\right\}\right). \] For varepsilonle c_0R with a universal c_0>0, the copy complexity is Θ((Γ+log(1/δ))/varepsilon^2). For the lower bound we use a weighted trace-norm witness to retain all spectral scales in a square-root perturbation family. For the upper bound, we give a direct spectral and concentration analysis of a modified existing purification-based procedure.

Invasive species in water bodies pose a major threat to the environment and biodiversity globally. Due to increased transportation and trade, non-native species have been introduced to new environments, causing damage to ecosystems and leading to economic losses in agriculture, forestry, and fisheries. Therefore, there is a pressing need for risk assessment and management techniques to mitigate the impact of these invasions. This study aims to develop a new physics-inspired model to forecast maritime shipping traffic and thus inform risk assessment of invasive species spread through global transportation networks. Inspired by the gravity model for international trades, our model considers various factors that influence the likelihood and impact of vessel activities, such as shipping flux density, distance between ports, trade flow, and centrality measures of transportation hubs. Additionally, by analyzing the risk network of invasive species, we provide a comprehensive framework for assessing the invasion threat level given a pair of origin and destination. Accordingly, this paper introduces transformers to gravity models to rebuild the short- and long-term dependencies that make the risk analysis feasible. Thus, we introduce a physics-inspired framework that achieves an 89% segmentation accuracy for existing and non-existing trajectories and an 84.8% accuracy for the number of vessels flowing between key port areas, representing more than 10% improvement over the traditional deep-gravity model. Along these lines, this research contributes to a better understanding of invasive species risk assessment. It allows policymakers, conservationists, and stakeholders to prioritize management actions by identifying high-risk invasion pathways. Besides, our model is versatile and can include new data sources, making it suitable for assessing species invasion risks in a changing global landscape.

Ball trajectory data are one of the most fundamental and useful information in the evaluation of players' performance and analysis of game strategies. Although vision-based object tracking techniques have been developed to analyze sport competition videos, it is still challenging to recognize and position a high-speed and tiny ball accurately. In this paper, we develop a deep learning network, called TrackNet, to track the tennis ball from broadcast videos in which the ball images are small, blurry, and sometimes with afterimage tracks or even invisible. The proposed heatmap-based deep learning network is trained to not only recognize the ball image from a single frame but also learn flying patterns from consecutive frames. TrackNet takes images with a size of 640times360 to generate a detection heatmap from either a single frame or several consecutive frames to position the ball and can achieve high precision even on public domain videos. The network is evaluated on the video of the men's singles final at the 2017 Summer Universiade, which is available on YouTube. The precision, recall, and F1-measure of TrackNet reach 99.7%, 97.3%, and 98.5%, respectively. To prevent overfitting, 9 additional videos are partially labeled together with a subset from the previous dataset to implement 10-fold cross-validation, and the precision, recall, and F1-measure are 95.3%, 75.7%, and 84.3%, respectively. A conventional image processing algorithm is also implemented to compare with TrackNet. Our experiments indicate that TrackNet outperforms conventional method by a big margin and achieves exceptional ball tracking performance. The dataset and demo video are available at https://nol.cs.nctu.edu.tw/ndo3je6av9/.

The shadow of an abstract simplicial complex K with vertices in R^N is a subset of R^N defined as the union of the convex hulls of simplices of K. The Vietoris--Rips complex of a metric space (S,d) at scale β is an abstract simplicial complex whose each k-simplex corresponds to (k+1) points of S within diameter β. In case Ssubsetmathbb R^2 and d(a,b)=|a-b| the standard Euclidean metric, the natural shadow projection of the Vietoris--Rips complex is already proved by Chambers et al. to induce isomorphisms on π_0 and π_1. We extend the result beyond the standard Euclidean distance on Ssubsetmathbb R^N to a family of path-based metrics, d^varepsilon_{S}. From the pairwise Euclidean distances of points in S, we introduce a family (parametrized by varepsilon) of path-based Vietoris--Rips complexes R^varepsilon_β(S) for a scale β>0. If SsubsetR^2 is Hausdorff-close to a planar Euclidean graph G, we provide quantitative bounds on scales β,varepsilon for the shadow projection map of the Vietoris--Rips complex of (S,d^varepsilon_S) at scale β to induce π_1-isomorphism. This paper first studies the homotopy-type recovery of Gsubsetmathbb R^N using the abstract Vietoris--Rips complex of a Hausdorff-close sample S under the d^varepsilon_S metric. Then, our result on the π_1-isomorphism induced by the shadow projection lends itself to providing also a geometrically close embedding for the reconstruction. Based on the length of the shortest loop and large-scale distortion of the embedding of G, we quantify the choice of a suitable sample density varepsilon and a scale β at which the shadow of R^varepsilon_β(S) is homotopy-equivalent and Hausdorff-close to G.

We study a class of optimization problems motivated by automating the design and update of AI systems like coding assistants, robots, and copilots. We propose an end-to-end optimization framework, Trace, which treats the computational workflow of an AI system as a graph akin to neural networks, based on a generalization of back-propagation. Optimization of computational workflows often involves rich feedback (e.g. console output or user's responses), heterogeneous parameters (e.g. prompts, hyper-parameters, codes), and intricate objectives (beyond maximizing a score). Moreover, its computation graph can change dynamically with the inputs and parameters. We frame a new mathematical setup of iterative optimization, Optimization with Trace Oracle (OPTO), to capture and abstract these properties so as to design optimizers that work across many domains. In OPTO, an optimizer receives an execution trace along with feedback on the computed output and updates parameters iteratively. Trace is the tool to implement OPTO in practice. Trace has a Python interface that efficiently converts a computational workflow into an OPTO instance using a PyTorch-like interface. Using Trace, we develop a general-purpose LLM-based optimizer called OptoPrime that can effectively solve OPTO problems. In empirical studies, we find that OptoPrime is capable of first-order numerical optimization, prompt optimization, hyper-parameter tuning, robot controller design, code debugging, etc., and is often competitive with specialized optimizers for each domain. We believe that Trace, OptoPrime and the OPTO framework will enable the next generation of interactive agents that automatically adapt using various kinds of feedback. Website: https://microsoft.github.io/Trace

Accurately detecting and tracking high-speed, small objects, such as balls in sports videos, is challenging due to factors like motion blur and occlusion. Although recent deep learning frameworks like TrackNetV1, V2, and V3 have advanced tennis ball and shuttlecock tracking, they often struggle in scenarios with partial occlusion or low visibility. This is primarily because these models rely heavily on visual features without explicitly incorporating motion information, which is crucial for precise tracking and trajectory prediction. In this paper, we introduce an enhancement to the TrackNet family by fusing high-level visual features with learnable motion attention maps through a motion-aware fusion mechanism, effectively emphasizing the moving ball's location and improving tracking performance. Our approach leverages frame differencing maps, modulated by a motion prompt layer, to highlight key motion regions over time. Experimental results on the tennis ball and shuttlecock datasets show that our method enhances the tracking performance of both TrackNetV2 and V3. We refer to our lightweight, plug-and-play solution, built on top of the existing TrackNet, as TrackNetV4.

Effective spatio-temporal representation is fundamental to modeling, understanding, and predicting dynamics in videos. The atomic unit of a video, the pixel, traces a continuous 3D trajectory over time, serving as the primitive element of dynamics. Based on this principle, we propose representing any video as a Trajectory Field: a dense mapping that assigns a continuous 3D trajectory function of time to each pixel in every frame. With this representation, we introduce Trace Anything, a neural network that predicts the entire trajectory field in a single feed-forward pass. Specifically, for each pixel in each frame, our model predicts a set of control points that parameterizes a trajectory (i.e., a B-spline), yielding its 3D position at arbitrary query time instants. We trained the Trace Anything model on large-scale 4D data, including data from our new platform, and our experiments demonstrate that: (i) Trace Anything achieves state-of-the-art performance on our new benchmark for trajectory field estimation and performs competitively on established point-tracking benchmarks; (ii) it offers significant efficiency gains thanks to its one-pass paradigm, without requiring iterative optimization or auxiliary estimators; and (iii) it exhibits emergent abilities, including goal-conditioned manipulation, motion forecasting, and spatio-temporal fusion. Project page: https://trace-anything.github.io/.

We present a method for 3D ball trajectory estimation from a 2D tracking sequence. To overcome the ambiguity in 3D from 2D estimation, we design an LSTM-based pipeline that utilizes a novel canonical 3D representation that is independent of the camera's location to handle arbitrary views and a series of intermediate representations that encourage crucial invariance and reprojection consistency. We evaluated our method on four synthetic and three real datasets and conducted extensive ablation studies on our design choices. Despite training solely on simulated data, our method achieves state-of-the-art performance and can generalize to real-world scenarios with multiple trajectories, opening up a range of applications in sport analysis and virtual replay. Please visit our page: https://where-is-the-ball.github.io.

Existing video frame interpolation (VFI) methods blindly predict where each object is at a specific timestep t ("time indexing"), which struggles to predict precise object movements. Given two images of a baseball, there are infinitely many possible trajectories: accelerating or decelerating, straight or curved. This often results in blurry frames as the method averages out these possibilities. Instead of forcing the network to learn this complicated time-to-location mapping implicitly together with predicting the frames, we provide the network with an explicit hint on how far the object has traveled between start and end frames, a novel approach termed "distance indexing". This method offers a clearer learning goal for models, reducing the uncertainty tied to object speeds. We further observed that, even with this extra guidance, objects can still be blurry especially when they are equally far from both input frames (i.e., halfway in-between), due to the directional ambiguity in long-range motion. To solve this, we propose an iterative reference-based estimation strategy that breaks down a long-range prediction into several short-range steps. When integrating our plug-and-play strategies into state-of-the-art learning-based models, they exhibit markedly sharper outputs and superior perceptual quality in arbitrary time interpolations, using a uniform distance indexing map in the same format as time indexing. Additionally, distance indexing can be specified pixel-wise, which enables temporal manipulation of each object independently, offering a novel tool for video editing tasks like re-timing.

In this work, we present a novel Sports Ball Detection and Tracking (SBDT) method that can be applied to various sports categories. Our approach is composed of (1) high-resolution feature extraction, (2) position-aware model training, and (3) inference considering temporal consistency, all of which are put together as a new SBDT baseline. Besides, to validate the wide-applicability of our approach, we compare our baseline with 6 state-of-the-art SBDT methods on 5 datasets from different sports categories. We achieve this by newly introducing two SBDT datasets, providing new ball annotations for two datasets, and re-implementing all the methods to ease extensive comparison. Experimental results demonstrate that our approach is substantially superior to existing methods on all the sports categories covered by the datasets. We believe our proposed method can play as a Widely Applicable Strong Baseline (WASB) of SBDT, and our datasets and codebase will promote future SBDT research. Datasets and codes are available at https://github.com/nttcom/WASB-SBDT .

Player attribution in American football remains an open problem due to the complex nature of twenty-two players interacting on the field, but the granularity of player tracking data provides ample opportunity for novel approaches. In this work, we introduce the first public framework to evaluate spatial and trajectory tracking data of players relative to a baseline distribution of "ghost" defenders. We demonstrate our framework in the context of modeling the nearest defender positioning at the moment of catch. In particular, we provide estimates of how much better or worse their observed positioning and trajectory compared to the expected play value of ghost defenders. Our framework leverages multi-dimensional tracking data features through flexible random forests for conditional density estimation in two ways: (1) to model the distribution of receiver yards gained enabling the estimation of within-play expected value, and (2) to model the 2D spatial distribution of baseline ghost defenders. We present novel metrics for measuring player and team performance based on tracking data, and discuss challenges that remain in extending our framework to other aspects of American football.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

Heavy-ball momentum with decaying learning rates is widely used with SGD for optimizing deep learning models. In contrast to its empirical popularity, the understanding of its theoretical property is still quite limited, especially under the standard anisotropic gradient noise condition for quadratic regression problems. Although it is widely conjectured that heavy-ball momentum method can provide accelerated convergence and should work well in large batch settings, there is no rigorous theoretical analysis. In this paper, we fill this theoretical gap by establishing a non-asymptotic convergence bound for stochastic heavy-ball methods with step decay scheduler on quadratic objectives, under the anisotropic gradient noise condition. As a direct implication, we show that heavy-ball momentum can provide mathcal{O}(kappa) accelerated convergence of the bias term of SGD while still achieving near-optimal convergence rate with respect to the stochastic variance term. The combined effect implies an overall convergence rate within log factors from the statistical minimax rate. This means SGD with heavy-ball momentum is useful in the large-batch settings such as distributed machine learning or federated learning, where a smaller number of iterations can significantly reduce the number of communication rounds, leading to acceleration in practice.

While theoretically appealing, the application of the Wasserstein distance to large-scale machine learning problems has been hampered by its prohibitive computational cost. The sliced Wasserstein distance and its variants improve the computational efficiency through the random projection, yet they suffer from low accuracy if the number of projections is not sufficiently large, because the majority of projections result in trivially small values. In this work, we propose a new family of distance metrics, called augmented sliced Wasserstein distances (ASWDs), constructed by first mapping samples to higher-dimensional hypersurfaces parameterized by neural networks. It is derived from a key observation that (random) linear projections of samples residing on these hypersurfaces would translate to much more flexible nonlinear projections in the original sample space, so they can capture complex structures of the data distribution. We show that the hypersurfaces can be optimized by gradient ascent efficiently. We provide the condition under which the ASWD is a valid metric and show that this can be obtained by an injective neural network architecture. Numerical results demonstrate that the ASWD significantly outperforms other Wasserstein variants for both synthetic and real-world problems.

Esports, despite its expanding interest, lacks fundamental sports analytics resources such as accessible data or proven and reproducible analytical frameworks. Even Counter-Strike: Global Offensive (CSGO), the second most popular esport, suffers from these problems. Thus, quantitative evaluation of CSGO players, a task important to teams, media, bettors and fans, is difficult. To address this, we introduce (1) a data model for CSGO with an open-source implementation; (2) a graph distance measure for defining distances in CSGO; and (3) a context-aware framework to value players' actions based on changes in their team's chances of winning. Using over 70 million in-game CSGO events, we demonstrate our framework's consistency and independence compared to existing valuation frameworks. We also provide use cases demonstrating high-impact play identification and uncertainty estimation.

