arXiv subjects

Weijia Wang

Publications and source records attributed to Weijia Wang.

At least 37 records · Page 2Linked to original sources

arXiv2025

The Diophantine Frobenius Problem revisited

Let $k\ge 2$ and $a_1, a_2, \cdots, a_k$ be positive integers with \[ \gcd(a_1, a_2, \cdots, a_k)=1. \] It is proved that there exists a positive integer $G_{a_1, a_2, \cdots, a_k}$ such that every integer $n$ strictly greater than it can be represented as the form \[ n=a_1x_1+a_2x_2+\cdots+a_kx_k, \quad (x_1, x_2, \cdots, x_k\in\mathbb{Z}_{\ge 0},~\gcd(x_1, x_2, \cdots, x_k)=1). \] We then investigate the size of $G_{a_1, a_2}$ explicitly. Our result strengthens the primality requirement of $x$'s in the classical Diophantine Frobenius Problem.

math.NT↗

arXiv2025

Solving generic parametric linear matrix inequalities

We consider linear matrix inequalities (LMIs) $A = A_0 + x_1 A_1 + ... + x_n A_n \succeq 0$ with the $A_i$'s being $m \times m$ symmetric matrices, with entries in a ring $\mathcal{R}$. When $\mathcal{R} = \mathbb{R}$, the feasibility problem consists in deciding whether the $x_i$'s can be instantiated to obtain a positive semidefinite matrix. When $\mathcal{R} = \mathbb{Q}[y_1, ... , y_t]$, the problem asks for a formula on the parameters $y_1, ..., y_t$, which describes the values of the parameters for which the specialized LMI is feasible. This problem can be solved using general quantifier elimination algorithms, with a complexity that is exponential in n. In this work, we leverage the LMI structure of the problem to design an algorithm that computes a formula $Φ$ describing a dense subset of the feasible region of parameters, under genericity assumptions. The complexity of this algorithm is exponential in n, m and t but becomes polynomial in $n$ when $m$ and $t$ are fixed. We apply the algorithm to a parametric sum-of-squares problem and to the convergence analyses of certain first-order optimization methods, which are both known to be equivalent to the feasibility of certain parametric LMIs, hence demonstrating its practical interest.

cs.SC↗

arXiv2025

Spacecraft Safe Robust Control Using Implicit Neural Representation for Geometrically Complex Targets in Proximity Operations

This study addresses the challenge of ensuring safe spacecraft proximity operations, focusing on collision avoidance between a chaser spacecraft and a complex-geometry target spacecraft under disturbances. To ensure safety in such scenarios, a safe robust control framework is proposed that leverages implicit neural representations. To handle arbitrary target geometries without explicit modeling, a neural signed distance function (SDF) is learned from point cloud data via a enhanced implicit geometric regularization method, which incorporates an over-apporximation strategy to create a conservative, safety-prioritized boundary. The target's surface is implicitly defined by the zero-level set of the learned neural SDF, while the values and gradients provide critical information for safety controller design. This neural SDF representation underpins a two-layer hierarchcial safe robust control framework: a safe velocity generation layer and a safe robust controller layer. In the first layer, a second-order cone program is formulated to generate safety-guaranteed reference velocity by explicitly incorporating the under-approximation error bound. Furthermore, a circulation inequality is introduced to mitigate the local minimum issues commonly encountered in control barrier function (CBF) methods. The second layer features an integrated disturbance observer and a smooth safety filter explicitly compensating for estimation error, bolstering robustness to external disturbances. Extensive numerical simulations and Monte Carlo analysis validate the proposed framework, demonstrating significantly improved safety margins and avoidance of local minima compared to conventional CBF approaches.

eess.SY↗

arXiv2025

Biclosed sets, quasitrivial semigroups and oriented matroid

In this paper, we establish a one-to-one correspondence between the set of biclosed sets in an irreducible root system of type $A_n$ and the set of quasitrivial semigroup structures on a set with $n+1$ elements. Building on this correspondence, we first generalize this bijection to provide a semigroup structural characterization of the biclosed sets in a standard parabolic subset. In particular, this allows us to derive an enumeration result for the elements in a parabolic weak order of type $A$. Secondly, we define an index for an arbitrary subset of the root system of type $A_n$, which quantifies their deviation from from being biclosed, and prove that such an index coincides with the associativity index of the associated quasitrivial magma. Thirdly, we define type $B_n$ quasitrivial semigroups, and prove that they are in bijective with biclosed sets in a type $B_n$ root system. Finally, by identifying certain biclosed sets with total preorders, we present a purely combinatorial proof that a root system of type $A$ possesses an oriented matroid structure.

