arXiv subjects
Publications and source records attributed to Xiaobo Liu.
In this paper, we give a complete description for the full isometry groups of a class of right invariant Riemannian metrics on the special unitary group $\mathrm{SU}(2^N)$. These metrics have been used by physicists to study Nielsen's geometric approach for complexities in quantum computations.
Lane-changing is a critical driving maneuver, and understanding its effects on traffic safety and efficiency is essential for effective traffic management and optimization. However, existing studies provide limited means to account for natural traffic dynamics when identifying disturbances associated with lane changes. Moreover, methods for measuring the spatial extent and duration of the impact of a single discretionary lane change, as well as comprehensive metrics for quantifying its overall spatiotemporal impact, remain underdeveloped. To address these gaps, this study proposes a comprehensive analytical framework to evaluate the spatiotemporal impact of a single discretionary lane change on upstream traffic. The framework identifies affected following vehicles and their impact durations from vehicle trajectories before and after the lane change. A Travel Distance Bias indicator is introduced to measure the motion deviation of following vehicles relative to local reference vehicles, while vehicle-specific pre-event fluctuation envelopes and run-length persistence filters are used to reduce the misattribution of background traffic fluctuations to the lane-change event. Based on the affected vehicles and affected intervals, two aggregate indicators, the Total Efficiency Impact Magnitude (TEIM) and the Total Safety Impact Magnitude (TSIM), are developed to quantify efficiency- and safety-related impact magnitudes. Matched no-lane-change controls and method-comparison experiments provide comparative evidence that the affected-vehicle identification procedure reduces, but does not eliminate, false-positive detections caused by background traffic fluctuations.
Circuit complexity for two-dimensional topological quantum field theories (2D TQFT) was defined by Couch, Fan, and Shashi in [12]. In this paper, we study complexity for the 2D TQFT given by quantum cohomology of compact symplectic manifolds. We will estimate the number of states with finite approximate complexity of arbitrarily small tolerance for Fano complete intersections and (co)minuscule homogeneous varieties. We will give an upper bound for the dimension of the space spanned by states with finite complexity. In the case of Gr(2, n), this bound is sharp and we also obtain a precise description for this subspace. For (co)minuscule homogeneous varieties, we prove a positivity result for the eigenvalues of quantum multiplication by the handle element (also called the quantum Euler class) divided by the class of a point.
Infrared-visible image fusion facilitates robust multimodal perception by integrating complementary textural nuances from visible sensors with thermal signatures from infrared systems. Due to the task's inherently ill-posed nature, existing methods heavily rely on structural priors but typically enforce rotation equivariance uniformly across all features. Such a holistic approach overlooks a critical distinction where low-frequency shared structures strictly adhere to equivariant constraints while high-frequency modality-specific details require greater flexibility to preserve unique information. To bridge this gap, we propose RoES, a Rotational Equivariant Selective-frequency fusion network. Instead of employing static decomposition, we introduce a trainable rotation-enhanced updater/predictor module to dynamically decouple low- and high-frequency components. The resulting representations are then processed through a dual-branch fusion module tailored for spectral consistency. Specifically, a rotation-equivariant Mamba is employed to capture long-range structural dependencies in the low-frequency domain, while a polar spectral attention-based Dual-Fourier block refines high-frequency details under explicit low-frequency guidance. Extensive experiments demonstrate that RoES consistently achieves state-of-the-art performance in both fusion quality and downstream object detection, establishing a robust solution for multimodal fusion by reconciling frequency-selective features with equivariant constraints. The source code is available at https://github.com/BryceLosky/RoES-Fusion.
We revisit established Krylov subspace methods for linear combinations of matrix $φ$-function actions from the viewpoint of the block triangular formulation of Al-Mohy and Liu [SIAM J. Sci. Comput., 48 (2026), pp. A726--A747]. In algorithms such as KIOPS [J. Comput. Phys., 372 (2018), pp. 236--255], one uses an augmentation approach based on evaluating the exponential of a slightly larger matrix that contains the operant vectors in its off-diagonal block, and its field of values may therefore grow substantially with these vectors. Typical convergence estimates for Krylov subspace methods result from bounding the error of polynomial approximations for the exponential on the field of values, so that only very pessimistic convergence estimates are available for these methods, in spite of their good practical performance. In contrast, the larger block formulation established by Al-Mohy and Liu involves an operator whose field of values is independent of the operant vectors, leading to more favorable convergence bounds. We work out the details of how these two approaches are connected to each other, which allows us to transfer the convergence bounds from the latter to the former, thus better explaining the observed performance.