Given a set of dissimilarity measurements amongst data points, determining what metric representation is most "consistent" with the input measurements or the metric that best captures the relevant geometric features of the data is a key step in many machine learning algorithms. Existing methods are restricted to specific kinds of metrics or small problem sizes because of the large number of metric constraints in such problems. In this paper, we provide an active set algorithm, Project and Forget, that uses Bregman projections, to solve metric constrained problems with many (possibly exponentially) inequality constraints. We provide a theoretical analysis of Project and Forget and prove that our algorithm converges to the global optimal solution and that the L_2 distance of the current iterate to the optimal solution decays asymptotically at an exponential rate. We demonstrate that using our method we can solve large problem instances of three types of metric constrained problems: general weight correlation clustering, metric nearness, and metric learning; in each case, out-performing the state of the art methods with respect to CPU times and problem sizes.

We study the online min-cost perfect matching with delay (MPMD) problem where m requests arrive in a metric space of n points. In MPMD, an algorithm can choose to match a request or to delay, and the objective is to minimise the sum of connection and delay costs. The connection cost of a match is the distance between the locations of two matched requests in the metric, and the increase of the delay cost is a function of the set of unmatched requests at every moment. In this paper, we study two different types of delay functions, size-based (MPMD-Size) and convex delays (MPMD-Convex). The study of MPMD-Size was initiated by Deryckere and Umboh (APPROX/RANDOM 2023) where the instantaneous delay increment is a non-negative monotone function of the number of unmatched requests. Our bounds are in terms of n, as opposed to Deryckere and Umboh's bounds that depend on m. Our results settle the deterministic competitive ratio (up to constants). At the heart of these results is a succinct encoding scheme of MPMD-Size on a given n-point metric as a metrical task system problem on a 2^{n-1}-point metric. We also consider MPMD-Convex proposed by Liu et al. (ISAAC 2018) where the delay cost incurred by each request is a uniform convex delay function of the time difference between its arrival time and the moment that it is matched by the algorithm. They focused on delay functions f that are unbounded, non-decreasing, continuous, and satisfy f(0)=f'(0)=0, and showed that the deterministic competitive ratio is Ω(n) for n-point uniform metrics. We show that, surprisingly, when f is a non-negative, monotone polynomial with f'(0)>0, there is an O(1)-competitive deterministic algorithm for uniform metrics. Our result completes our understanding of MPMD-Convex on uniform metrics for a broad class of functions.

Computer vision models have seen increased usage in sports, and reinforcement learning (RL) is famous for beating humans in strategic games such as Chess and Go. In this paper, we are interested in building upon these advances and examining the game of classic 8-ball pool. We introduce pix2pockets, a foundation for an RL-assisted pool coach. Given a single image of a pool table, we first aim to detect the table and the balls and then propose the optimal shot suggestion. For the first task, we build a dataset with 195 diverse images where we manually annotate all balls and table dots, leading to 5748 object segmentation masks. For the second task, we build a standardized RL environment that allows easy development and benchmarking of any RL algorithm. Our object detection model yields an AP50 of 91.2 while our ball location pipeline obtains an error of only 0.4 cm. Furthermore, we compare standard RL algorithms to set a baseline for the shot suggestion task and we show that all of them fail to pocket all balls without making a foul move. We also present a simple baseline that achieves a per-shot success rate of 94.7% and clears a full game in a single turn 30% of the time.

Recent deep learning-based object detection approaches have led to significant progress in multi-object tracking (MOT) algorithms. The current MOT methods mainly focus on pedestrian or vehicle scenes, but basketball sports scenes are usually accompanied by three or more object occlusion problems with similar appearances and high-intensity complex motions, which we call complex multi-object occlusion (CMOO). Here, we propose an online and robust MOT approach, named Basketball-SORT, which focuses on the CMOO problems in basketball videos. To overcome the CMOO problem, instead of using the intersection-over-union-based (IoU-based) approach, we use the trajectories of neighboring frames based on the projected positions of the players. Our method designs the basketball game restriction (BGR) and reacquiring Long-Lost IDs (RLLI) based on the characteristics of basketball scenes, and we also solve the occlusion problem based on the player trajectories and appearance features. Experimental results show that our method achieves a Higher Order Tracking Accuracy (HOTA) score of 63.48% on the basketball fixed video dataset and outperforms other recent popular approaches. Overall, our approach solved the CMOO problem more effectively than recent MOT algorithms.

Retrieving objects from dense clutter requires rearrangement during which the manipulator can occlude objects while moving them. Repeated arm withdrawals to restore visibility interrupt execution. We introduce TRACE, a plan-conditioned imitation framework for retrieval under self-occlusion. A single unoccluded observation initializes a digital twin, where a privileged teacher generates a fixed nominal rollout. A recurrent student combines local rollout context, partial object observations, and proprioception to select actions that can correct deviations from the prediction. Behavior cloning initializes the student; DAgger refines it with teacher labels on student-visited states. The rollout remains fixed throughout execution, so the deployed student needs neither online teacher queries nor additional simulator rollouts during pushing. On 511 simulation test scenes, TRACE achieves 90.7% success versus 43.4% for nominal replay and 96.7% for the privileged closed-loop teacher. At a matched 26,373-label budget, student-state supervision achieves 87.8% versus 66.7% for expert-only cloning, demonstrating gains beyond additional labels. On a UR5e, TRACE achieves 90.0% success versus 95.0% for the closed-loop teacher, while reducing total execution time from 192.7 s to 67.3 s. It avoids the teacher's 16.8 sensing-related arm retractions per trial during pushing, retaining a final withdrawal for graspability evaluation. Code and data will be released at: https://trace-retrieval.github.io.

We present a novel geometric perspective on the latent space of diffusion models. We first show that the standard pullback approach, utilizing the deterministic probability flow ODE decoder, is fundamentally flawed. It provably forces geodesics to decode as straight segments in data space, effectively ignoring any intrinsic data geometry beyond the ambient Euclidean space. Complementing this view, diffusion also admits a stochastic decoder via the reverse SDE, which enables an information geometric treatment with the Fisher-Rao metric. However, a choice of x_T as the latent representation collapses this metric due to memorylessness. We address this by introducing a latent spacetime z=(x_t,t) that indexes the family of denoising distributions p(x_0 | x_t) across all noise scales, yielding a nontrivial geometric structure. We prove these distributions form an exponential family and derive simulation-free estimators for curve lengths, enabling efficient geodesic computation. The resulting structure induces a principled Diffusion Edit Distance, where geodesics trace minimal sequences of noise and denoise edits between data. We also demonstrate benefits for transition path sampling in molecular systems, including constrained variants such as low-variance transitions and region avoidance. Code is available at: https://github.com/rafalkarczewski/spacetime-geometry.

Tracking and identifying athletes on the pitch holds a central role in collecting essential insights from the game, such as estimating the total distance covered by players or understanding team tactics. This tracking and identification process is crucial for reconstructing the game state, defined by the athletes' positions and identities on a 2D top-view of the pitch, (i.e. a minimap). However, reconstructing the game state from videos captured by a single camera is challenging. It requires understanding the position of the athletes and the viewpoint of the camera to localize and identify players within the field. In this work, we formalize the task of Game State Reconstruction and introduce SoccerNet-GSR, a novel Game State Reconstruction dataset focusing on football videos. SoccerNet-GSR is composed of 200 video sequences of 30 seconds, annotated with 9.37 million line points for pitch localization and camera calibration, as well as over 2.36 million athlete positions on the pitch with their respective role, team, and jersey number. Furthermore, we introduce GS-HOTA, a novel metric to evaluate game state reconstruction methods. Finally, we propose and release an end-to-end baseline for game state reconstruction, bootstrapping the research on this task. Our experiments show that GSR is a challenging novel task, which opens the field for future research. Our dataset and codebase are publicly available at https://github.com/SoccerNet/sn-gamestate.

Modern generative models typically rely on an adversarial critic, a prescribed noise-to-data path, or an autoregressive factorization. Instead, we show that a proper distributional energy can induce sample-level motion and provide direct regression supervision for a one-step generator. Three-Body Scattering Modeling (TBSM) for generation turns the energy distance into a constant-size per-projectile interaction: each projectile is attracted toward one real source and repelled from one independently generated source. Conditioned on the projectile and its condition, its expectation equals the 2-Wasserstein gradient-flow velocity of frac12D_E^2(P_θ,Q). A batch of B frozen-target events yields O(B) sample-level losses, each using one reference for its condition instead of the minibatch-wide all-pairs field used by methods such as Drifting Models. Tracking this conditional expectation online can reduce field noise. Using scattering in frozen image features, TBSM trains one-step generators on ImageNet-256, achieving FID{}=2.23 with pixel-space PixelDiT-XL and FID{}=1.63 with latent-space DiT-XL at NFE{}=1. We provide a design map relating diffusion-related supervision, Drift-like dynamics, and GAN-like objectives. These results establish tracked scattering as a route to high-dimensional one-step generation. Code: https://github.com/sp12138/TBSM.

Real-world video benchmarks provide broad coverage, but their fixed clips entangle event count, rate, duration, and visual complexity, making failure modes hard to isolate. While existing programmatic benchmarks offer better control, they score only the final answer rather than auditing reported events against executable ground truth. To bridge this gap, we introduce trace-grounded parametric profiling for event counting in three controlled video tasks: bouncing-ball wall contacts, visual blinks, and categorical state transitions. Across 2,190 videos, we vary event count N and frequency F while holding rendering fixed. Each video includes an executable event trace for capability-surface estimation and timestamp-level evaluation. Our results reveal a staged temporal failure. At an 80% reliability threshold, Gemini 3.6 Flash reliably counts persistent state transitions up to 12 events at 0.5 and 1.0 Hz, yet demonstrates no reliable positive-count region for transient blinking events. Thus, event representation dictates whether a model initially accesses evidence -- a limitation that compounds as count and frequency increase. In the high-count, high-frequency regime, only 0.2% of final counts are correct and the model recovers just 18.1% of true events. To test if visual access is the primary bottleneck, we increase sampling rate. Although this boosts Bounce Ball accuracy from 19.6% to 29.3%, the reported sequence agrees with ground truth only 3.7% of the time. Extra frames can therefore inflate final scores without producing faithful event recovery. Different prompting strategies yield similarly limited gains, and real-world video evaluations show the same concentration of success at low event counts. Ultimately, trace-grounded profiling shifts video evaluation from aggregate accuracy metrics to a detailed diagnostic of where temporal reasoning fails.

Existing active reconstruction systems with Gaussian-splatting maps select observations greedily, optimizing a single next-best-view (NBV) at each step and connecting the chosen views by short-horizon path planning. This greedy decoupling disregards the global structure of scene information, producing inefficient trajectories that waste sensing capacity in transit between selected views. In this work, we study active reconstruction as an ergodic coverage problem: the time-averaged spatial statistics of the sensor trajectory should match a target information distribution induced by the current map. Our approach derives this target distribution online from uncertainty and visibility, and calculates ergodic trajectories via a kernel-ergodic horizon planner with gradient flow and footprint depletion, closing the loop between mapping and trajectory optimization. We thoroughly evaluate TRACE on the Replica dataset against the Next-Best-View (NBV) baselines, improving PSNR by 1.5 dB. Code: https://github.com/spikelab-jhu/trace-active-reconstruction.

Geolocating images of a ground-level scene entails estimating the location on Earth where the picture was taken, in absence of GPS or other location metadata. Typically, methods are evaluated by measuring the Great Circle Distance (GCD) between a predicted location and ground truth. However, this measurement is limited because it only evaluates a single point, not estimates of regions or score heatmaps. This is especially important in applications to rural, wilderness and under-sampled areas, where finding the exact location may not be possible, and when used in aggregate systems that progressively narrow down locations. In this paper, we introduce a novel metric, Recall vs Area (RvA), which measures the accuracy of estimated distributions of locations. RvA treats image geolocation results similarly to document retrieval, measuring recall as a function of area: For a ranked list of (possibly non-contiguous) predicted regions, we measure the accumulated area required for the region to contain the ground truth coordinate. This produces a curve similar to a precision-recall curve, where "precision" is replaced by square kilometers area, allowing evaluation of performance for different downstream search area budgets. Following directly from this view of the problem, we then examine a simple ensembling approach to global-scale image geolocation, which incorporates information from multiple sources to help address domain shift, and can readily incorporate multiple models, attribute predictors, and data sources. We study its effectiveness by combining the geolocation models GeoEstimation and the current SOTA GeoCLIP, with attribute predictors based on ORNL LandScan and ESA-CCI Land Cover. We find significant improvements in image geolocation for areas that are under-represented in the training set, particularly non-urban areas, on both Im2GPS3k and Street View images.

We address the problem of distance metric learning (DML), defined as learning a distance consistent with a notion of semantic similarity. Traditionally, for this problem supervision is expressed in the form of sets of points that follow an ordinal relationship -- an anchor point x is similar to a set of positive points Y, and dissimilar to a set of negative points Z, and a loss defined over these distances is minimized. While the specifics of the optimization differ, in this work we collectively call this type of supervision Triplets and all methods that follow this pattern Triplet-Based methods. These methods are challenging to optimize. A main issue is the need for finding informative triplets, which is usually achieved by a variety of tricks such as increasing the batch size, hard or semi-hard triplet mining, etc. Even with these tricks, the convergence rate of such methods is slow. In this paper we propose to optimize the triplet loss on a different space of triplets, consisting of an anchor data point and similar and dissimilar proxy points which are learned as well. These proxies approximate the original data points, so that a triplet loss over the proxies is a tight upper bound of the original loss. This proxy-based loss is empirically better behaved. As a result, the proxy-loss improves on state-of-art results for three standard zero-shot learning datasets, by up to 15% points, while converging three times as fast as other triplet-based losses.

We present two new classes of algorithms for efficient field integration on graphs encoding point clouds. The first class, SeparatorFactorization(SF), leverages the bounded genus of point cloud mesh graphs, while the second class, RFDiffusion(RFD), uses popular epsilon-nearest-neighbor graph representations for point clouds. Both can be viewed as providing the functionality of Fast Multipole Methods (FMMs), which have had a tremendous impact on efficient integration, but for non-Euclidean spaces. We focus on geometries induced by distributions of walk lengths between points (e.g., shortest-path distance). We provide an extensive theoretical analysis of our algorithms, obtaining new results in structural graph theory as a byproduct. We also perform exhaustive empirical evaluation, including on-surface interpolation for rigid and deformable objects (particularly for mesh-dynamics modeling), Wasserstein distance computations for point clouds, and the Gromov-Wasserstein variant.

