Gradient-flow analyses show that simplified linear transformers can learn the in-context linear-regression algorithm, but they do not explain the finite-step behavior of gradient descent at large learning rates. Motivated by empirical work on high-learning-rate transformer instabilities and by the cubic-map phase diagram for quadratic regression, we study an exactly reducible one-prompt linear-transformer training problem. After normalization, the dynamics reduce to a two-factor product map with an effective step-size parameter \(μ\). On the balanced slice, this map recovers the known scalar cubic transition from monotone convergence to catapult convergence, periodic and chaotic bounded nonconvergence, and divergence. We then analyze the full two-dimensional system and show that, for \(0<μ<2\), it has an explicit invariant Chebyshev ellipse separating forward-invariant regions; this ellipse carries off-balanced chaotic dynamics but is transversely repelling, while balanced scalar attractors can be transversely attracting. These results show that large constant learning rates can change the training attractor of the learned transformer rather than merely accelerating convergence: beyond sharp stability thresholds, finite-step training may settle into cycles, bounded chaos, or divergence instead of a single in-context linear-regression solution. We also discuss the consequences for mini-batch gradient descent based training methods.

Eye tracking has become increasingly important in virtual and augmented reality applications; however, the current gaze accuracy falls short of meeting the requirements for spatial computing. We designed a gaze collection framework and utilized high-precision equipment to gather the first precise benchmark dataset, GazeTrack, encompassing diverse ethnicities, ages, and visual acuity conditions for pupil localization and gaze tracking. We propose a novel shape error regularization method to constrain pupil ellipse fitting and train on open-source datasets, enhancing semantic segmentation and pupil position prediction accuracy. Additionally, we invent a novel coordinate transformation method similar to paper unfolding to accurately predict gaze vectors on the GazeTrack dataset. Finally, we built a gaze vector generation model that achieves reduced gaze angle error with lower computational complexity compared to other methods.Please refer to our project page for more details: https://github.com/---(please

Abstract--Circular targets are widely used in LiDAR-camera extrinsic calibration due to their geometric consistency and ease of detection. However, achieving accurate 3D-2D circular center correspondence remains challenging. Existing methods often fail due to decoupled 3D fitting and erroneous 2D ellipse-center estimation. T o address this, we propose a geometrically principled framework featuring two innovations: (i) a robust 3D circle center estimator based on conformal geometric algebra and RANSAC; and (ii) a chord-length variance minimization method to recover the true 2D projected center, resolving its dual-minima ambiguity via homography validation or a quasi-RANSAC fallback. Evaluated on synthetic and real-world datasets, our framework significantly outperforms state-of-the-art approaches. It reduces extrinsic estimation error and enables robust calibration across diverse sensors and target types, including natural circular objects. Our code will be publicly released for reproducibility. USING Light Detection and Ranging (LiDAR) and camera data is critical for robust perception in autonomous systems. This approach combines LiDAR's precise 3D geometry with the rich photometric detail from 2D imagery.

Interactive applications demand believable characters that respond naturally to dynamic environments. Traditional character animation techniques often struggle to handle arbitrary situations, leading to a growing trend of dynamically selecting motion-captured animations based on predefined features. While Motion Matching has proven effective for locomotion by aligning to target trajectories, animating environment interactions and crowd behaviors remains challenging due to the need to consider surrounding elements. Existing approaches often involve manual setup or lack the naturalism of motion capture. Furthermore, in crowd animation, body animation is frequently treated as a separate process from trajectory planning, leading to inconsistencies between body pose and root motion. To address these limitations, we present Environment-aware Motion Matching, a novel real-time system for full-body character animation that dynamically adapts to obstacles and other agents, emphasizing the bidirectional relationship between pose and trajectory. In a preprocessing step, we extract shape, pose, and trajectory features from a motion capture database. At runtime, we perform an efficient search that matches user input and current pose while penalizing collisions with a dynamic environment. Our method allows characters to naturally adjust their pose and trajectory to navigate crowded scenes.

Conjunction analysis and maneuver planning for spacecraft collision avoidance remains a manual and time-consuming process, typically involving repeated forward simulations of hand-designed maneuvers. With the growing density of satellites in low-Earth orbit (LEO), autonomy is becoming essential for efficiently evaluating and mitigating collisions. In this work, we present an algorithm to design low-thrust collision-avoidance maneuvers for short-term conjunction events. We first formulate the problem as a nonconvex quadratically-constrained quadratic program (QCQP), which we then relax into a convex semidefinite program (SDP) using Shor's relaxation. We demonstrate empirically that the relaxation is tight, which enables the recovery of globally optimal solutions to the original nonconvex problem. Our formulation produces a minimum-energy solution while ensuring a desired probability of collision at the time of closest approach. Finally, if the desired probability of collision cannot be satisfied, we relax this constraint into a penalty, yielding a minimum-risk solution. We validate our algorithm with a high-fidelity simulation of a satellite conjunction in low-Earth orbit with a simulated conjunction data message (CDM), demonstrating its effectiveness in reducing collision risk.