Learner agency and autonomy are foundational to personal development, yet a pervasive "jingle-jangle" fallacy (i.e. identical terms denoting different constructs, distinct terms denoting identical ones) has substantially hindered cumulative knowledge. Treating meaning as a phenomenon constituted through use in linguistic practice, we extracted 8,954 definitions and 2,700 scale items from over 14,000 publications, to investigate how researchers actually used learner agency and autonomy with a semantic analysis pipeline. The definitional landscape of two constructs resolves into three dimensions: regulation and control of learning (task), intrinsic motivation and internal decision-making (person), and social-relational action (sociocultural), thereby empirically quantifying the jingle-jangle fallacy. Existing scales, however, systematically underrepresent the sociocultural dimension. Critically, current generative AI research in education concentrates on learning regulation and control, narrowing the behavioral repertoire that AI-mediated learning environments are designed to cultivate. Beyond conceptual clarification, this work carries direct implications for conceptualization, measurement, and practice towards supporting the multidimensional learner agency and autonomy.
Motivated by the increasing availability of low- and mixed-precision arithmetic on modern hardware, we develop mixed-precision variants of Lloyd's algorithm for k-means clustering. The main ingredient is a family of mixed-precision kernels for Euclidean distance computation. These kernels are guided by rounding-error analysis and use a simple reliability test to decide whether the expanded distance formula can be evaluated safely with low precision or a higher-precision correction by the direct distance formula is required. Thus, most distance computations can be carried out with low precision, while high-precision arithmetic is used selectively when cancellation may lead to a loss of accuracy. We evaluate the proposed methods on large-scale distance-computation benchmarks, synthetic clustering problems, and image-segmentation tasks. The experiments verify that the mixed-precision kernels on GPUs can substantially improve performance while retaining the accuracy and convergence behavior of higher-precision baselines. In particular, our CUDA implementations achieve orders-of-magnitude speedups over the CPU implementation in \texttt{scikit-learn} and up to $4\times$ faster than the IEEE double-precision \texttt{cdist} routine of \texttt{PyTorch} on NVIDIA A100 GPU, while providing improved numerical robustness in cancellation-prone regimes. The resulting mixed-precision k-means methods are effective for clustering and image segmentation, although the observed gains depend on the dataset, feature dimension, and number of clusters. These results demonstrate that mixed-precision distance kernels can offer a useful trade-off between performance and accuracy for k-means clustering and suggest that similar ideas may be beneficial for other distance-based machine learning methods.
LiDAR model selection is a critical issue in roadside sensing systems, as it directly determines both perception capability and deployment cost. However, the lack of empirical benchmarks for comparing perception performance across different LiDAR configurations has greatly constrained scientific sensor selection and deployment planning. To address this gap, we present MR-LiDAR, a controlled multi-resolution LiDAR benchmark for roadside perception diagnostics. Using 16-, 32-, 80-, and 128-beam LiDARs in identical roadside scenarios, we collect point clouds and ground-truth annotations for diverse traffic participants, including vehicles and vulnerable road users (VRUs), across varying distances. This controlled design isolates intrinsic LiDAR specifications, particularly beam count and beam distribution, as the key variables for precise performance diagnostics. Based on MR-LiDAR, we conduct systematic empirical analyses to examine how beam count, beam distribution, target distance, object category, and vehicle occlusion affect LiDAR perception performance. The results reveal that all of these factors have substantial impacts. In particular, contrary to the common assumption that higher beam counts always yield better perception, we show that an 80-beam LiDAR with optimized beam distribution can match or even outperform a 128-beam LiDAR with uniform beam distribution. In addition, we provide a practical reference guide for LiDAR selection, including target point-count statistics and detection performance comparisons based on two widely used detection algorithms. This work offers a diagnostic benchmark and practical guidance for determining cost-effective LiDAR configurations in roadside perception applications.