Autonomous LLM agents can pursue hidden malicious objectives through sequences of individually benign actions, making sabotage difficult to detect using standard trajectory-level monitoring. Existing approaches either evaluate complete trajectories in a single pass or partition them into independently scored windows, limiting their ability to connect evidence across temporally distant actions. We propose TRACE, a monitoring framework for long-horizon LLM agent trajectories. TRACE operates through a TIJ (Triage-Inspect-Judge) loop that identifies high-signal regions, performs targeted inspection while maintaining accumulated evidence across reasoning steps, and synthesizes a trajectory-level verdict. We evaluate TRACE on ten task domains from SHADE-Arena against state-of-the-art baselines. TRACE achieves an aggregate F1 of 0.713 and recall of 0.844, with the largest gains on tasks requiring long-range evidence linking.

Training large neural networks (NNs) requires optimizing high-dimensional data-dependent loss functions. The optimization landscape of these functions is often highly complex and textured, even fractal-like, with many spurious local minima, ill-conditioned valleys, degenerate points, and saddle points. Complicating things further is the fact that these landscape characteristics are a function of the data, meaning that noise in the training data can propagate forward and give rise to unrepresentative small-scale geometry. This poses a difficulty for gradient-based optimization methods, which rely on local geometry to compute updates and are, therefore, vulnerable to being derailed by noisy data. In practice,this translates to a strong dependence of the optimization dynamics on the noise in the data, i.e., poor generalization performance. To remediate this problem, we propose a new optimization procedure: Rolling Ball Optimizer (RBO), that breaks this spatial locality by incorporating information from a larger region of the loss landscape in its updates. We achieve this by simulating the motion of a rigid sphere of finite radius rolling on the loss landscape, a straightforward generalization of Gradient Descent (GD) that simplifies into it in the infinitesimal limit. The radius serves as a hyperparameter that determines the scale at which RBO sees the loss landscape, allowing control over the granularity of its interaction therewith. We are motivated by the intuition that the large-scale geometry of the loss landscape is less data-specific than its fine-grained structure, and that it is easier to optimize. We support this intuition by proving that our algorithm has a smoothing effect on the loss function. Evaluation against SGD, SAM, and Entropy-SGD, on MNIST and CIFAR-10/100 demonstrates promising results in terms of convergence speed, training accuracy, and generalization performance.

Tracking objects in soccer videos is extremely important to gather both player and team statistics, whether it is to estimate the total distance run, the ball possession or the team formation. Video processing can help automating the extraction of those information, without the need of any invasive sensor, hence applicable to any team on any stadium. Yet, the availability of datasets to train learnable models and benchmarks to evaluate methods on a common testbed is very limited. In this work, we propose a novel dataset for multiple object tracking composed of 200 sequences of 30s each, representative of challenging soccer scenarios, and a complete 45-minutes half-time for long-term tracking. The dataset is fully annotated with bounding boxes and tracklet IDs, enabling the training of MOT baselines in the soccer domain and a full benchmarking of those methods on our segregated challenge sets. Our analysis shows that multiple player, referee and ball tracking in soccer videos is far from being solved, with several improvement required in case of fast motion or in scenarios of severe occlusion.

Objects in post-fire environments often undergo irreversible physical transformations that change their geometry, material state, and visual appearance. Detecting and identifying these remnants is critical for locating hazards, reconstructing pre-incident contents, and inventorying losses. Unlike standard image corruptions, these degradations affect the physical structure of the object itself. To study this setting, we introduce TRACE, a transformation-aware benchmark for post-fire object understanding. TRACE contains 21.4K real-image-grounded synthetic scenes and paired object-level pristine-to-degraded progressions spanning 499 object identities across 189 categories. We define five tasks targeting localization and pre-degradation understanding: degraded-object detection, pristine-state recovery and retrieval, original material recovery, pristine description generation, and functional reasoning. Existing models degrade sharply with severity. From the least to the most severe level, RF-DETR mAP decreases by 71% relative, while InternVL3.5 retrieval R@1 falls from 93.85 to 28.11. To address this, we propose the Feature Recovery Module (FRM), a plug-and-play module that maps degraded encoder features to pristine-aligned representations while keeping the host frozen. Trained only with paired feature supervision, FRM improves scene-level detection, CLIP/SigLIP2 feature recovery, and all four object-level VLM tasks, with larger gains under more severe degradation. Across VLM hosts and severity levels, relative gains average 12.5% for retrieval, 20.1% for material recovery, 13.2% for description generation, and 12.4% for functional reasoning.

Recent Large Vision-Language Models (LVLMs) demonstrate remarkable capabilities in image understanding and natural language generation. However, current approaches focus predominantly on global image understanding, struggling to simulate human visual attention trajectories and explain associations between descriptions and specific regions. We propose TraceVision, a unified vision-language model integrating trajectory-aware spatial understanding in an end-to-end framework. TraceVision employs a Trajectory-aware Visual Perception (TVP) module for bidirectional fusion of visual features and trajectory information. We design geometric simplification to extract semantic keypoints from raw trajectories and propose a three-stage training pipeline where trajectories guide description generation and region localization. We extend TraceVision to trajectory-guided segmentation and video scene understanding, enabling cross-frame tracking and temporal attention analysis. We construct the Reasoning-based Interactive Localized Narratives (RILN) dataset to enhance logical reasoning and interpretability. Extensive experiments on trajectory-guided captioning, text-guided trajectory prediction, understanding, and segmentation demonstrate that TraceVision achieves state-of-the-art performance, establishing a foundation for intuitive spatial interaction and interpretable visual understanding.

Optimal transport (OT) theory has been been used in machine learning to study and characterize maps that can push-forward efficiently a probability measure onto another. Recent works have drawn inspiration from Brenier's theorem, which states that when the ground cost is the squared-Euclidean distance, the ``best'' map to morph a continuous measure in P(Rd) into another must be the gradient of a convex function. To exploit that result, [Makkuva+ 2020, Korotin+2020] consider maps T=nabla f_theta, where f_theta is an input convex neural network (ICNN), as defined by Amos+2017, and fit theta with SGD using samples. Despite their mathematical elegance, fitting OT maps with ICNNs raises many challenges, due notably to the many constraints imposed on theta; the need to approximate the conjugate of f_theta; or the limitation that they only work for the squared-Euclidean cost. More generally, we question the relevance of using Brenier's result, which only applies to densities, to constrain the architecture of candidate maps fitted on samples. Motivated by these limitations, we propose a radically different approach to estimating OT maps: Given a cost c and a reference measure rho, we introduce a regularizer, the Monge gap M^c_{rho}(T) of a map T. That gap quantifies how far a map T deviates from the ideal properties we expect from a c-OT map. In practice, we drop all architecture requirements for T and simply minimize a distance (e.g., the Sinkhorn divergence) between Tsharpmu and nu, regularized by M^c_rho(T). We study M^c_{rho}, and show how our simple pipeline outperforms significantly other baselines in practice.

The Abstraction and Reasoning Corpus (ARC) tests whether a model can infer an unseen transformation from a few input-output examples and apply it to a new grid. Looped visual reasoners refine predictions over multiple iterations, but conventional training constrains only the final output, leaving intermediate refinements unconstrained. We propose that these refinements should instead follow the transformation step by step. We introduce TraceViT, a looped visual reasoner trained with semantically monotonic transformation chains. We obtain these chains by rewriting and verifying programmatic task implementations, decomposing each solution into intermediate grid states. Each iteration is grounded by a task reference derived from the few-shot demonstrations and an object workspace representing the current grid state. Because these chains may differ in length from the loop, soft trace alignment enforces only their ordering, letting the model allocate iterations freely. TraceViT achieves 67.8% pass@2 on ARC-AGI-1 and 24.3% on ARC-AGI-2. Controlled ablations on ARC-AGI-1 show that trace supervision becomes beneficial only when paired with grounding. Code and data will be available at https://github.com/LiuBinnan/TraceViT.

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

In time-varying D-optimal subset selection, the number of replaced coordinates does not measure physical displacement: when the optimal selection for a set of mobile sensors changes, a single sensor may have to cross the entire graph. We therefore measure movement by a bottleneck matching distance on a fixed mobility graph, and define the distance margin at radius P as the smallest old-objective loss among configurations farther than P from the current maximizer. Our main result shows that if the projector drift is small and the distance margin at a chosen radius exceeds an explicit threshold, then every maximizer of the new objective lies within that radius, without tie-breaking rules or uniform conditioning assumptions over subsets. Moreover, we show that separation information of this kind is genuinely needed: a weighted-path construction shows that, at a fixed baseline eigengap, an arbitrarily small perturbation can move an optimal sensor from one end of the graph to the other, so no bound depending only on perturbation size and eigengap can vanish with the perturbation. We also give a finite certificate, conditions under which local and Voronoi restrictions retain a global optimum, and oversampled and regularized extensions, and we identify the learned dictionary criteria to which the results apply.

Consider a random uniform sample of n points in a compact region A of Euclidean d-space, d geq 2, with a smooth or (when d=2) polygonal boundary. Fix k bf N. Let T_{n,k} be the threshold r at which the geometric graph on these n vertices with distance parameter r becomes k-connected. We show that if d=2 then n (pi/|A|) T_{n,1}^2 - log n is asymptotically standard Gumbel. For (d,k) neq (2,1), it is n (theta_d/|A|) T_{n,k}^d - (2-2/d) log n - (4-2k-2/d) log log n that converges in distribution to a nondegenerate limit, where theta_d is the volume of the unit ball. The limit is Gumbel with scale parameter 2 except when (d,k)=(2,2) where the limit is two component extreme value distributed. The different cases reflect the fact that boundary effects are more more important in some cases than others. We also give similar results for the largest k-nearest neighbour link U_{n,k} in the sample, and show T_{n,k}=U_{n,k} with high probability. We provide estimates on rates of convergence and give similar results for Poisson samples in A. Finally, we give similar results even for non-uniform samples, with a less explicit sequence of centring constants.

In this paper, it is shown that if a sequence of resistance metric spaces equipped with measures converges with respect to the local Gromov-Hausdorff-vague topology, and certain non-explosion and metric-entropy conditions are satisfied, then the associated stochastic processes and their local times also converge. The metric-entropy condition can be checked by applying volume estimates of balls. Whilst similar results have been proved previously, the approach of this article is more widely applicable. Indeed, we recover various known conclusions for scaling limits of some deterministic self-similar fractal graphs, critical Galton-Watson trees, the critical Erdos-R\'enyi random graph and the configuration model (in the latter two cases, we prove for the first time the convergence of the models with respect to the resistance metric and also, for the configuration model, we overcome an error in the existing proof of local time convergence). Moreover, we derive new ones for scaling limits of uniform spanning trees and random recursive fractals. The metric-entropy condition also implies convergence of associated Gaussian processes.

A significant gap exists between theory and practice in deep learning. Generalization and approximation error bounds are often derived for simplified models or are too loose to be informative. Many rely on the manifold hypothesis and on geometric regularity such as intrinsic dimension, curvature, and reach. Progress requires insight into data-manifold geometry and suitable benchmarks, yet existing options are polarized: analytic manifolds with known geometry but limited applicability, or real-world datasets where geometry is only coarsely estimable. We introduce a benchmarking framework for studying data geometry. We repurpose and extend dSprites and COIL-20 with additional transformation dimensions and dense, axis-aligned sampling, and pair them with finite-difference estimators that recover curvature, reach, and volume at near-ground-truth accuracy in a regime where general-purpose estimators are unreliable or difficult to deploy. The framework is intended as a controlled testbed, useful as a calibration environment for geometric estimators and a sandbox for probing theoretical assumptions. To illustrate its use, we present two application studies, namely assessing the scaling behavior of the bounds of Genovese et al. and Fefferman et al., and tracking the layer-wise geometry of a β-VAE, highlighting the behavior of current bounds and the value of controlled benchmarks for guiding and validating future theory. A reference implementation is available at https://github.com/koulakis/manifold-microscope.

Symmetric Positive Definite (SPD) matrices are ubiquitous in data analysis under the form of covariance matrices or correlation matrices. Several O(n)-invariant Riemannian metrics were defined on the SPD cone, in particular the kernel metrics introduced by Hiai and Petz. The class of kernel metrics interpolates between many classical O(n)-invariant metrics and it satisfies key results of stability and completeness. However, it does not contain all the classical O(n)-invariant metrics. Therefore in this work, we investigate super-classes of kernel metrics and we study which key results remain true. We also introduce an additional key result called cometric-stability, a crucial property to implement geodesics with a Hamiltonian formulation. Our method to build intermediate embedded classes between O(n)-invariant metrics and kernel metrics is to give a characterization of the whole class of O(n)-invariant metrics on SPD matrices and to specify requirements on metrics one by one until we reach kernel metrics. As a secondary contribution, we synthesize the literature on the main O(n)-invariant metrics, we provide the complete formula of the sectional curvature of the affine-invariant metric and the formula of the geodesic parallel transport between commuting matrices for the Bures-Wasserstein metric.

We present TT4D, a large-scale, high-fidelity table tennis dataset. It provides 140+ hours of reconstructed singles and doubles gameplay from monocular broadcast videos, featuring multimodal annotations like high-quality camera calibrations, precise 3D ball positions, ball spin, time segmentation, and 3D human meshes over time. This rich data provides a new foundation for virtual replay, in-depth player analysis, and robot learning. The dataset's combination of scale and precision is achieved through a novel reconstruction pipeline. Prior methods first partition a game sequence into individual shot segments based on the 2D ball track, and only then attempt reconstruction. However, 2D-based time segmentation collapses under occlusion and varied camera viewpoints, preventing reliable reconstruction. We invert this paradigm by first lifting the entire unsegmented 2D ball track to 3D through a learned lifting network. This 3D trajectory then allows us to reliably perform time segmentation. The learned lifting network also infers the ball's spin, handles unreliable ball detections, and successfully reconstructs the ball trajectory in cases of high occlusion. This lift-first design is necessary, as our pipeline is the only method capable of reconstructing table tennis gameplay from general-view broadcast monocular videos. We demonstrate the dataset's fidelity through two downstream tasks: estimating the racket's pose \& velocity at impact, and training a generative model of competitive rallies.