math.GR↗

arXiv2024

Order types of reflection orders on affine Weyl groups

In this paper, we investigate the order types of reflection orders on irreducible affine Weyl groups. We show that they are intimately related to the Catalan combinatorics. We explicitly describe all of those order types and show that these order types can be parameterized by deformed Dyck words, which can be enumerated by Catalan numbers.

math.RT↗

arXiv2024

Deep Point Cloud Normal Estimation via Triplet Learning

Normal estimation on 3D point clouds is a fundamental problem in 3D vision and graphics. Current methods often show limited accuracy in predicting normals at sharp features (e.g., edges and corners) and less robustness to noise. In this paper, we propose a novel normal estimation method for point clouds. It consists of two phases: (a) feature encoding which learns representations of local patches, and (b) normal estimation that takes the learned representation as input and regresses the normal vector. We are motivated that local patches on isotropic and anisotropic surfaces have similar or distinct normals, and that separable features or representations can be learned to facilitate normal estimation. To realise this, we first construct triplets of local patches on 3D point cloud data, and design a triplet network with a triplet loss for feature encoding. We then design a simple network with several MLPs and a loss function to regress the normal vector. Despite having a smaller network size compared to most other methods, experimental results show that our method preserves sharp features and achieves better normal estimation results on CAD-like shapes.

cs.CV↗

arXiv2023

On derivatives of zeta and $L$-functions near the 1-line

We study the conditional upper bounds and extreme values of derivatives of the Riemann zeta function and Dirichlet $L$-functions near the 1-line. Let $\ell$ be a fixed natural number. We show that, if $|σ-1|\ll1/\log_2t$, then $|ζ^{(\ell)}(σ+ it)|$ has the same maximal order (up to the leading coefficients) as $|ζ^{(\ell)}(1+ it)|$ when $t\to\infty$. The range $1-σ\ll1/\log_2t$ is wide enough, since we also show that $(1-σ) \log_2t \to \infty\; (t \to \infty)$ implies $\limsup_{t\to\infty}|ζ^{(\ell)}(σ+ it)| / (\log_2t)^{\ell+1} = \infty$. Similar results can be obtained for Dirichlet $L$-functions $L^{(\ell)}(σ,χ)$ with $χ\pmod q$.

math.NT↗

arXiv2023

Large zeta sums

In this article, we investigate the behaviour of values of zeta sums $\sum_{n\le x}n^{it}$ when $t$ is large. We show some asymptotic behaviour and Omega results of zeta sums, which are analogous to previous results of large character sums $\sum_{n\le x}χ(n)$.

math.NT↗

arXiv2023

Photoplethysmography based atrial fibrillation detection: an updated review from July 2019

Atrial fibrillation (AF) is a prevalent cardiac arrhythmia associated with significant health ramifications, including an elevated susceptibility to ischemic stroke, heart disease, and heightened mortality. Photoplethysmography (PPG) has emerged as a promising technology for continuous AF monitoring for its cost-effectiveness and widespread integration into wearable devices. Our team previously conducted an exhaustive review on PPG-based AF detection before June 2019. However, since then, more advanced technologies have emerged in this field. This paper offers a comprehensive review of the latest advancements in PPG-based AF detection, utilizing digital health and artificial intelligence (AI) solutions, within the timeframe spanning from July 2019 to December 2022. Through extensive exploration of scientific databases, we have identified 59 pertinent studies. Our comprehensive review encompasses an in-depth assessment of the statistical methodologies, traditional machine learning techniques, and deep learning approaches employed in these studies. In addition, we address the challenges encountered in the domain of PPG-based AF detection. Furthermore, we maintain a dedicated website to curate the latest research in this area, with regular updates on a regular basis.

eess.SP↗

arXiv2023

Adaptive Reduced-Attitude Control for Spacecraft Boresight Alignment with Safety Constraints and Accuracy Requirements

