recover edge orientations

Existing directional statistics tools are seldom adopted in engineering and computer science due to terminological barriers and lack of practical interfaces for modeling orientation data—such as angles, unit vectors, rotation matrices, and quaternions—in applications ranging from robotics to 3D vision. Method: We introduce the first comprehensive, practitioner-oriented reference guide for probability distributions over multi-degree-of-freedom orientation domains (1D–3D), employing a unified, engineering-friendly notation. The guide systematically presents density functions, maximum-likelihood parameter estimation procedures, and inverse-transform or rejection-sampling algorithms for six canonical directional distributions. Contribution/Results: We release an open-source Python library (built on NumPy/SciPy) supporting distribution fitting and random sampling. Empirical validation on robot pose calibration and 3D point cloud normal estimation demonstrates its practical efficacy, substantially bridging the gap between theoretical directional statistics and real-world engineering deployment.

Existing topological methods struggle to jointly model directed (e.g., water flow) and undirected (e.g., pipe diameter) edge signals, while failing to distinguish the intrinsic directionality of edges themselves. To address this, we propose a direction-aware edge signal modeling paradigm that formally defines and simultaneously satisfies both direction equivariance and direction invariance—establishing the first principled framework for unified edge-level representation learning. Based on this, we introduce EIGN, the first general-purpose edge-level topological graph neural network, equipped with an algebraic-topology-inspired, direction-aware edge-level graph shift operator that rigorously preserves theoretical guarantees. Extensive experiments across traffic, hydrological, and power systems demonstrate consistent superiority over state-of-the-art methods: EIGN achieves up to 23.5% reduction in RMSE for flow simulation tasks.

This paper addresses the unsolved problem of holistic 3D shape orientation estimation—jointly predicting the side, up, and forward axes to achieve canonical coordinate alignment. We propose the first two-stage deep learning framework trained and evaluated on the full ShapeNet dataset. Methodologically, we theoretically analyze orientation ambiguities arising from rotational symmetries and circumvent them via geometric invariance modeling and axis-decoupled regression. Our contributions are threefold: (1) the first systematic solution to ambiguity in joint three-axis estimation; (2) state-of-the-art performance on up-axis prediction; and (3) significantly improved holistic orientation accuracy and cross-category generalization, establishing a robust foundation for 3D shape standardization and alignment.

本文研究了通过随机游走方法在动态边定向问题中维持小的最大出度和更新时间的问题，该方法从树扩展到了外平面图及其他类型图。

The C-Orientation problem seeks to orient an undirected phylogenetic network into a directed network belonging to a specified class (e.g., tree-child networks) to support visual modeling of evolutionary relationships. It is particularly critical for undirected graphs produced by distance-based methods such as Neighbor-Net, yet its computational complexity remained unresolved, and no practical algorithms existed. This paper presents the first fixed-parameter tractable (FPT) exact algorithm for arbitrary network classes, parameterized by hybridization number and fundamental cycle size. Additionally, we propose an efficient heuristic for tree-child orientation, leveraging cycle decomposition and strategic vertex placement to drastically reduce the search space. Experiments demonstrate that our FPT algorithm significantly outperforms existing exponential-time approaches; meanwhile, the heuristic achieves both high speed and accuracy for instances with low-to-moderate hybridization numbers, exhibiting clear biological applicability.

This work proposes a novel paradigm that unifies the entire statistical inference pipeline through a probabilistic language, aiming to coherently bridge observed data, inferential targets, and real-world decision-making. By treating probability and stochastic processes as a central “translation language,” the framework integrates tools from probability measures, likelihood theory, weak convergence, empirical processes, functional data analysis, M- and Z-estimation, kernel methods, and event-time processes into a common syntax. This synthesis connects classical theoretical foundations with modern data structures and practical applications. The approach is validated through historical and biomedical case studies, demonstrating its capacity to provide systematic modeling pathways for complex data while substantially enhancing inferential stability and predictive performance.

This study addresses the challenge of accurately modeling how sensory conditions influence directional errors in spatial orientation, particularly when covariates comprise a mixture of continuous and categorical variables. To this end, the authors propose a novel nonparametric circular regression framework that integrates product kernel estimation to handle mixed-type covariates and introduces a bootstrap-based bandwidth selection criterion tailored to the cosine loss function inherent to circular responses. This approach extends nonparametric circular regression to mixed-covariate settings for the first time and constructs simultaneous confidence bands to quantify estimation uncertainty. Evaluations on both simulated and real-world data—including participants with blindness, low vision, and normal vision—demonstrate the method’s superior bias-variance trade-off, its robustness in uncovering nonlinear patterns of directional error across sensory conditions, and its reliability for statistical inference.

This study addresses the graph edge orientation problem, which involves assigning directions to edges such that each vertex satisfies a prescribed local out-degree constraint. By modeling the problem as a restricted SAT instance—where each edge variable appears in exactly two vertex constraints—the authors establish a complete computational complexity dichotomy for symmetric vertex types parameterized by required in-degree. They precisely characterize the boundary between polynomial-time solvability and NP-completeness on both planar and non-planar graphs. As a consequence, they resolve the long-standing open problem of KPlumber, presenting a new polynomial-time algorithm for it. Furthermore, their framework simplifies the existing NP-hardness proof for triomino tiling and provides the first proof of NP-completeness for tetromino tiling.

This study addresses the challenge of distinguishing directional asymmetry from tail-ratio deviations in multivariate distributions by proposing a quantile-based projection diagnostic framework that avoids reliance on higher-order moments. The method integrates directional skewness and tail-ratio measures through one-dimensional projections, sparse rank-one computations, and directional search to robustly classify heavy-tailed multivariate distributions into four categories: symmetric baseline tails, symmetric tail deviations, skewed baseline tails, and skewed tail deviations. Theoretical analysis establishes population-level properties, finite-sample uniform bounds, and classifier consistency, while revealing the complementary roles of coordinate and random directions in high dimensions, thereby offering a reliable foundation for multivariate modeling choices.

Existing nonparametric statistical testing methods based on surrogate data are primarily designed for undirected graphs and are ill-suited for directed graph structures. This work extends such approaches to the directed graph setting for the first time by defining wide-sense stationary signals through the eigendecomposition of graph shift operators and constructing a surrogate signal generation framework that preserves the covariance structure. Evaluated on real-world data, the proposed method significantly outperforms conventional undirected-graph approaches and naive permutation strategies, offering enhanced statistical power while maintaining test validity and practical feasibility.