Detecting tiny objects is a very challenging problem since a tiny object only contains a few pixels in size. We demonstrate that state-of-the-art detectors do not produce satisfactory results on tiny objects due to the lack of appearance information. Our key observation is that Intersection over Union (IoU) based metrics such as IoU itself and its extensions are very sensitive to the location deviation of the tiny objects, and drastically deteriorate the detection performance when used in anchor-based detectors. To alleviate this, we propose a new evaluation metric using Wasserstein distance for tiny object detection. Specifically, we first model the bounding boxes as 2D Gaussian distributions and then propose a new metric dubbed Normalized Wasserstein Distance (NWD) to compute the similarity between them by their corresponding Gaussian distributions. The proposed NWD metric can be easily embedded into the assignment, non-maximum suppression, and loss function of any anchor-based detector to replace the commonly used IoU metric. We evaluate our metric on a new dataset for tiny object detection (AI-TOD) in which the average object size is much smaller than existing object detection datasets. Extensive experiments show that, when equipped with NWD metric, our approach yields performance that is 6.7 AP points higher than a standard fine-tuning baseline, and 6.0 AP points higher than state-of-the-art competitors. Codes are available at: https://github.com/jwwangchn/NWD.

The Distance Geometry Problem (DGP) seeks to find positions for a set of points in geometric space when some distances between pairs of these points are known. The so-called discretization assumptions allow to discretize the search space of DGP instances. In this paper, we study the Discretizable Molecular Distance Geometry Problem whose feasible solutions provide a discretization scheme for the DGP. We propose the first constraint programming formulations as well as a set of checks for proving infeasibility, domain reduction techniques, symmetry breaking constraints and valid inequalities. Our computational results indicate that our formulations outperform the state-of-the-art integer programming formulations, both for feasible and infeasible instances.

The "maximum similarity correlation" definition introduced in this study is motivated by the seminal work of Szekely et al on "distance covariance" (Ann. Statist. 2007, 35: 2769-2794; Ann. Appl. Stat. 2009, 3: 1236-1265). Instead of using Euclidean distances "d" as in Szekely et al, we use "similarity", which can be defined as "exp(-d/s)", where the scaling parameter s>0 controls how rapidly the similarity falls off with distance. Scale parameters are chosen by maximizing the similarity correlation. The motivation for using "similarity" originates in spectral clustering theory (see e.g. Ng et al 2001, Advances in Neural Information Processing Systems 14: 849-856). We show that a particular form of similarity correlation is asymptotically equivalent to distance correlation for large values of the scale parameter. Furthermore, we extend similarity correlation to coherence between complex valued vectors, including its partitioning into real and imaginary contributions. Several toy examples are used for comparing distance and similarity correlations. For instance, points on a noiseless straight line give distance and similarity correlation values equal to 1; but points on a noiseless circle produces near zero distance correlation (dCorr=0.02) while the similarity correlation is distinctly non zero (sCorr=0.36). In distinction to the distance approach, similarity gives more importance to small distances, which emphasizes the local properties of functional relations. This paper represents a preliminary empirical study, showing that the novel similarity association has some distinct practical advantages over distance based association.For the sake of reproducible research, the software code implementing all methods here (using lazarus free-pascal "www.lazarus.freepascal.org"), including all test data, are freely available at: "sites.google.com/site/pascualmarqui/home/similaritycovariance".

Multi-object tracking in sports broadcasts is hard for two reasons: athletes move fast and change direction abruptly, and the camera pans and zooms at the same time. We present ARC-SORT (Adaptive, Robust, Compensated SORT), a tracking-by-detection tracker that trains nothing and runs in real time. Its design rule: the tracker measures its own uncertainty, and that measurement decides how it acts. An Interacting Multiple Model (IMM) Kalman filter gives every track a predictive covariance, and we turn it into a per-track search region: the less reliable the motion prediction, the larger the region in which a detection may match (uncertainty-adaptive Expansion IoU, EIoU). On identical inputs, this continuous coupling is what matters: the same covariance used as a hard Mahalanobis gate hurts at every threshold we swept, and richer covariance summaries add nothing measurable. When two tracks cross, the assignment cost itself reports whether the decision was close. ARC-SORT keeps such pairs under review and applies at most one corrective identity swap once post-crossing appearance evidence shows the first decision was wrong; this adds +1.0 HOTA (higher order tracking accuracy) on both the SportsMOT and DanceTrack test sets at under a millisecond per frame. For players who leave the frame for seconds, motion extrapolation carries no information, so our offline re-identification drops the usual motion gate and links tracklets by appearance alone. ARC-SORT reaches 81.82 HOTA on the SportsMOT test set with the fewest identity switches of the 69 leaderboard entries as of August 2026 and tied-third multi-object tracking accuracy (MOTA, 97.09), and 62.1 HOTA on DanceTrack test with the same association settings. On Deep-EIoU's public detections and embeddings it reaches 80.31 HOTA, +3.1 over Deep-EIoU. It runs at about 50 frames per second on a laptop RTX 3060 GPU. Code will be released.

Shot charts in basketball analytics provide an indispensable tool for evaluating players' shooting performance by visually representing the distribution of field goal attempts across different court locations. However, conventional methods often overlook the bounded nature of the basketball court, leading to inaccurate representations, particularly along the boundaries and corners. In this paper, we propose a novel model-based approach to shot chart estimation and visualization that explicitly considers the physical boundaries of the basketball court. By employing Gaussian mixtures for bounded data, our methodology allows to obtain more accurate estimation of shot density distributions for both made and missed shots. Bayes' rule is then applied to derive estimates for the probability of successful shooting from any given locations, and to identify the regions with the highest expected scores. To illustrate the efficacy of our proposal, we apply it to data from the 2022-23 NBA regular season, showing its usefulness through detailed analyses of shot patterns for two prominent players.

Multi-object tracking, player identification, and pose estimation are fundamental components of sports analytics, essential for analyzing player movements, performance, and tactical strategies. However, existing datasets and methodologies primarily target mainstream team sports such as soccer and conventional 5-on-5 basketball, often overlooking scenarios involving fixed-camera setups commonly used at amateur levels, less mainstream sports, or datasets that explicitly incorporate pose annotations. In this paper, we propose the TrackID3x3 dataset, the first publicly available comprehensive dataset specifically designed for multi-player tracking, player identification, and pose estimation in 3x3 basketball scenarios. The dataset comprises three distinct subsets (Indoor fixed-camera, Outdoor fixed-camera, and Drone camera footage), capturing diverse full-court camera perspectives and environments. We also introduce the Track-ID task, a simplified variant of the game state reconstruction task that excludes field detection and focuses exclusively on fixed-camera scenarios. To evaluate performance, we propose a baseline algorithm called Track-ID algorithm, tailored to assess tracking and identification quality. Furthermore, our benchmark experiments, utilizing recent multi-object tracking algorithms (e.g., BoT-SORT-ReID) and top-down pose estimation methods (HRNet, RTMPose, and SwinPose), demonstrate robust results and highlight remaining challenges. Our dataset and evaluation benchmarks provide a solid foundation for advancing automated analytics in 3x3 basketball. Dataset and code will be available at https://github.com/open-starlab/TrackID3x3.

We study the classic Text-to-Pattern Hamming Distances problem: given a pattern P of length m and a text T of length n, both over a polynomial-size alphabet, compute the Hamming distance between P and T[i, ., . , i+m-1] for every shift i, under the standard Word-RAM model with Theta(log n)-bit words. - We provide an O(nm) time Las Vegas randomized algorithm for this problem, beating the decades-old O(n m log m) running time [Abrahamson, SICOMP 1987]. We also obtain a deterministic algorithm, with a slightly higher O(nm(log mloglog m)^{1/4}) running time. Our randomized algorithm extends to the k-bounded setting, with running time Obig(n+nk{m}big), removing all the extra logarithmic factors from earlier algorithms [Gawrychowski and Uzna\'{n}ski, ICALP 2018; Chan, Golan, Kociumaka, Kopelowitz and Porat, STOC 2020]. - For the (1+epsilon)-approximate version of Text-to-Pattern Hamming Distances, we give an O(epsilon^{-0.93}n) time Monte Carlo randomized algorithm, beating the previous O(epsilon^{-1}n) running time [Kopelowitz and Porat, FOCS 2015; Kopelowitz and Porat, SOSA 2018]. Our approximation algorithm exploits a connection with 3SUM, and uses a combination of Fredman's trick, equality matrix product, and random sampling; in particular, we obtain new results on approximate counting versions of 3SUM and Exact Triangle, which may be of independent interest. Our exact algorithms use a novel combination of hashing, bit-packed FFT, and recursion; in particular, we obtain a faster algorithm for computing the sumset of two integer sets, in the regime when the universe size is close to quadratic in the number of elements. We also prove a fine-grained equivalence between the exact Text-to-Pattern Hamming Distances problem and a range-restricted, counting version of 3SUM.

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank k tangent subbundle on R^d, k<d, which we call a principal subbundle. This determines a sub-Riemannian metric on R^d. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold M, construction of a representation of the point-cloud in R^k, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.

Sports analytics benefits from recent advances in machine learning providing a competitive advantage for teams or individuals. One important task in this context is the performance measurement of individual players to provide reports and log files for subsequent analysis. During sport events like basketball, this involves the re-identification of players during a match either from multiple camera viewpoints or from a single camera viewpoint at different times. In this work, we investigate whether it is possible to transfer the out-standing zero-shot performance of pre-trained CLIP models to the domain of player re-identification. For this purpose we reformulate the contrastive language-to-image pre-training approach from CLIP to a contrastive image-to-image training approach using the InfoNCE loss as training objective. Unlike previous work, our approach is entirely class-agnostic and benefits from large-scale pre-training. With a fine-tuned CLIP ViT-L/14 model we achieve 98.44 % mAP on the MMSports 2022 Player Re-Identification challenge. Furthermore we show that the CLIP Vision Transformers have already strong OCR capabilities to identify useful player features like shirt numbers in a zero-shot manner without any fine-tuning on the dataset. By applying the Score-CAM algorithm we visualise the most important image regions that our fine-tuned model identifies when calculating the similarity score between two images of a player.

Computing the optimal transport distance between statistical distributions is a fundamental task in machine learning. One remarkable recent advancement is entropic regularization and the Sinkhorn algorithm, which utilizes only matrix scaling and guarantees an approximated solution with near-linear runtime. Despite the success of the Sinkhorn algorithm, its runtime may still be slow due to the potentially large number of iterations needed for convergence. To achieve possibly super-exponential convergence, we present Sinkhorn-Newton-Sparse (SNS), an extension to the Sinkhorn algorithm, by introducing early stopping for the matrix scaling steps and a second stage featuring a Newton-type subroutine. Adopting the variational viewpoint that the Sinkhorn algorithm maximizes a concave Lyapunov potential, we offer the insight that the Hessian matrix of the potential function is approximately sparse. Sparsification of the Hessian results in a fast O(n^2) per-iteration complexity, the same as the Sinkhorn algorithm. In terms of total iteration count, we observe that the SNS algorithm converges orders of magnitude faster across a wide range of practical cases, including optimal transportation between empirical distributions and calculating the Wasserstein W_1, W_2 distance of discretized densities. The empirical performance is corroborated by a rigorous bound on the approximate sparsity of the Hessian matrix.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

Measuring geometric similarity between high-dimensional network representations is a topic of longstanding interest to neuroscience and deep learning. Although many methods have been proposed, only a few works have rigorously analyzed their statistical efficiency or quantified estimator uncertainty in data-limited regimes. Here, we derive upper and lower bounds on the worst-case convergence of standard estimators of shape distancex2014a measure of representational dissimilarity proposed by Williams et al. (2021).These bounds reveal the challenging nature of the problem in high-dimensional feature spaces. To overcome these challenges, we introduce a new method-of-moments estimator with a tunable bias-variance tradeoff. We show that this estimator achieves substantially lower bias than standard estimators in simulation and on neural data, particularly in high-dimensional settings. Thus, we lay the foundation for a rigorous statistical theory for high-dimensional shape analysis, and we contribute a new estimation method that is well-suited to practical scientific settings.

We study sparse locally private discrete mechanisms generated by a nonnegative kernel and an input-dependent admissible output support. The support set is intentionally small relative to the ambient alphabet and acts as a mechanism-design parameter. This formulation covers metric-ball supports, sparse exponential mechanisms, sparse staircase mechanisms, sparse randomized response, sparse additive kernels such as Skellam perturbations, and Poisson-binomial-inspired bounded-support channels. We give a general exact characterization of pure and approximate local differential privacy for this sparse-kernel class. Pure local differential privacy forces all supports to coincide, so genuinely input-dependent sparse supports are incompatible with pure privacy. In the approximate regime, the privacy defect decomposes exactly into support leakage and overlap excess loss. We instantiate the formula for several discrete mechanism families. For sparse staircase mechanisms, a shell-ratio condition eliminates overlap excess loss. For graph-metric supports, overlap of metric balls is necessary for nontrivial privacy. For sparse randomized response, the support-leakage term has a closed form in terms of support mismatch. For generic radius-truncated additive kernels and, in particular, sparse Skellam mechanisms, we obtain exact finite-support formulas in terms of the Skellam CDF and an exact Bessel-ratio overlap condition. Numerical evaluations illustrate how support radius and kernel shape separately affect support leakage and overlap excess loss.

In this paper, we explore the application of mean field theory, a technique from statistical physics, to deep metric learning and address the high training complexity commonly associated with conventional metric learning loss functions. By adapting mean field theory for deep metric learning, we develop an approach to design classification-based loss functions from pair-based ones, which can be considered complementary to the proxy-based approach. Applying the mean field theory to two pair-based loss functions, we derive two new loss functions, MeanFieldContrastive and MeanFieldClassWiseMultiSimilarity losses, with reduced training complexity. We extensively evaluate these derived loss functions on three image-retrieval datasets and demonstrate that our loss functions outperform baseline methods in two out of the three datasets.

We present anyakrakusuma, an open-source Python library that solves the discrete static Schrödinger bridge problem, the entropically regularized counterpart of optimal transport, through a log-domain Sinkhorn--Knopp iteration and reconstructs the entropic interpolation between two empirical point clouds. The solver is paired with a diagnostic pipeline that characterizes the optimal coupling and the intermediate distributions through information-theoretic and geometric measures. We exercise the library on four idealized planar cases spanning a circle-to-circle dilation, a spiral-to-mixture fragmentation, a rigid reorientation of two moons, and a Lissajous-to-trefoil deformation. The log-domain formulation is necessary rather than merely convenient at the parameters studied, where the cost-to-regularization ratio reaches four hundred and the Gibbs kernel underflows double precision across most of its range; the iteration nonetheless attains a marginal residual of 10^{-9} and unit marginal fidelity in every case. Residual histories decay geometrically over approximately eight decades at per-iteration contraction factors between 0.966 and 0.976, which are local rates near the fixed point that lie many orders of magnitude below the worst-case Hilbert-metric bound. The covariance analysis recovers an imposed ninety-degree reorientation to within 0.07^circ, roughly forty times smaller than its uncertainty, across a masked interval of near-isotropy on which the principal axis is unobservable. The diagnostics are reported with explicit attention to the regimes in which each is well defined, including the differential entropy, which is meaningful only on the open interpolation interval. The presented cases are constructed rather than measured; quantitative application to empirical point clouds requires further study.

