arXiv subjects

Xiaobo Liu

Publications and source records attributed to Xiaobo Liu.

At least 19 recordsLinked to original sources

arXiv2026

Reduced and Mixed-Precision Block Householder QR Factorization

Recent advances in hardware architecture have highlighted the importance of reduced-precision arithmetic for improving performance and reducing energy consumption in large-scale scientific applications. However, uniformly low-precision methods can suffer from numerical instability, especially in fundamental linear algebra routines. Mixed-precision strategies therefore play an important role in balancing efficiency, stability, and accuracy. In this paper, we focus on the block Householder QR decomposition using the WY representation and study its behavior in reduced- and mixed-precision settings. Starting from a uniform half-precision variant, we show that performing norm computations with fp32 accumulation is necessary to avoid overflow and underflow on large matrices. Based on this observation, we introduce mpWYQR, a mixed-precision algorithm that increases the precision of only selected computations and avoids most of the precision conversions required by the existing mixed-precision blocked QR algorithm (mpBQR3) introduced in [L. M. Yang, A. Fox, and G. Sanders, SIAM J. Sci. Comp., 43.3 (2021), pp. A1723--A1753]. Our analysis shows that mpWYQR accuracy is governed by low-precision unit roundoff, and that its error bound exceeds that of mpBQR3 by only a constant factor, with identical heuristic bounds. In experiments across a range of matrix sizes and condition numbers, we show that mpWYQR retains mpBQR3 accuracy while running up to $18\times$ faster and using less memory. We also discuss the performance cost of computing inner products in higher precision, and the library support needed for an efficient GPU implementation.

math.NA↗

arXiv2026

MR-LiDAR: A Multi-Resolution Roadside LiDAR Benchmark for Perception Diagnostics and Deployment Guidance

LiDAR configuration selection in roadside sensing remains constrained by limited controlled evidence across sensors. To address this gap, we present MR-LiDAR, a controlled cross-configuration benchmark comparing four LiDAR configurations from 16 to 128 beams on dynamic vehicle and vulnerable road user (VRU) targets. Its cross-configuration main set covers matched daytime, nighttime, and vehicle-occlusion conditions, while three deployment and operating condition subsets examine mounting height, target speed, and curved-road geometry. By reducing confounding from scene and trajectory variation, MR-LiDAR links sensor configuration to target point acquisition and algorithmic detection outcomes. We characterize distance-dependent point loss, define effective beam count as the number of vertical beams intersecting the target's angular extent, and introduce a point-based Occlusion Ratio. The results show that 16- and 32-beam configurations degrade earlier with distance, while VRUs exhibit greater point sparsity than motor vehicles. Nominal beam count alone is insufficient to explain roadside scanning capability. In the cross-configuration main set, the 80- and 128-beam configurations achieve comparable results, consistent with their nearly identical downward beam layouts. A uniform 80-beam counterfactual yields only about one third of the effective beam count of the actual non-uniform configuration. Illumination and the tested speed range have little influence on target-point-count patterns, whereas mounting height and near-field road geometry can substantially alter configuration ordering. Finally, an empirical minimal-sufficient configuration matrix combines frame-level geometric sufficiency with matched-box quality and includes threshold sensitivity analysis, providing bounded configuration guidance for the tested conditions.

cs.RO↗

arXiv2026

Isoparametric Submanifolds in Products of Space Forms

In this paper, we obtain a classification of isoparametric submanifolds with arbitrary codimensions in products of simply-connected space forms, except for isoparametric hypersurfaces in $\mathbb{S}^n \times \mathbb{S}^m$ and $\mathbb{H}^n \times\mathbb{H}^m$.

math.DG↗

arXiv2026

Isometry Groups of Right Invariant Metrics in Geometric Quantum Complexity

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.

math.DG↗

arXiv2026

On the quantification of the spatiotemporal impact of a single discretionary lane change on upstream traffic safety and efficiency

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.

math.NA↗

arXiv2026

Complexity of quantum cohomology

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.

math.AG↗

arXiv2026

RoES: Rotational Equivariant Selective-frequency Fusion for Multimodal Images

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.

cs.CV↗

arXiv2026

Field-of-values analysis of augmented Krylov methods for matrix $φ$-function actions

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.

math.NA↗

arXiv2026

Large-scale semantic mapping of learner agency and autonomy reveals what measurement and generative AI research overlook

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.

cs.AI↗

arXiv2026

Computing k-means in mixed precision

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.

math.NA↗

arXiv2026

Generalizing Reduced Rank Extrapolation to Low-Rank Matrix Sequences

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.

math.NA↗

arXiv2026

Mixed-precision iterative refinement for low-rank Lyapunov 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.

math.NA↗

arXiv2026

MLE-Toolbox: An Open-Source Toolbox for Comprehensive EEG and MEG Data Analysis

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.

q-bio.NC↗

arXiv2026

Mixed-precision algorithms for solving the Sylvester matrix equation

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.

math.NA↗

arXiv2026

Unpacking Interaction Profiles and Strategies in Human-AI Collaborative Problem Solving: A Cognitive Distribution and Regulation Perspective

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.

cs.HC↗

arXiv2026

Reduced rank extrapolation for multi-term Sylvester equations

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.

math.NA↗

arXiv2026

MindWatcher: Toward Smarter Multimodal Tool-Integrated Reasoning

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.

cs.AI↗

arXiv2025

Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces

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.

math.DG↗