Reduced rank extrapolation (RRE) is an acceleration method typically used to accelerate the iterative solution of nonlinear systems of equations using a fixed-point process. In this context, the iterates are vectors generated from a fixed-point mapping function. However, when considering the iterative solution of large-scale matrix equations, the iterates are low-rank matrices generated from a fixed-point process for which, generally, the mapping function changes in each iteration. To enable acceleration of the iterative solution for these problems, we propose two novel generalizations of RRE. First, we show how to effectively compute RRE for sequences of low-rank matrices. Second, we derive a formulation of RRE that is suitable for fixed-point processes for which the mapping function changes each iteration. We demonstrate the potential of the methods on several numerical examples involving the iterative solution of large-scale Lyapunov and Riccati matrix equations.
We develop a mixed-precision iterative refinement framework for solving low-rank Lyapunov matrix equations $AX + XA^T + W =0$, where $W=LL^T$ or $W=LSL^T$. Via rounding error analysis of the algorithms we derive sufficient conditions for the attainable normwise residuals in different precision settings and show how the algorithmic parameters should be chosen. These conditions are independent of the choice of inner solver, provided that the prescribed residual accuracy is attained in the inner solves. Using the sign-function Newton iteration as the solver, we demonstrate that reduced precisions, such as half precision with unit roundoff $u_s$, can be used efficiently for Lyapunov equations with condition numbers of order $1/u_s$ without compromising the attainable solution quality. This provides an algorithmic framework towards exploiting native low-precision hardware to accelerate Lyapunov solvers without sacrificing accuracy.
MLE-Toolbox is a comprehensive open-source MATLAB toolbox for end-to-end analysis of magnetoencephalography (MEG) and electroencephalography (EEG) data. Inspired by widely used neuroimaging platforms such as Brainstorm and FieldTrip, it integrates the full analysis pipeline within a unified and user-friendly graphical interface (GUI), covering raw data import, preprocessing, source localization, functional connectivity, oscillatory analysis, and machine learning-based classification. The toolbox includes automated artifact rejection methods, including independent component analysis (ICA), signal-space projection (SSP), and signal-space separation (SSS); multiple source localization approaches, including minimum norm estimation (MNE), dynamic statistical parametric mapping (dSPM), standardized low-resolution brain electromagnetic tomography (sLORETA), and beamforming; multi-atlas parcellation with anatomical visualization; spectral power analysis with frequency-band brain mapping; phase-amplitude coupling (PAC); graph-theoretic brain network analysis; and integrated machine learning and deep learning classifiers. MLE-Toolbox also provides native interoperability with Brainstorm, FieldTrip, EEGLAB, and FreeSurfer, allowing researchers to build on established workflows while benefiting from additional automation, interactive visualization, and one-click academic report generation. Freely available for non-commercial use, MLE-Toolbox is designed to lower the barrier to rigorous, reproducible MEG/EEG research.
We consider the solution of the Sylvester equation $AX+XB=C$ in mixed precision. We derive a new iterative refinement scheme to solve perturbed quasi-triangular Sylvester equations; our rounding error analysis provides sufficient conditions for convergence and a bound on the attainable relative residual. We leverage this iterative scheme to solve the general Sylvester equation. The new algorithms compute the Schur decomposition of the coefficient matrices $A$ and $B$ in lower than working precision, use the low-precision Schur factors to obtain an approximate solution to the perturbed quasi-triangular equation, and iteratively refine it to obtain a working-precision solution. In order to solve the original equation to working precision, the unitary Schur factors of the coefficient matrices must be unitary to working precision, but this is not the case if the Schur decomposition is computed in low precision. We propose two effective approaches to address this: one is based on re-orthonormalization in working precision, and the other on explicit inversion of the almost-unitary factors. The two mixed-precision algorithms thus obtained are tested on various Sylvester and Lyapunov equations from the literature. Our numerical experiments show that, for both types of equations, the new algorithms are at least as accurate as existing ones. Our cost analysis, on the other hand, suggests that they would typically be faster than mono-precision alternatives if implemented on hardware that natively supports low precision.
This study adopts an integrated distributed cognition and regulation of learning perspective to examine the collaboration patterns and dynamics of human-AI collaboration when college students collaborating with AI for complex problem-solving. Through cluster analysis, three distinct collaborative problem-solving modes were identified in this study: Delegated Reasoning (DR), Concerted Interpretation (CI), and Delegated Elaboration (DE). This study found that the DR group achieved the highest task performance, significantly outperforming the CI group. Additionally, the semantic similarity between human and AI discourse was notably the highest in the DR group. In contrast, the CI group reported significantly greater use of self-regulation strategies. These findings uncover a critical tension between the efficiency of the distributed system and the depth of human learners regulatory engagement. Insights from this study offer valuable implications for the future design of AI-empowered educational tools and student-AI collaborative learning frameworks.