Badminton, known for having the fastest ball speeds among all sports, presents significant challenges to the field of computer vision, including player identification, court line detection, shuttlecock trajectory tracking, and player stroke-type classification. In this paper, we introduce a novel video clipping strategy to extract frames of each player's racket swing in a badminton broadcast match. These clipped frames are then processed by three existing models: one for Human Pose Estimation to obtain human skeletal joints, another for shuttlecock trajectory tracking, and the other for court line detection to determine player positions on the court. Leveraging these data as inputs, we propose Badminton Stroke-type Transformer (BST) to classify player stroke-types in singles. To the best of our knowledge, experimental results demonstrate that our method outperforms the previous state-of-the-art on the largest publicly available badminton video dataset (ShuttleSet), another badminton dataset (BadmintonDB), and a tennis dataset (TenniSet). These results suggest that effectively leveraging ball trajectory is a promising direction for action recognition in racket sports.

Players and ball detection are among the first required steps on a football analytics platform. Until recently, the existing open datasets on which the evaluations of most models were based, were not sufficient. In this work, we point out their weaknesses, and with the advent of the SoccerNet v3, we propose and deliver to the community an edited part of its dataset, in YOLO normalized annotation format for training and evaluation. The code of the methods and metrics are provided so that they can be used as a benchmark in future comparisons. The recent YOLO8n model proves better than FootAndBall in long-shot real-time detection of the ball and players on football fields.

Standard simulations of Turing machines suggest a linear relationship between the temporal duration t of a run and the amount of information that must be stored by known simulations to certify, verify, or regenerate the configuration at time t. For deterministic multitape Turing machines over a fixed finite alphabet, this apparent linear dependence is not intrinsic: any length-t run can be simulated using O(t) work-tape cells via a Height Compression Theorem for succinct computation trees together with an Algebraic Replay Engine. In this paper we recast that construction in geometric and information-theoretic language. We interpret the execution trace as a spacetime DAG of local update events and exhibit a family of recursively defined holographic boundary summaries such that, along the square-root-space simulation, the total description length of all boundary data stored at any time is O(t). Using Kolmogorov complexity, we prove that every internal configuration has constant conditional description complexity given the appropriate boundary summary and time index, establishing that the spacetime bulk carries no additional algorithmic information beyond its boundary. We express this as a one-dimensional computational area law: there exists a simulation in which the information capacity of the active "holographic screen'' needed to generate a spacetime region of volume proportional to t is bounded by O(t). In this precise sense, deterministic computation on a one-dimensional work tape admits a holographic representation, with the bulk history algebraically determined by data residing on a lower-dimensional boundary screen.

Recent advances in deep generative models have lead to remarkable progress in synthesizing high quality images. Following their successful application in image processing and representation learning, an important next step is to consider videos. Learning generative models of video is a much harder task, requiring a model to capture the temporal dynamics of a scene, in addition to the visual presentation of objects. While recent attempts at formulating generative models of video have had some success, current progress is hampered by (1) the lack of qualitative metrics that consider visual quality, temporal coherence, and diversity of samples, and (2) the wide gap between purely synthetic video data sets and challenging real-world data sets in terms of complexity. To this extent we propose Fr\'{e}chet Video Distance (FVD), a new metric for generative models of video, and StarCraft 2 Videos (SCV), a benchmark of game play from custom starcraft 2 scenarios that challenge the current capabilities of generative models of video. We contribute a large-scale human study, which confirms that FVD correlates well with qualitative human judgment of generated videos, and provide initial benchmark results on SCV.

Distances between probability distributions that take into account the geometry of their sample space,like the Wasserstein or the Maximum Mean Discrepancy (MMD) distances have received a lot of attention in machine learning as they can, for instance, be used to compare probability distributions with disjoint supports. In this paper, we study a class of statistical Hilbert distances that we term the Schoenberg-Rao distances, a generalization of the MMD that allows one to consider a broader class of kernels, namely the conditionally negative semi-definite kernels. In particular, we introduce a principled way to construct such kernels and derive novel closed-form distances between mixtures of Gaussian distributions. These distances, derived from the concave Rao's quadratic entropy, enjoy nice theoretical properties and possess interpretable hyperparameters which can be tuned for specific applications. Our method constitutes a practical alternative to Wasserstein distances and we illustrate its efficiency on a broad range of machine learning tasks such as density estimation, generative modeling and mixture simplification.

Intersection over Union (IoU) is the most popular evaluation metric used in the object detection benchmarks. However, there is a gap between optimizing the commonly used distance losses for regressing the parameters of a bounding box and maximizing this metric value. The optimal objective for a metric is the metric itself. In the case of axis-aligned 2D bounding boxes, it can be shown that IoU can be directly used as a regression loss. However, IoU has a plateau making it infeasible to optimize in the case of non-overlapping bounding boxes. In this paper, we address the weaknesses of IoU by introducing a generalized version as both a new loss and a new metric. By incorporating this generalized IoU (GIoU) as a loss into the state-of-the art object detection frameworks, we show a consistent improvement on their performance using both the standard, IoU based, and new, GIoU based, performance measures on popular object detection benchmarks such as PASCAL VOC and MS COCO.

In this paper we study holomorphic properties of infinite dimensional spin factors. Among the infinite dimensional Banach spaces with homogeneous open unit balls, we show that the spin factors are natural outlier spaces in which to ask the question (as was proved in the early 1970s for Hilbert spaces): Do biholomorphic automorphisms g of the open unit ball B have fixed points in overline B? In this paper, for infinite dimensional spin factors, we provide reasonable conditions on g that allow us to explicitly construct fixed points of g lying on partial B. En route, we also prove that every spin factor has the density property. In another direction, we focus on (compact) holomorphic maps f:Brightarrow B, having no fixed point in B and examine the sequence of iterates (f^n). As (f^n) does not generally converge, we instead trace the target set T(f) of f, that is, the images of all accumulation points of (f^n)_n, for any topology finer than the topology of pointwise convergence on B. We prove for a spin factor that T(f) lies on the boundary of a single bidisc unique to f.

The sliced Wasserstein (SW) distances between two probability measures are defined as the expectation of the Wasserstein distance between two one-dimensional projections of the two measures. The randomness comes from a projecting direction that is used to project the two input measures to one dimension. Due to the intractability of the expectation, Monte Carlo integration is performed to estimate the value of the SW distance. Despite having various variants, there has been no prior work that improves the Monte Carlo estimation scheme for the SW distance in terms of controlling its variance. To bridge the literature on variance reduction and the literature on the SW distance, we propose computationally efficient control variates to reduce the variance of the empirical estimation of the SW distance. The key idea is to first find Gaussian approximations of projected one-dimensional measures, then we utilize the closed-form of the Wasserstein-2 distance between two Gaussian distributions to design the control variates. In particular, we propose using a lower bound and an upper bound of the Wasserstein-2 distance between two fitted Gaussians as two computationally efficient control variates. We empirically show that the proposed control variate estimators can help to reduce the variance considerably when comparing measures over images and point-clouds. Finally, we demonstrate the favorable performance of the proposed control variate estimators in gradient flows to interpolate between two point-clouds and in deep generative modeling on standard image datasets, such as CIFAR10 and CelebA.

The increasing availability of large-scale global datasets has generated a demand for scalable spatial prediction methods defined on spherical domains. Classical spatial models that rely on Euclidean distance representations are inappropriate for spherical data because planar projections distort geodesic distances and spatial neighborhood structures, while traditional kriging-based prediction methods are often computationally prohibitive for massive datasets. To address these challenges, we propose a Spherical DeepKriging framework for spatial prediction on S^2. The proposed approach constructs a flexible prediction model by integrating thin-plate spline (TPS) basis functions defined intrinsically on the sphere. Simulation studies and real data analyses are presented to demonstrate the superior predictive performance of the proposed method.

Probability estimation models play an important role in various fields, such as weather forecasting, recommendation systems, and sports analysis. Among several models estimating probabilities, it is difficult to evaluate which model gives reliable probabilities since the ground-truth probabilities are not available. The win probability estimation model for esports, which calculates the win probability under a certain game state, is also one of the fields being actively studied in probability estimation. However, most of the previous works evaluated their models using accuracy, a metric that only can measure the performance of discrimination. In this work, we firstly investigate the Brier score and the Expected Calibration Error (ECE) as a replacement of accuracy used as a performance evaluation metric for win probability estimation models in esports field. Based on the analysis, we propose a novel metric called Balance score which is a simple yet effective metric in terms of six good properties that probability estimation metric should have. Under the general condition, we also found that the Balance score can be an effective approximation of the true expected calibration error which has been imperfectly approximated by ECE using the binning technique. Extensive evaluations using simulation studies and real game snapshot data demonstrate the promising potential to adopt the proposed metric not only for the win probability estimation model for esports but also for evaluating general probability estimation models.

Learning new robot tasks on new platforms and in new scenes from only a handful of demonstrations remains challenging. While videos of other embodiments - humans and different robots - are abundant, differences in embodiment, camera, and environment hinder their direct use. We address the small-data problem by introducing a unifying, symbolic representation - a compact 3D "trace-space" of scene-level trajectories - that enables learning from cross-embodiment, cross-environment, and cross-task videos. We present TraceGen, a world model that predicts future motion in trace-space rather than pixel space, abstracting away appearance while retaining the geometric structure needed for manipulation. To train TraceGen at scale, we develop TraceForge, a data pipeline that transforms heterogeneous human and robot videos into consistent 3D traces, yielding a corpus of 123K videos and 1.8M observation-trace-language triplets. Pretraining on this corpus produces a transferable 3D motion prior that adapts efficiently: with just five target robot videos, TraceGen attains 80% success across four tasks while offering 50-600x faster inference than state-of-the-art video-based world models. In the more challenging case where only five uncalibrated human demonstration videos captured on a handheld phone are available, it still reaches 67.5% success on a real robot, highlighting TraceGen's ability to adapt across embodiments without relying on object detectors or heavy pixel-space generation.

We introduce a flexible framework that produces high-quality almost-exact matches for causal inference. Most prior work in matching uses ad-hoc distance metrics, often leading to poor quality matches, particularly when there are irrelevant covariates. In this work, we learn an interpretable distance metric for matching, which leads to substantially higher quality matches. The learned distance metric stretches the covariate space according to each covariate's contribution to outcome prediction: this stretching means that mismatches on important covariates carry a larger penalty than mismatches on irrelevant covariates. Our ability to learn flexible distance metrics leads to matches that are interpretable and useful for the estimation of conditional average treatment effects.

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

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

Spatial tracing, as a fundamental embodied interaction ability for robots, is inherently challenging as it requires multi-step metric-grounded reasoning compounded with complex spatial referring and real-world metric measurement. However, existing methods struggle with this compositional task. To this end, we propose RoboTracer, a 3D-aware VLM that first achieves both 3D spatial referring and measuring via a universal spatial encoder and a regression-supervised decoder to enhance scale awareness during supervised fine-tuning (SFT). Moreover, RoboTracer advances multi-step metric-grounded reasoning via reinforcement fine-tuning (RFT) with metric-sensitive process rewards, supervising key intermediate perceptual cues to accurately generate spatial traces. To support SFT and RFT training, we introduce TraceSpatial, a large-scale dataset of 30M QA pairs, spanning outdoor/indoor/tabletop scenes and supporting complex reasoning processes (up to 9 steps). We further present TraceSpatial-Bench, a challenging benchmark filling the gap to evaluate spatial tracing. Experimental results show that RoboTracer surpasses baselines in spatial understanding, measuring, and referring, with an average success rate of 79.1%, and also achieves SOTA performance on TraceSpatial-Bench by a large margin, exceeding Gemini-2.5-Pro by 36% accuracy. Notably, RoboTracer can be integrated with various control policies to execute long-horizon, dynamic tasks across diverse robots (UR5, G1 humanoid) in cluttered real-world scenes.

In this paper, we present a real-time egocentric trajectory prediction system for table tennis using event cameras. Unlike standard cameras, which suffer from high latency and motion blur at fast ball speeds, event cameras provide higher temporal resolution, allowing more frequent state updates, greater robustness to outliers, and accurate trajectory predictions using just a short time window after the opponent's impact. We collect a dataset of ping-pong game sequences, including 3D ground-truth trajectories of the ball, synchronized with sensor data from the Meta Project Aria glasses and event streams. Our system leverages foveated vision, using eye-gaze data from the glasses to process only events in the viewer's fovea. This biologically inspired approach improves ball detection performance and significantly reduces computational latency, as it efficiently allocates resources to the most perceptually relevant regions, achieving a reduction factor of 10.81 on the collected trajectories. Our detection pipeline has a worst-case total latency of 4.5 ms, including computation and perception - significantly lower than a frame-based 30 FPS system, which, in the worst case, takes 66 ms solely for perception. Finally, we fit a trajectory prediction model to the estimated states of the ball, enabling 3D trajectory forecasting in the future. To the best of our knowledge, this is the first approach to predict table tennis trajectories from an egocentric perspective using event cameras.

We study the chemical distance of supercritical Bernoulli percolation on Z^d. Recently, Dembin [Dem22] showed that the chemical distance exhibits sublinear variance, a phenomenon now referred to as superconcentration. In this article, we establish an equivalence between this phenomenon and chaotic behavior of geodesics under small perturbations of the configuration, thereby confirming Chatterjee's general principle relating anomalous fluctuations to chaos in the context of Bernoulli percolation. Our methods rely on a dynamical version of the effective radius, refining the notion first proposed in [CN25], in order to measure the co-influence of a given edge whose weight may be infinite. Together with techniques from the theory of lattice animals, this approach allows us to quantify the total co-influence of edges in terms of the overlap between original and perturbed geodesics.