This paper investigates the boresight alignment control problem under safety constraints and performance requirements, involving pointing-forbidden constraint, attitude angular velocity limitation, and pointing accuracy requirement. Meanwhile, the parameter uncertainty issue is taken into account simultaneously. To address this problem, we propose a modified composite framework integrating the Artificial Potential Field (APF) methodology and the Prescribed Performance Control (PPC) scheme. The APF scheme ensures safety, while the PPC scheme is employed to realize an accuracy-guaranteed control. A Switched Prescribed Performance Function (SPPF) is proposed to facilitate the integration, which monitors various constraints and further establishes compatibility between safety and performance concerns by leveraging a special PPC freezing mechanism. To further address the parameter uncertainty, we introduce the Immersion-and-Invariance (I\&I) adaptive control technique to derive an adaptive APF-PPC composite controller, further guaranteeing the closed-loop system's asymptotic convergence. Finally, numerical simulations are carried out to validate the effectiveness of the proposed scheme.

eess.SY↗

arXiv2023

Structure of large quadratic character sums

In this article, we study the distribution of large quadratic character sums. Based on the recent work of Lamzouri~\cite{La2022}, we obtain the structure results of quadratic characters with large character sums.

math.NT↗

arXiv2023

A Practical Recipe for Federated Learning Under Statistical Heterogeneity Experimental Design

Federated Learning (FL) has been an area of active research in recent years. There have been numerous studies in FL to make it more successful in the presence of data heterogeneity. However, despite the existence of many publications, the state of progress in the field is unknown. Many of the works use inconsistent experimental settings and there are no comprehensive studies on the effect of FL-specific experimental variables on the results and practical insights for a more comparable and consistent FL experimental setup. Furthermore, the existence of several benchmarks and confounding variables has further complicated the issue of inconsistency and ambiguity. In this work, we present the first comprehensive study on the effect of FL-specific experimental variables in relation to each other and performance results, bringing several insights and recommendations for designing a meaningful and well-incentivized FL experimental setup. We further aid the community by releasing FedZoo-Bench, an open-source library based on PyTorch with pre-implementation of 22 state-of-the-art methods, and a broad set of standardized and customizable features available at https://github.com/MMorafah/FedZoo-Bench. We also provide a comprehensive comparison of several state-of-the-art (SOTA) methods to better understand the current state of the field and existing limitations.

cs.LG↗

arXiv2023

Safety Guaranteed Control for Spacecraft Inspection Mission

This paper investigates the safety guaranteed problem in spacecraft inspection missions, considering multiple position obstacles and logical attitude forbidden zones. In order to address this issue, we propose a control strategy based on control barrier functions, summarized as "safety check on kinematics" and "velocity tracking on dynamics" approach. The proposed approach employs control barrier functions to describe the obstacles and to generate safe velocities via the solution of a quadratic programming problem. Subsequently, we design a proportional-like controller based on the generated velocity, which, despite its simplicity, can ensure safety even in the presence of velocity tracking errors. The stability and safety of the system are rigorously analyzed in this paper. Furthermore, to account for model uncertainties and external disturbances, we incorporate an immersion and invariance-based disturbance observer in our design. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed control strategy.

eess.SY↗

arXiv2023

Adaptive Compatible Performance Control for Spacecraft Attitude Control under Motion Constraints with Guaranteed Accuracy

This paper focuses on the problem of spacecraft attitude control in the presence of time-varying parameter uncertainties and multiple constraints, accounting for angular velocity limitation, performance requirements, and input saturation. To tackle this problem, we propose a modified framework called Compatible Performance Control (CPC), which integrates the Prescribed Performance Control (PPC) scheme with a contradiction detection and alleviation strategy. Firstly, by introducing the Zeroing Barrier Function (ZBF) concept, we propose a detection strategy to yield judgment on the compatibility between the angular velocity constraint and the performance envelope constraint. Subsequently, we propose a projection operator-governed dynamical system with a varying upper bound to generate an appropriate bounded performance envelope-modification signal if a contradiction exists, thereby alleviating the contradiction and promoting compatibility within the system. Next, a dynamical filter technique is introduced to construct a bounded reference velocity signal to address the angular velocity limitation. Furthermore, we employ a time-varying gain technique to address the challenge posed by time-varying parameter uncertainties, further developing an adaptive strategy that exhibits robustness on disturbance rejection. By utilizing the proposed CPC scheme and time-varying gain adaptive strategy, we construct an adaptive CPC controller, which guarantees the ultimate boundedness of the system, and all constraints are satisfied simultaneously during the whole control process. Finally, numerical simulation results are presented to show the effectiveness of the proposed framework.