We investigate the acceleration of stationary iterations for multi-term Sylvester equation by means of reduced rank extrapolation (RRE). Theoretical convergence results and implementations are provided for both small and large-scale problems. For the large-scale problems, an inexact non-stationary iteration is discussed, which makes use of low-rank matrix approximations. Numerical experiments illustrate the potential of the RRE acceleration which often leads to a substantial gain in convergence speed and therefore reducing the consumption of storage and computing time.
Traditional workflow-based agents exhibit limited intelligence when addressing real-world problems requiring tool invocation. Tool-integrated reasoning (TIR) agents capable of autonomous reasoning and tool invocation are rapidly emerging as a powerful approach for complex decision-making tasks involving multi-step interactions with external environments. In this work, we introduce MindWatcher, a TIR agent integrating interleaved thinking and multimodal chain-of-thought (CoT) reasoning. MindWatcher can autonomously decide whether and how to invoke diverse tools and coordinate their use, without relying on human prompts or workflows. The interleaved thinking paradigm enables the model to switch between thinking and tool calling at any intermediate stage, while its multimodal CoT capability allows manipulation of images during reasoning to yield more precise search results. We implement automated data auditing and evaluation pipelines, complemented by manually curated high-quality datasets for training, and we construct a benchmark, called MindWatcher-Evaluate Bench (MWE-Bench), to evaluate its performance. MindWatcher is equipped with a comprehensive suite of auxiliary reasoning tools, enabling it to address broad-domain multimodal problems. A large-scale, high-quality local image retrieval database, covering eight categories including cars, animals, and plants, endows model with robust object recognition despite its small size. Finally, we design a more efficient training infrastructure for MindWatcher, enhancing training speed and hardware utilization. Experiments not only demonstrate that MindWatcher matches or exceeds the performance of larger or more recent models through superior tool invocation, but also uncover critical insights for agent training, such as the genetic inheritance phenomenon in agentic RL.
Mean curvature flows of isoparametric submanifolds in Euclidean spaces and spheres have been studied by Liu and Terng. In particular, it was proved that such flows always have ancient solutions. This is also true for mean curvature flows of isoparametric hypersurfaces in hyperbolic spaces by a result of Reis and Tenenblat. In this paper, we study mean curvature flows of isoparametric submanifolds in hyperbolic spaces with arbitrary codimension. In particular, we will show that they always have ancient solutions and study their limiting behaviors.
A novel approach is presented for computing flexible body dynamics based on conventional structural dynamics models. This approach innovatively captures the rigid body motion component embedded within a flexible body's movement, generates and synthesizes single-step responses in a sequence of reconfigured inertial frames that follow the rigid body motion. By doing so, it effectively bypasses the complexities associated with modeling flexibilities and formulating highly nonlinear coupled motion equations. In addition to improving predictive accuracy, this approach offers valuable insights into the interaction between rigid body motion and structural vibration. By bridging these two aspects, it advances the understanding of flexible body dynamics and delivers a precise, efficient simulation framework for a wide range of engineering applications.
A new scaling and recovering algorithm is proposed for simultaneously computing the matrix $φ$-functions that arise in exponential integrator methods for the numerical solution of certain first-order systems of ordinary differential equations. The algorithm initially scales the input matrix down by a nonnegative integer power of two, and then evaluates the $[m/m]$ diagonal Padé approximant to $φ_p$, where $p$ is the largest index of interest. The remaining $[m+p{-}j/m]$ Padé approximants to $φ_j$, $0 \le j < p$, are obtained implicitly via a recurrence relation. The effect of scaling is subsequently recovered using the double-argument formula. A rigorous backward error analysis, based on the $[m+p/m]$ Padé approximant to the exponential, enables sharp bounds on the relative backward errors. These bounds are expressed in terms of the sequence $\|A^k\|^{1/k}$, which can be much smaller than $\|A\|$ for nonnormal matrices. The scaling parameter and the degrees of the Padé approximants are selected to minimize the overall computational cost, which benefits from the sharp bounds and the optimal evaluation schemes for diagonal Padé approximants. Furthermore, if the input matrix is (quasi-)triangular, the algorithm exploits its structure in the recovering phase. Numerical experiments demonstrate the superiority of the proposed algorithm over existing alternatives in both accuracy and efficiency.