Large Language Models (LLMs) often generate code with subtle but critical bugs, especially for complex tasks. Existing automated repair methods typically rely on superficial pass/fail signals, offering limited visibility into program behavior and hindering precise error localization. In addition, without a way to learn from prior failures, repair processes often fall into repetitive and inefficient cycles. To overcome these challenges, we present TraceCoder, a collaborative multi-agent framework that emulates the observe-analyze-repair process of human experts. The framework first instruments the code with diagnostic probes to capture fine-grained runtime traces, enabling deep insight into its internal execution. It then conducts causal analysis on these traces to accurately identify the root cause of the failure. This process is further enhanced by a novel Historical Lesson Learning Mechanism (HLLM), which distills insights from prior failed repair attempts to inform subsequent correction strategies and prevent recurrence of similar mistakes. To ensure stable convergence, a Rollback Mechanism enforces that each repair iteration constitutes a strict improvement toward the correct solution. Comprehensive experiments across multiple benchmarks show that TraceCoder achieves up to a 34.43\% relative improvement in Pass@1 accuracy over existing advanced baselines. Ablation studies verify the significance of each system component, with the iterative repair process alone contributing a 65.61\% relative gain in accuracy. Furthermore, TraceCoder significantly outperforms leading iterative methods in terms of both accuracy and cost-efficiency.

Mathematical programming formulations are routinely modified through algebraic reformulations, perturbations of constraint functions, and the addition or removal of constraints. Some of these modifications preserve the feasible set, whereas others reflect newly imposed modeling requirements and therefore change the optimization problem itself. This paper develops a metric framework for quantifying the robustness of formulation-dependent properties. Formulations are modeled as typed collections of constraint functions. Within each fixed typed-cardinality class, their distance is defined by optimally matching constraints of the same type and measuring the resulting discrepancies in a norm on the underlying function space. As a concrete demonstration of the framework, we study the Linear Independence Constraint Qualification (LICQ). We introduce the pointwise LICQ radius, defined as the distance from a feasible formulation to the set of formulations that remain feasible at a prescribed point but fail LICQ there. Under suitable assumptions, we derive an explicit formula for this radius and characterize the effect of adding a single equality or inequality constraint.

Reconstructing population dynamics is a central problem in the physical and data sciences. Often, the dynamics are modeled as a Wasserstein gradient flow (WGF): a curve of distributions driven by an energy functional. Though there are multiple mathematical characterizations of a WGF, the dominant algorithmic approach relies on the Jordan--Kinderlehrer--Otto (JKO) scheme. JKO-based methods are inflexible to time discretisation and require solving costly optimal transport problems. We take a residual approach, enforcing the continuity equations via a non-negative loss function whose minimum is the WGF. Combined with a data-fitting divergence, this gives a single global objective. This perspective unifies several existing methods and leads to a new particle-based method, stitching, that is simulation-free and robust to large gaps between observations. We demonstrate that the stitching method achieves state-of-the-art performance across trajectory inference benchmarks. For code see github.com/BasisResearch/wasserstein-residuals.

We present a new regular grid search algorithm for quick fixed-radius nearest-neighbor lookup developed in Python. This module indexes a set of k-dimensional points in a regular grid, with optional periodic conditions, providing a fast approach for nearest neighbors queries. In this first installment we provide three types of queries: bubble, shell and the nth-nearest; as well as three different metrics of interest in astronomy: the euclidean and two distance functions in spherical coordinates of varying precision, haversine and Vincenty; and the possibility of providing a custom distance function. This package results particularly useful for large datasets where a brute-force search turns impractical.

Multi-object tracking (MOT) is crucial for various multi-agent analyses such as evaluating team sports tactics and player movements and performance. While pedestrian tracking has advanced with Tracking-by-Detection MOT, team sports like basketball pose unique challenges. These challenges include players' unpredictable movements, frequent close interactions, and visual similarities that complicate pose labeling and lead to significant occlusions, frequent ID switches, and high manual annotation costs. To address these challenges, we propose a novel pose-based virtual marker (VM) MOT method for team sports, named Sports-vmTracking. This method builds on the vmTracking approach developed for multi-animal tracking with active learning. First, we constructed a 3x3 basketball pose dataset for VMs and applied active learning to enhance model performance in generating VMs. Then, we overlaid the VMs on video to identify players, extract their poses with unique IDs, and convert these into bounding boxes for comparison with automated MOT methods. Using our 3x3 basketball dataset, we demonstrated that our VM configuration has been highly effective, and reduced the need for manual corrections and labeling during pose model training while maintaining high accuracy. Our approach achieved an average HOTA score of 72.3%, over 10 points higher than other state-of-the-art methods without VM, and resulted in 0 ID switches. Beyond improving performance in handling occlusions and minimizing ID switches, our framework could substantially increase the time and cost efficiency compared to traditional manual annotation.

Maximum mean discrepancy (MMD) flows suffer from high computational costs in large scale computations. In this paper, we show that MMD flows with Riesz kernels K(x,y) = - |x-y|^r, r in (0,2) have exceptional properties which allow their efficient computation. We prove that the MMD of Riesz kernels, which is also known as energy distance, coincides with the MMD of their sliced version. As a consequence, the computation of gradients of MMDs can be performed in the one-dimensional setting. Here, for r=1, a simple sorting algorithm can be applied to reduce the complexity from O(MN+N^2) to O((M+N)log(M+N)) for two measures with M and N support points. As another interesting follow-up result, the MMD of compactly supported measures can be estimated from above and below by the Wasserstein-1 distance. For the implementations we approximate the gradient of the sliced MMD by using only a finite number P of slices. We show that the resulting error has complexity O(d/P), where d is the data dimension. These results enable us to train generative models by approximating MMD gradient flows by neural networks even for image applications. We demonstrate the efficiency of our model by image generation on MNIST, FashionMNIST and CIFAR10.

Large Language Models (LLMs) deployed in agentic environments must exercise multiple capabilities across different task instances, where a capability is performing one or more actions in a trajectory that are necessary for successfully solving a subset of tasks in the environment. Many existing approaches either rely on synthetic training data that is not targeted to the model's actual capability deficits in the target environment or train directly on the target environment, where the model needs to implicitly learn the capabilities across tasks. We introduce TRACE (Turning Recurrent Agent failures into Capability-targeted training Environments), an end-to-end system for environment-specific agent self-improvement. TRACE contrasts successful and failed trajectories to automatically identify lacking capabilities, synthesizes a targeted training environment for each that rewards whether the capability was exercised, and trains a LoRA adapter via RL on each synthetic environment, routing to the relevant adapter at inference. Empirically, TRACE generalizes across different environments, improving over the base agent by +14.1 points on τ^2-bench (customer service) and +7 perfect scores on ToolSandbox (tool use), outperforming the strongest baseline by +7.4 points and +4 perfect scores, respectively. Given the same number of rollouts, TRACE scales more efficiently than baselines, outperforming GRPO and GEPA by +9.2 and +7.4 points on τ^2-bench.

Residual connections are the de facto mechanism for training deep neural networks stably. Geodesic Normalization (GeoNorm) recasts them on a Riemannian manifold, orthogonalizing each layer output against the current hidden state and applying the resulting update through the Riemannian exponential map. Every hidden state thus keeps a constant ell_{2}-norm, confining the residual stream to a hypersphere. The exponential map, however, is only one member of a broad family of retraction maps. We show that on the hypersphere this entire family collapses to a single scalar design choice. What distinguishes one retraction from another is only how the magnitude of an update is converted into a rotation angle within the plane spanned by the hidden state and the update. This view places Euclidean residual connections and GeoNorm in one framework. Instantiating it with the metric projection retraction and the Cayley retraction yields Proj-SpheretNorm and Cay-SpheretNorm, which are exactly norm-preserving yet require only algebraic operations. Both prove to be members of a one-parameter family of angular retractions, p-SpheretNorm, whose rotation angle saturates rather than growing without bound. The two methods above are recovered exactly at p = 1 and p = 2, while the identity map and GeoNorm arise only as limits at either end. On nanoGPT, all three methods outperform existing lightweight deep connection schemes, and the best validation loss is attained at finite p, indicating that the exponential map is not the preferred retraction for spherical residual streams but merely one end of a spectrum.

We study the relation between equal-weight circle quadrature, Gauss--Chebyshev quadrature, and planar Minkowski reconstruction. The regular N-point grid is an mathbb S^1-design of strength N-1; a half-step rotation of the 2M-point grid projects to the classical M-point Gauss--Chebyshev rule. A common family of quadrature-error functionals gives exact identities for integration error, mixed-area error, and support-function error. For Nge3, the centered Minkowski polygon P_N with perimeter 2π has the exact Hausdorff distance from the unit disk B^2 \[ d_H(P_N,B^2)=1-\fracπ{N}\cot\fracπ{N} =π^2{3N^2}+O(N^{-4}). \] With Steiner centering, we obtain d_H(K_μ,K_ν)leπW_1(μ,ν) for balanced, nondegenerate probability measures, where W_1 uses geodesic distance. Its exponent is sharp even for smooth positive densities near the uniform measure. The identity W_1(ν_N,σ)=π/(2N) yields a general O(N^{-1}) bound for the regular grid. The sharper O(N^{-2}) rate follows from the sparse discrepancy spectrum and the inverse multiplier (1-k^2)^{-1}. Finally, a perturbation estimate gives sufficient conditions for preserving the quadratic rate and its leading constant.

Hyperbolic space has proven to be well-suited for capturing hierarchical relations in data, such as trees and directed acyclic graphs. Prior work introduced the concept of entailment cones, which uses partial orders defined by nested cones in the Poincar\'e ball to model hierarchies. Here, we introduce the ``shadow cones" framework, a physics-inspired entailment cone construction. Specifically, we model partial orders as subset relations between shadows formed by a light source and opaque objects in hyperbolic space. The shadow cones framework generalizes entailment cones to a broad class of formulations and hyperbolic space models beyond the Poincar\'e ball. This results in clear advantages over existing constructions: for example, shadow cones possess better optimization properties over constructions limited to the Poincar\'e ball. Our experiments on datasets of various sizes and hierarchical structures show that shadow cones consistently and significantly outperform existing entailment cone constructions. These results indicate that shadow cones are an effective way to model partial orders in hyperbolic space, offering physically intuitive and novel insights about the nature of such structures.

Metric learning aims to learn a highly discriminative model encouraging the embeddings of similar classes to be close in the chosen metrics and pushed apart for dissimilar ones. The common recipe is to use an encoder to extract embeddings and a distance-based loss function to match the representations -- usually, the Euclidean distance is utilized. An emerging interest in learning hyperbolic data embeddings suggests that hyperbolic geometry can be beneficial for natural data. Following this line of work, we propose a new hyperbolic-based model for metric learning. At the core of our method is a vision transformer with output embeddings mapped to hyperbolic space. These embeddings are directly optimized using modified pairwise cross-entropy loss. We evaluate the proposed model with six different formulations on four datasets achieving the new state-of-the-art performance. The source code is available at https://github.com/htdt/hyp_metric.

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

We introduce a neural network (NN) strictly governed by Newton's Law, with the nature required basis functions derived from the fundamental classic mechanics. Then, by classifying the training model as a quick procedure of 'force pattern' recognition, we developed the Newton physics-based NS scheme. Once the force pattern is confirmed, the neuro network simply does the checking of the 'pattern stability' instead of the continuous fitting by computational resource consuming big data-driven processing. In the given physics's law system, once the field is confirmed, the mathematics bases for the force field description actually are not diverged but denumerable, which can save the function representations from the exhaustible available mathematics bases. In this work, we endorsed Newton's Law into the deep learning technology and proposed Newton Scheme (NS). Under NS, the user first identifies the path pattern, like the constant acceleration movement.The object recognition technology first loads mass information, then, the NS finds the matched physical pattern and describe and predict the trajectory of the movements with nearly zero error. We compare the major contribution of this NS with the TCN, GRU and other physics inspired 'FIND-PDE' methods to demonstrate fundamental and extended applications of how the NS works for the free-falling, pendulum and curve soccer balls.The NS methodology provides more opportunity for the future deep learning advances.

Classical convergence guarantees for gradient-based learning in games require the pseudo-gradient to be (strongly) monotone in Euclidean geometry as shown by rosen(1965), a condition that often fails even in simple games with strong cross-player couplings. We introduce Small-Gain Nash (SGN), a block small-gain condition in a custom block-weighted geometry. SGN converts local curvature and cross-player Lipschitz coupling bounds into a tractable certificate of contraction. It constructs a weighted block metric in which the pseudo-gradient becomes strongly monotone on any region where these bounds hold, even when it is non-monotone in the Euclidean sense. The continuous flow is exponentially contracting in this designed geometry, and projected Euler and RK4 discretizations converge under explicit step-size bounds derived from the SGN margin and a local Lipschitz constant. Our analysis reveals a certified ``timescale band'', a non-asymptotic, metric-based certificate that plays a TTUR-like role: rather than forcing asymptotic timescale separation via vanishing, unequal step sizes, SGN identifies a finite band of relative metric weights for which a single-step-size dynamics is provably contractive. We validate the framework on quadratic games where Euclidean monotonicity analysis fails to predict convergence, but SGN successfully certifies it, and extend the construction to mirror/Fisher geometries for entropy-regularized policy gradient in Markov games. The result is an offline certification pipeline that estimates curvature, coupling, and Lipschitz parameters on compact regions, optimizes block weights to enlarge the SGN margin, and returns a structural, computable convergence certificate consisting of a metric, contraction rate, and safe step-sizes for non-monotone games.

GUI agents accumulate high-resolution screenshots as the trajectory unfolds, increasing inference latency and memory usage. Training-free visual token pruning can reduce this cost, but cache reuse introduces a fundamental constraint. Once tokens are discarded, the corresponding visual evidence cannot be recovered without re-encoding. Pruning therefore becomes an irreversible admission decision that must remain useful for unknown future targets while preserving coverage of operable regions under tight budgets. To address these challenges, we propose \method{}, a training-free framework for \textbf{Trajectory-robust Admission and Coverage-aware Evidence ordering}. Specifically, we combine a query-independent layout-derived interaction prior with instruction relevance and feature novelty to rank visual evidence according to both potential future utility and diversity. Then, we reserve part of the budget for native visual tokens distributed across the screen, repairing missing spatial coverage without breaking the ordering. Together, these mechanisms produce a nested token order, allowing retained visual evidence to shrink monotonically across budgets while remaining reusable throughout the trajectory. Finally, our monotone KV contraction incrementally contracts retired frames into compact session state, avoiding repeated visual encoding or pruning. Extensive experiments across six GUI benchmarks and diverse models verify the effectiveness of our proposed under tight budgets. The source code will be released.