eess.SY↗

arXiv2023

Composite Triggered Intermittent Control for Constrained Spacecraft Attitude Tracking

This paper focuses on the spacecraft attitude control problem with intermittent actuator activation, taking into account the attitude rotation rate limitation and input saturation issue simultaneously. To address this problem, we first propose a composite event-trigger mechanism, which composed of two state-dependent trigger that governing the activation and deactivation of actuators. Subsequently, by introducing the cascaded decomposition of Backstepping control philosophy, the designed trigger mechanism is then applied to the decomposed dynamical subsystem, providing a layered intermittent stabilization strategy. Further, the basic intermittent attitude controller is extended to a "constrained version" by introducing a strictly bounded virtual control law and an input saturation compensation auxiliary system. By analyzing the local boundedness of the system on each inter-event time interval, a uniformly, strictly decreasing upper boundary of the lumped system is further characterized, thereby completing the proof of the system's uniformly ultimately boundedness (UUB). Finally, numerical simulation results are illustrated to demonstrate the effectiveness of the proposed scheme.

eess.SY↗

arXiv2023

Weighted Point Cloud Normal Estimation

Existing normal estimation methods for point clouds are often less robust to severe noise and complex geometric structures. Also, they usually ignore the contributions of different neighbouring points during normal estimation, which leads to less accurate results. In this paper, we introduce a weighted normal estimation method for 3D point cloud data. We innovate in two key points: 1) we develop a novel weighted normal regression technique that predicts point-wise weights from local point patches and use them for robust, feature-preserving normal regression; 2) we propose to conduct contrastive learning between point patches and the corresponding ground-truth normals of the patches' central points as a pre-training process to facilitate normal regression. Comprehensive experiments demonstrate that our method can robustly handle noisy and complex point clouds, achieving state-of-the-art performance on both synthetic and real-world datasets.

cs.CV↗

arXiv2023

Segmentation-Driven Feature-Preserving Mesh Denoising

Feature-preserving mesh denoising has received noticeable attention in visual media, with the aim of recovering high-fidelity, clean mesh shapes from the ones that are contaminated by noise. Existing denoising methods often design smaller weights for anisotropic surfaces and larger weights for isotropic surfaces in order to preserve sharp features, such as edges or corners, on the mesh shapes. However, they often disregard the fact that such small weights on anisotropic surfaces still pose negative impacts on the denoising outcomes and detail preservation results on the shapes. In this paper, we propose a novel segmentation-driven mesh denoising method which performs region-wise denoising, and thus avoids the disturbance of anisotropic neighbour faces for better feature preservation results. Also, our backbone can be easily embedded into commonly-used mesh denoising frameworks. Extensive experiments have demonstrated that our method can enhance the denoising results on a wide range of synthetic and real mesh models, both quantitatively and visually.

cs.GR↗

arXiv2022

When Do Curricula Work in Federated Learning?

An oft-cited open problem of federated learning is the existence of data heterogeneity at the clients. One pathway to understanding the drastic accuracy drop in federated learning is by scrutinizing the behavior of the clients' deep models on data with different levels of "difficulty", which has been left unaddressed. In this paper, we investigate a different and rarely studied dimension of FL: ordered learning. Specifically, we aim to investigate how ordered learning principles can contribute to alleviating the heterogeneity effects in FL. We present theoretical analysis and conduct extensive empirical studies on the efficacy of orderings spanning three kinds of learning: curriculum, anti-curriculum, and random curriculum. We find that curriculum learning largely alleviates non-IIDness. Interestingly, the more disparate the data distributions across clients the more they benefit from ordered learning. We provide analysis explaining this phenomenon, specifically indicating how curriculum training appears to make the objective landscape progressively less convex, suggesting fast converging iterations at the beginning of the training procedure. We derive quantitative results of convergence for both convex and nonconvex objectives by modeling the curriculum training on federated devices as local SGD with locally biased stochastic gradients. Also, inspired by ordered learning, we propose a novel client selection technique that benefits from the real-world disparity in the clients. Our proposed approach to client selection has a synergic effect when applied together with ordered learning in FL.

cs.LG↗