Achieving persistent tracking of multiple dynamic targets over a large spatial area poses significant challenges for a single-robot system with constrained sensing capabilities. As the robot moves to track different targets, the ones outside the field of view accumulate uncertainty, making them progressively harder to track. An effective path planning algorithm must manage uncertainty over a long horizon and account for the risk of permanently losing track of targets that remain unseen for too long. However, most existing approaches rely on short planning horizons and assume small, bounded environments, resulting in poor tracking performance and target loss in large-scale scenarios. In this paper, we present a hierarchical planner for tracking multiple moving targets with an aerial vehicle. To address the challenge of tracking non-static targets, our method incorporates motion models and uncertainty propagation during path execution, allowing for more informed decision-making. We decompose the multi-target tracking task into sub-tasks of single target search and detection, and our proposed pipeline consists a novel low-level coverage planner that enables searching for a target in an evolving belief area, and an estimation method to assess the likelihood of success for each sub-task, making it possible to convert the active target tracking task to a Markov decision process (MDP) that we solve with a tree-based algorithm to determine the sequence of sub-tasks. We validate our approach in simulation, demonstrating its effectiveness compared to existing planners for active target tracking tasks, and our proposed planner outperforms existing approaches, achieving a reduction of 11-70% in final uncertainty across different environments.

The ubiquity of closed-weight language models with public-facing APIs has generated interest in forensic methods, both for extracting hidden model details (e.g., parameters) and for identifying models by their outputs. One successful approach to these goals has been to exploit the geometric constraints imposed by the language model architecture and parameters. In this work, we show that a lesser-known geometric constraint--namely, that language model outputs lie on the surface of a high-dimensional ellipse--functions as a signature for the model and can be used to identify the source model of a given output. This ellipse signature has unique properties that distinguish it from existing model-output association methods like language model fingerprints. In particular, the signature is hard to forge: without direct access to model parameters, it is practically infeasible to produce log-probabilities (logprobs) on the ellipse. Secondly, the signature is naturally occurring, since all language models have these elliptical constraints. Thirdly, the signature is self-contained, in that it is detectable without access to the model inputs or the full weights. Finally, the signature is compact and redundant, as it is independently detectable in each logprob output from the model. We evaluate a novel technique for extracting the ellipse from small models and discuss the practical hurdles that make it infeasible for production-scale models. Finally, we use ellipse signatures to propose a protocol for language model output verification, analogous to cryptographic symmetric-key message authentication systems.

This paper presents the classification of a general quadric into an axisymmetric quadric (AQ) and the solution to the problem of the proximity of a given point to an AQ. The problem of proximity in $R^3$ is reduced to the same in $R^2$, which is not found in the literature. A new method to solve the problem in $R^2$ is used based on the geometrical properties of the conics, such as sub-normal, length of the semi-major axis, eccentricity, slope and radius. Furthermore, the problem in $R^2$ is categorised into two and three more sub-cases for parabola and ellipse/hyperbola, respectively, depending on the location of the point, which is a novel approach as per the authors' knowledge. The proposed method is suitable for implementation in a common programming language, such as C and proved to be faster than a commercial library, namely, Bullet.

We introduce CSAR, an algorithm for inducing morphemes from emergent language corpora of parallel utterances and meanings. It is a greedy algorithm that (1) weights morphemes based on mutual information between forms and meanings, (2) selects the highest-weighted pair, (3) removes it from the corpus, and (4) repeats the process to induce further morphemes (i.e., Count, Select, Ablate, Repeat). The effectiveness of CSAR is first validated on procedurally generated datasets and compared against baselines for related tasks. Second, we validate CSAR's performance on human language data to show that the algorithm makes reasonable predictions in adjacent domains. Finally, we analyze a handful of emergent languages, quantifying linguistic characteristics like degree of synonymy and polysemy.

The Fast Multipole Method (FMM) is an efficient numerical algorithm for computation of long-ranged forces in $N$-body problems within gravitational and electrostatic fields. This method utilizes multipole expansions of the Green's function inherent to the underlying dynamical systems. Despite its widespread application in physics and engineering, the integration of FMM with modern machine learning architectures remains underexplored. In this work, we propose a novel neural network architecture, the Neural FMM, that integrates the information flow of the FMM into a hierarchical machine learning framework for learning the Green's operator of an Elliptic PDE. Our Neural FMM architecture leverages a hierarchical computation flow of the FMM method to split up the local and far-field interactions and efficiently learn their respective representations.