Language-conditioned humanoid motion generation enables users to create and refine robot movements through natural language. The central challenge is to connect an underspecified request to concrete motion structure that can subsequently be revised relative to an accepted reference. We present TRACE, a generation and editing framework for the Unitree G1 humanoid robot built around typed, measurable kinematic quantities. Our insight is to ground generation conditions in paired training motions and editing goals in the current motion. A language model learns to predict support, root, and hand events from motion-derived supervision; their time-aligned encodings complement the original text in a diffusion generator. Gravity-referenced states and trajectory--contact objectives improve kinematic consistency. For editing, typed requests define body--time scopes and source-relative goals. A separate reference-conditioned diffusion editor uses these goals and revised events to realize changes, while sampling-time projection preserves protected state coordinates. Quantitative evaluation and human assessment show improved semantic alignment and motion quality over the compared methods. Multi-turn editing evaluations demonstrate more reliable execution of successive relative requests under shared preservation constraints. Code is available at https://github.com/Multimodal-Perception-Action/TRACE.

Recent advances in flow-based offline reinforcement learning (RL) have achieved strong performance by parameterizing policies via flow matching. However, they still face critical trade-offs among expressiveness, optimality, and efficiency. In particular, existing flow policies interpret the L_2 regularization as an upper bound of the 2-Wasserstein distance (W_2), which can be problematic in offline settings. This issue stems from a fundamental geometric mismatch: the behavioral policy manifold is inherently anisotropic, whereas the L_2 (or upper bound of W_2) regularization is isotropic and density-insensitive, leading to systematically misaligned optimization directions. To address this, we revisit offline RL from a geometric perspective and show that policy refinement can be formulated as a local transport map: an initial flow policy augmented by a residual displacement. By analyzing the induced density transformation, we derive a local quadratic approximation of the KL-constrained objective governed by the Fisher information matrix, enabling a tractable anisotropic optimization formulation. By leveraging the score function embedded in the flow velocity, we obtain a corresponding quadratic constraint for efficient optimization. Our results reveal that the optimality gap in prior methods arises from their isotropic approximation. In contrast, our framework achieves a controllable approximation error within a provable neighborhood of the optimal solution. Extensive experiments demonstrate state-of-the-art performance across diverse offline RL benchmarks. See project page: https://github.com/ARC0127/Fisher-Decorator.

Multi-Object Tracking (MOT) plays a critical role in analyzing player behavior from videos, enabling performance evaluation. Current MOT methods are often evaluated using publicly available datasets. However, most of these focus on everyday scenarios such as pedestrian tracking or are tailored to specific sports, including soccer and basketball. Despite the inherent challenges of tracking players in American football, such as frequent occlusion and physical contact, no standardized dataset has been publicly available, making fair comparisons between methods difficult. To address this gap, we constructed the first dedicated detection and tracking dataset for the American football players and conducted a comparative evaluation of various detection and tracking methods. Our results demonstrate that accurate detection and tracking can be achieved even in crowded scenarios. Fine-tuning detection models improved performance over pre-trained models. Furthermore, when these fine-tuned detectors and re-identification models were integrated into tracking systems, we observed notable improvements in tracking accuracy compared to existing approaches. This work thus enables robust detection and tracking of American football players in challenging, high-density scenarios previously underserved by conventional methods.

In the realm of computer vision and graphics, accurately establishing correspondences between geometric 3D shapes is pivotal for applications like object tracking, registration, texture transfer, and statistical shape analysis. Moving beyond traditional hand-crafted and data-driven feature learning methods, we incorporate spectral methods with deep learning, focusing on functional maps (FMs) and optimal transport (OT). Traditional OT-based approaches, often reliant on entropy regularization OT in learning-based framework, face computational challenges due to their quadratic cost. Our key contribution is to employ the sliced Wasserstein distance (SWD) for OT, which is a valid fast optimal transport metric in an unsupervised shape matching framework. This unsupervised framework integrates functional map regularizers with a novel OT-based loss derived from SWD, enhancing feature alignment between shapes treated as discrete probability measures. We also introduce an adaptive refinement process utilizing entropy regularized OT, further refining feature alignments for accurate point-to-point correspondences. Our method demonstrates superior performance in non-rigid shape matching, including near-isometric and non-isometric scenarios, and excels in downstream tasks like segmentation transfer. The empirical results on diverse datasets highlight our framework's effectiveness and generalization capabilities, setting new standards in non-rigid shape matching with efficient OT metrics and an adaptive refinement module.

This paper introduces SFSORT, the world's fastest multi-object tracking system based on experiments conducted on MOT Challenge datasets. To achieve an accurate and computationally efficient tracker, this paper employs a tracking-by-detection method, following the online real-time tracking approach established in prior literature. By introducing a novel cost function called the Bounding Box Similarity Index, this work eliminates the Kalman Filter, leading to reduced computational requirements. Additionally, this paper demonstrates the impact of scene features on enhancing object-track association and improving track post-processing. Using a 2.2 GHz Intel Xeon CPU, the proposed method achieves an HOTA of 61.7\% with a processing speed of 2242 Hz on the MOT17 dataset and an HOTA of 60.9\% with a processing speed of 304 Hz on the MOT20 dataset. The tracker's source code, fine-tuned object detection model, and tutorials are available at https://github.com/gitmehrdad/SFSORT.

Collections of probability distributions arise in a variety of applications ranging from user activity pattern analysis to brain connectomics. In practice these distributions can be defined over diverse domain types including finite intervals, circles, cylinders, spheres, other manifolds, and graphs. This paper introduces an approach for detecting differences between two collections of distributions over such general domains. To this end, we propose the intrinsic slicing construction that yields a novel class of Wasserstein distances on manifolds and graphs. These distances are Hilbert embeddable, allowing us to reduce the distribution collection comparison problem to a more familiar mean testing problem in a Hilbert space. We provide two testing procedures one based on resampling and another on combining p-values from coordinate-wise tests. Our experiments in various synthetic and real data settings show that the resulting tests are powerful and the p-values are well-calibrated.

Finding meaningful representations and distances of hierarchical data is important in many fields. This paper presents a new method for hierarchical data embedding and distance. Our method relies on combining diffusion geometry, a central approach to manifold learning, and hyperbolic geometry. Specifically, using diffusion geometry, we build multi-scale densities on the data, aimed to reveal their hierarchical structure, and then embed them into a product of hyperbolic spaces. We show theoretically that our embedding and distance recover the underlying hierarchical structure. In addition, we demonstrate the efficacy of the proposed method and its advantages compared to existing methods on graph embedding benchmarks and hierarchical datasets.

The field of image generation has made significant progress thanks to the introduction of Diffusion Models, which learn to progressively reverse a given image corruption. Recently, a few studies introduced alternative ways of corrupting images in Diffusion Models, with an emphasis on blurring. However, these studies are purely empirical and it remains unclear what is the optimal procedure for corrupting an image. In this work, we hypothesize that the optimal procedure minimizes the length of the path taken when corrupting an image towards a given final state. We propose the Fisher metric for the path length, measured in the space of probability distributions. We compute the shortest path according to this metric, and we show that it corresponds to a combination of image sharpening, rather than blurring, and noise deblurring. While the corruption was chosen arbitrarily in previous work, our Shortest Path Diffusion (SPD) determines uniquely the entire spatiotemporal structure of the corruption. We show that SPD improves on strong baselines without any hyperparameter tuning, and outperforms all previous Diffusion Models based on image blurring. Furthermore, any small deviation from the shortest path leads to worse performance, suggesting that SPD provides the optimal procedure to corrupt images. Our work sheds new light on observations made in recent works and provides a new approach to improve diffusion models on images and other types of data.

Metric space magnitude, an active field of research in algebraic topology, is a scalar quantity that summarizes the effective number of distinct points that live in a general metric space. The {\em weighting vector} is a closely-related concept that captures, in a nontrivial way, much of the underlying geometry of the original metric space. Recent work has demonstrated that when the metric space is Euclidean, the weighting vector serves as an effective tool for boundary detection. We recast this result and show the weighting vector may be viewed as a solution to a kernelized SVM. As one consequence, we apply this new insight to the task of outlier detection, and we demonstrate performance that is competitive or exceeds performance of state-of-the-art techniques on benchmark data sets. Under mild assumptions, we show the weighting vector, which has computational cost of matrix inversion, can be efficiently approximated in linear time. We show how nearest neighbor methods can approximate solutions to the minimization problems defined by SVMs.

Visibility information is critical in game AI applications, but the computational cost of raycasting-based methods poses a challenge for real-time systems. To address this challenge, we propose a novel method that represents a partitioned game scene as neural Omnidirectional Distance Fields (ODFs), allowing scalable and efficient visibility approximation between positions without raycasting. For each position of interest, we map its omnidirectional distance data from the spherical surface onto a UV plane. We then use multi-resolution grids and bilinearly interpolated features to encode directions. This allows us to use a compact multi-layer perceptron (MLP) to reconstruct the high-frequency directional distance data at these positions, ensuring fast inference speed. We demonstrate the effectiveness of our method through offline experiments and in-game evaluation. For in-game evaluation, we conduct a side-by-side comparison with raycasting-based visibility tests in three different scenes. Using a compact MLP (128 neurons and 2 layers), our method achieves an average cold start speedup of 9.35 times and warm start speedup of 4.8 times across these scenes. In addition, unlike the raycasting-based method, whose evaluation time is affected by the characteristics of the scenes, our method's evaluation time remains constant.

Cross-entropy loss has long been the standard choice for training deep neural networks, yet it suffers from interpretability limitations, unbounded weight growth, and inefficiencies that can contribute to costly training dynamics. The harmonic loss is a distance-based alternative grounded in Euclidean geometry that improves interpretability and mitigates phenomena such as grokking, or delayed generalization on the test set. However, the study of harmonic loss remains narrow: only Euclidean distance is explored, and no systematic evaluation of computational efficiency or sustainability was conducted. We extend harmonic loss by systematically investigating a broad spectrum of distance metrics as replacements for the Euclidean distance. We comprehensively evaluate distance-tailored harmonic losses on both vision backbones and large language models. Our analysis is framed around a three-way evaluation of model performance, interpretability, and sustainability. On vision tasks, cosine distances provide the most favorable trade-off, consistently improving accuracy while lowering carbon emissions, whereas Bray-Curtis and Mahalanobis further enhance interpretability at varying efficiency costs. On language models, cosine-based harmonic losses improve gradient and learning stability, strengthen representation structure, and reduce emissions relative to cross-entropy and Euclidean heads. Our code is available at: https://anonymous.4open.science/r/rethinking-harmonic-loss-5BAB/.

Recent findings suggest that the consecutive layers of ReLU neural networks can be understood geometrically as space folding transformations of the input space, revealing patterns of self-similarity. In this paper, we present the first quantitative analysis of this space folding phenomenon in ReLU neural networks. Our approach focuses on examining how straight paths in the Euclidean input space are mapped to their counterparts in the Hamming activation space. In this process, the convexity of straight lines is generally lost, giving rise to non-convex folding behavior. To quantify this effect, we introduce a novel measure based on range metrics, similar to those used in the study of random walks, and provide the proof for the equivalence of convexity notions between the input and activation spaces. Furthermore, we provide empirical analysis on a geometrical analysis benchmark (CantorNet) as well as an image classification benchmark (MNIST). Our work advances the understanding of the activation space in ReLU neural networks by leveraging the phenomena of geometric folding, providing valuable insights on how these models process input information.

For any integer d geq 2 and prime power q, we construct unexpectedly large induced matchings in the point-line incidence graph of F_{q}^{d} by leveraging a new connection with the Furstenberg-Sárközy problem from arithmetic combinatorics. In particular, we significantly improve the previously well-known baselines when q is prime, showing that F_{q}^{2} contains matchings of size q^{1.233} and F_{q}^{d} contains matchings of size q^{d-o_{d}(1)}. These results and their proofs have several applications. First, we also obtain new constructions for finite field Nikodym sets in dimension d geq 2, improving recent results of Tao by polynomial factors. For example, when q is prime, we show the existence of Nikodym sets in F_q^d of size q^d - q^{d - o_d(1)}. Second, we construct a new minimal blocking set in PG(2,q), solving a longstanding problem in finite geometry. Third, we obtain new constructions for the minimal distance problem (in R^{2} and also in higher dimensions), improving a recent result of Logunov-Zakharov. We also obtain analogous results for general finite fields with large characteristics. In particular, in one of our constructions we introduce a new special set of points inside the norm hypersurface in F_{q}^{d}, which directly generalizes the classical Hermitian unital and which may be of independent interest for applications.

We propose conditional flows of the maximum mean discrepancy (MMD) with the negative distance kernel for posterior sampling and conditional generative modeling. This MMD, which is also known as energy distance, has several advantageous properties like efficient computation via slicing and sorting. We approximate the joint distribution of the ground truth and the observations using discrete Wasserstein gradient flows and establish an error bound for the posterior distributions. Further, we prove that our particle flow is indeed a Wasserstein gradient flow of an appropriate functional. The power of our method is demonstrated by numerical examples including conditional image generation and inverse problems like superresolution, inpainting and computed tomography in low-dose and limited-angle settings.

The increasing availability of spatio-temporal tracking data has created new opportunities for analysing tactical behaviour in football. However, many existing approaches evaluate passes primarily through outcome-based metrics such as scoring probability or possession value, providing limited insight into how passes influence the defensive organisation of the opponent. This paper introduces a structural framework for analysing football passes based on their interaction with defensive structure. Using synchronised tracking/event data, we derive three complementary structural metrics, Line Bypass Score, Space Gain Metric, and Structural Disruption Index, that quantify how passes alter the spatial configuration of defenders. These metrics are combined into a composite measure termed Tactical Impact Value (TIV), which captures the structural influence of individual passes. Using tracking and event data from the 2022 FIFA World Cup, we analyse structural passing behaviour across multiple tactical levels. Unsupervised clustering of structural features reveals four interpretable pass archetypes: circulatory, destabilising, line-breaking, and space-expanding passes. Empirical results show that passes with higher TIV are significantly more likely to lead to territorial progression, particularly entries into the final third and penalty box. Spatial, team-level analyses further reveal distinctive structural passing styles across teams, while player-level analysis highlights the role of build-up defenders as key drivers of structural progression. In addition, analysing passer-receiver interactions identifies structurally impactful passing partnerships that amplify tactical progression within teams. Overall, the proposed framework demonstrates how structural representations derived from tracking data can reveal interpretable tactical patterns in football.

Devising indicative evaluation metrics for the image generation task remains an open problem. The most widely used metric for measuring the similarity between real and generated images has been the Fr\'echet Inception Distance (FID) score. Because it does not differentiate the fidelity and diversity aspects of the generated images, recent papers have introduced variants of precision and recall metrics to diagnose those properties separately. In this paper, we show that even the latest version of the precision and recall metrics are not reliable yet. For example, they fail to detect the match between two identical distributions, they are not robust against outliers, and the evaluation hyperparameters are selected arbitrarily. We propose density and coverage metrics that solve the above issues. We analytically and experimentally show that density and coverage provide more interpretable and reliable signals for practitioners than the existing metrics. Code: https://github.com/clovaai/generative-evaluation-prdc.

This paper develops a hierarchical clustering algorithm for weighted projective spaces P_{q}, utilizing a Finsler metric d_F([z], [w]) and its rational analogue d_{F,Q}([z], [w]) to define distances that preserve the non-Euclidean geometry of these quotient manifolds. Defined via geodesic integrals of a scaling invariant Finsler norm weighted by the grades q = (q_0, q_1, dots, q_n), these metrics satisfy true metric properties including the triangle inequality, overcoming the limitations of the non-metric dissimilarity measure from prior work.

High-fidelity simulations and complex inverse problems, such as detector modeling and unfolding, are computationally intensive bottlenecks across subatomic physics, yet essential for accurate physical interpretation. While Conditional Flow Matching (CFM) offers a robust acceleration approach, we demonstrate its standard training loss is fundamentally misleading. Specifically, utilizing a Jefferson Lab Nuclear Physics (NP) kinematic dataset (γp to ρ^0 p to π^+π^- p), we expose that CFM loss plateaus prematurely, obscuring ongoing physical refinement. To verify this disconnect is a dataset-agnostic pathology, we introduce ScatterPrism, an efficient generative surrogate evaluated against both the NP data and synthetic stress tests modeling challenging 1D distribution topologies. Coupling these benchmarks, we establish that physics-informed metrics continue improving long after standard loss converges. Consequently, we propose a multi-metric diagnostic protocol to ensure true kinematic fidelity without data memorization. Driven by NP challenges relevant to the forthcoming Electron-Ion Collider (EIC), this unified machinery has strong potential to extend to High-Energy Physics (HEP) applications, such as jet modeling. Furthermore, the framework holds promise for broader domains requiring rigorous generative reliability, including medical imaging, astrophysics, and quantitative finance.

We consider a number of graph kernels and proximity measures including commute time kernel, regularized Laplacian kernel, heat kernel, exponential diffusion kernel (also called "communicability"), etc., and the corresponding distances as applied to clustering nodes in random graphs and several well-known datasets. The model of generating random graphs involves edge probabilities for the pairs of nodes that belong to the same class or different predefined classes of nodes. It turns out that in most cases, logarithmic measures (i.e., measures resulting after taking logarithm of the proximities) perform better while distinguishing underlying classes than the "plain" measures. A comparison in terms of reject curves of inter-class and intra-class distances confirms this conclusion. A similar conclusion can be made for several well-known datasets. A possible origin of this effect is that most kernels have a multiplicative nature, while the nature of distances used in cluster algorithms is an additive one (cf. the triangle inequality). The logarithmic transformation is a tool to transform the first nature to the second one. Moreover, some distances corresponding to the logarithmic measures possess a meaningful cutpoint additivity property. In our experiments, the leader is usually the logarithmic Communicability measure. However, we indicate some more complicated cases in which other measures, typically, Communicability and plain Walk, can be the winners.

When dealing with electro or magnetoencephalography records, many supervised prediction tasks are solved by working with covariance matrices to summarize the signals. Learning with these matrices requires using Riemanian geometry to account for their structure. In this paper, we propose a new method to deal with distributions of covariance matrices and demonstrate its computational efficiency on M/EEG multivariate time series. More specifically, we define a Sliced-Wasserstein distance between measures of symmetric positive definite matrices that comes with strong theoretical guarantees. Then, we take advantage of its properties and kernel methods to apply this distance to brain-age prediction from MEG data and compare it to state-of-the-art algorithms based on Riemannian geometry. Finally, we show that it is an efficient surrogate to the Wasserstein distance in domain adaptation for Brain Computer Interface applications.

In this paper, we use Stein's method to obtain optimal bounds, both in Kolmogorov and in Wasserstein distance, in the normal approximation for the empirical distribution of the ground state of a many-interacting-worlds harmonic oscillator proposed by Hall, Deckert, and Wiseman [Phys. Rev. X. (2014)]. Our bounds on the Wasserstein distance solve a conjecture of McKeague and Levin [Ann. Appl. Probab. (2016)].

We present a novel method, called NeTO, for capturing 3D geometry of solid transparent objects from 2D images via volume rendering. Reconstructing transparent objects is a very challenging task, which is ill-suited for general-purpose reconstruction techniques due to the specular light transport phenomena. Although existing refraction-tracing based methods, designed specially for this task, achieve impressive results, they still suffer from unstable optimization and loss of fine details, since the explicit surface representation they adopted is difficult to be optimized, and the self-occlusion problem is ignored for refraction-tracing. In this paper, we propose to leverage implicit Signed Distance Function (SDF) as surface representation, and optimize the SDF field via volume rendering with a self-occlusion aware refractive ray tracing. The implicit representation enables our method to be capable of reconstructing high-quality reconstruction even with a limited set of images, and the self-occlusion aware strategy makes it possible for our method to accurately reconstruct the self-occluded regions. Experiments show that our method achieves faithful reconstruction results and outperforms prior works by a large margin. Visit our project page at https://www.xxlong.site/NeTO/

Evaluating open-ended outputs from large language models (LLMs) remains challenging due to the absence of ground truth. Existing metrics rely on final-answer accuracy or surface-level statistics, leaving the reasoning process itself unexamined. We introduce TRACE (Toulmin-based Reasoning Assessment through Constructive Elements), a metric that analyzes Chain-of-Thought (CoT) reasoning processes. Rather than judging outcomes, TRACE inspects how arguments are constructed by integrating Toulmin's argumentation theory with Flavell's metacognitive framework to assess reasoning structure. Experiments on 26.3K QA samples across 7 reasoning models show strong correlation with benchmark accuracy (r=0.74). Furthermore, TRACE is effective as a reinforcement learning reward signal, outperforming accuracy-only baselines. Together, these results indicate that logically sound reasoning leads to higher-quality answers. TRACE thus serves as a complementary metric for evaluating open-ended outputs. Code is available at https://github.com/hyyangkisti/trace.

Trackers and video generators solve closely related problems: the former analyze motion, while the latter synthesize it. We show that this connection enables pretrained video diffusion models to perform zero-shot point tracking by simply prompting them to visually mark points as they move over time. We place a distinctively colored marker at the query point, then regenerate the rest of the video from an intermediate noise level. This propagates the marker across frames, tracing the point's trajectory. To ensure that the marker remains visible in this counterfactual generation, despite such markers being unlikely in natural videos, we use the unedited initial frame as a negative prompt. Through experiments with multiple image-conditioned video diffusion models, we find that these "emergent" tracks outperform those of prior zero-shot methods and persist through occlusions, often obtaining performance that is competitive with specialized self-supervised models.

We study object motion path editing in videos, where the goal is to alter a target object's trajectory while preserving the original scene content. Unlike prior video editing methods that primarily manipulate appearance or rely on point-track-based trajectory control, which is often challenging for users to provide during inference, especially in videos with camera motion, we offer a practical, easy-to-use approach to controllable object-centric motion editing. We present Trace, a framework that enables users to design the desired trajectory in a single anchor frame and then synthesizes a temporally consistent edited video. Our approach addresses this task with a two-stage pipeline: a cross-view motion transformation module that maps first-frame path design to frame-aligned box trajectories under camera motion, and a motion-conditioned video re-synthesis module that follows these trajectories to regenerate the object while preserving the remaining content of the input video. Experiments on diverse real-world videos show that our method produces more coherent, realistic, and controllable motion edits than recent image-to-video and video-to-video methods.

We verify that persistent observers in causally invariant hypergraph substrates satisfy the conditions of the Conant-Ashby Good Regulator Theorem. Building on Wolfram's hypergraph physics and Vanchurin's neural network cosmology, we formalize persistent observers as entities that minimize prediction error at their boundary with the environment. Applying a modern reformulation of the Conant-Ashby theorem, we demonstrate that hypergraph observers satisfy Good Regulator conditions, requiring them to maintain internal models. Once an internal model with loss function exists, the emergence of a Fisher information metric follows from standard information geometry. Invoking Amari's uniqueness theorem for reparameterization-invariant gradients, we show that natural gradient descent is the unique admissible learning rule. Under the ansatz M=F^2 for exponential family observers and one specific convergence time functional, we derive a closed-form formula for the regime parameter alpha in Vanchurin's Type II framework, with a quantum-classical threshold at kappa(F)=2. However, three alternative convergence models do not reproduce this result, so this prediction is strongly model-dependent. We further introduce the directional regime parameter alpha_{v_k} and the trace-free deviation tensor, showing that a single observer can simultaneously occupy different Vanchurin regimes along different eigendirections of the Fisher metric. This connects Wolfram and Vanchurin frameworks through established theorems, providing approximately 25-30% novel contribution.

Learned simulators for deformable linear objects (DLOs) such as cables have to predict the motion of cables they were not trained on and stay stable over long rollouts. Most of their error occurs where the cable touches itself or the floor. Attention over all pairs of cable segments can represent contact between parts of the cable that are far apart along its length, but attention has no notion of geometry. A cable has two pairwise distances, which agree only while it is straight: the arc-length distance along the cable, which governs elastic forces, and the Euclidean distance in space, which governs contact. We add a physical attention bias, an additive term on the attention logits with a learned rate, and ask which distance it should use. We compare no bias, each distance alone, and both distances on disjoint sets of heads, keeping the rest of the model and the training protocol fixed. A physical bias improves prediction on unseen cables. The gain is largest when attention is the only mechanism that connects distant segments: there, the arc-length bias reduces prediction error by 15% and more than halves the drift in segment length. The Euclidean bias alone stays close to unbiased attention, while assigning both distances across heads is best or near-best on every metric we report. Code and per-run records: https://github.com/avihaig/dlogps.

We introduce a principled way of computing the Wasserstein distance between two distributions in a federated manner. Namely, we show how to estimate the Wasserstein distance between two samples stored and kept on different devices/clients whilst a central entity/server orchestrates the computations (again, without having access to the samples). To achieve this feat, we take advantage of the geometric properties of the Wasserstein distance -- in particular, the triangle inequality -- and that of the associated {\em geodesics}: our algorithm, FedWad (for Federated Wasserstein Distance), iteratively approximates the Wasserstein distance by manipulating and exchanging distributions from the space of geodesics in lieu of the input samples. In addition to establishing the convergence properties of FedWad, we provide empirical results on federated coresets and federate optimal transport dataset distance, that we respectively exploit for building a novel federated model and for boosting performance of popular federated learning algorithms.

We study a space-time Brownian motion with drift B(t)=(t_0+t,y_0+W(t)+t) killed at the moving boundary of the cone {(t,x):0<x<t}. This article determines the parabolic Martin boundary and all harmonic functions associated with this process. To that end, the asymptotics of Green's functions are determined along all directions. We also find the exit probabilities at the edges, the probability of remaining in the cone forever and the laws of the exit point and exit time. From this, we derive an explicit formula for the transition kernel of the process. These results arise from two different methods initially introduced to study random walks. An analytical approach, developed in the 1970s by Malyshev and based on the steepest descent method on a Riemann surface, is used to determine the asymptotics of the Green's functions. A recursive compensation approach, inspired by the method developed in the 1990s by Adan, Wessels and Zijm, is used to determine the harmonic functions.

Deep metric learning objectives (e.g., triplet loss) require storing and comparing high-dimensional embeddings, making the per-batch loss buffer scale as O(Scdot D), where S is the number of samples in a batch and D is the feature dimension, thus limiting training on memory-constrained hardware. We propose Shadow Loss, a proxy-free, parameter-free objective that measures similarity via scalar projections onto the anchor direction, reducing the loss-specific buffer from O(Scdot D) to O(S) while preserving the triplet structure. We analyze gradients, provide a Lipschitz continuity bound, and show that Shadow Loss penalizes trivial collapse for stable optimization. Across fine-grained retrieval (CUB-200, CARS196), large-scale product retrieval (Stanford Online Products, In-Shop Clothes), and standard/medical benchmarks (CIFAR-10/100, Tiny-ImageNet, HAM-10K, ODIR-5K), Shadow Loss consistently outperforms recent objectives (Triplet, Soft-Margin Triplet, Angular Triplet, SoftTriple, Multi-Similarity). It also converges in approx 1.5-2times fewer epochs under identical backbones and mining. Furthermore, it improves representation separability as measured by higher silhouette scores. The design is architecture-agnostic and vectorized for efficient implementation. By decoupling discriminative power from embedding dimensionality and reusing batch dot-products, Shadow Loss enables memory-linear training and faster convergence, making deep metric learning practical on both edge and large-scale systems.

Multi-turn agents solve complex tasks through extended sequences of tool interactions before producing a final answer, making credit assignment a fundamental challenge during post-training. Outcome rewards provide reliable supervision for short-horizon reasoning, but become sparse and high-variance as trajectories grow to tens or hundreds of tool calls. They can also be misleading: a failed rollout may contain many useful actions that move the agent closer to the goal, yet outcome-only training assigns them the same negative advantage as the eventual mistake. We propose TRACE (Turn-level Reward Assignment via Credit Estimation), a dense credit-assignment method for agentic reinforcement learning. TRACE represents rollouts as state transitions at tool-call boundaries, obtains gold-answer log-probabilities from a frozen reference model, transforms them into log-ratio state values, and derives per-action rewards as Temporal-Difference changes in those values. This requires no additional critic or process-label training, and its one-step log-ratio TD component telescopes across redundant tool calls. On long-horizon complex search, TRACE substantially improves base-model tool-use ability using pure RL, without a cold-start supervised fine-tuning stage, an agentic mid-training stage, or training on live-web data. On the closed-web BrowseComp-Plus benchmark, it raises Qwen3-4B from 7.2 to 35.6 and Qwen3-30B-A3B from 8.4 to 42.6. The learned search behavior also transfers to open-web benchmarks, and the learning curves show earlier improvement and faster convergence during RL training.