arXiv subjects

Haoran Guan

Publications and source records attributed to Haoran Guan.

7 recordsLinked to original sources

arXiv2026

RCLUPPr: a new randomized CholeskyQR with LU preconditioning

In this work, we present the comprehensive rounding error analysis of RCLUPPr proposed in \cite{RCLUPP}, which is a novel randomized CholeskyQR-type algorithm performing LU decomposition with partial pivoting (LUPP decomposition) directly on the tall-skinny $X\in\mathbb{R}^{m\times n}$ with $m \ge n$ and $\mbox{rank}(X)=n$. In contrast to the existing RCLUPP in \cite{RCLUPP}, which applies matrix sketching before LUPP decomposition, RCLUPPr places LUPP decomposition as a preconditioning step first, significantly reducing error propagation. Our analysis rigorously proves that RCLUPPr enjoys markedly better applicability to the ill-conditioned matrices than the existing CholeskyQR-type algorithms and remains stable and accurate in the mixed-precision arithmetic. We further propose practical acceleration strategies in the real implementations of RCLUPPr. Extensive numerical experiments on the real-world problems confirm the theoretical results in this work, demonstrating the robustness and practicality of RCLUPPr in the single, double, and the mixed-precision architecture.

math.NA↗

arXiv2025

Analysis of randomized CholeskyQR for sparse matrices

This work is about rounding error analysis of randomized CholeskyQR-type algorithms for sparse matrices. We often encounter QR factorization of the sparse matrices in many real problems. In this work, we focus on some typical CholeskyQR-type algorithms with matrix sketching, which is a popular randomized technique in recent years. We build a new model of the sparse matrices and provide rounding error analysis of randomized CholeskyQR-type algorithms for the sparse cases with this model. We make comparison between the bounds with different models of sparsity both theoretically and experimentally. Numerical experiments show some new phenomena of randomized CholeskyQR-type algorithms for the sparse cases, which do not occur in the common sparse cases. We also test the applicability, accuracy, efficiency and robustness of randomized CholeskyQR-type algorithms for sparse matrices.

math.NA↗

arXiv2025

Shifted CholeskyQR for sparse matrices

In this work, we focus on Shifted CholeskyQR (SCholeskyQR) for sparse matrices. We provide a new shifted item $s$ for Shifted CholeskyQR3 (SCholeskyQR3) based on the number of non-zero elements (nnze) and the element with the largest absolute value of the input sparse $X \in \mathbb{R}^{m\times n}$ with $m \ge n$. We do rounding error analysis of SCholeskyQR3 with such an $s$ and show that SCholeskyQR3 is accurate in this case. Therefore, an alternative choice of $s$ can be taken for SCholeskyQR3 with the comparison between our new $s$ and the $s$ shown in the previous work when the input $X$ is sparse, improving the applicability and residual of the algorithm for the ill-conditioned cases. Numerical experiments demonstrate the advantage of SCholeskyQR3 with our alternative choice of $s$ in both applicablity and accuracy over the case with the original $s$, together with the same level of efficiency. This work is also the first to build connections between sparsity and numerical algorithms with detailed rounding error analysis to the best of our knowledge.

math.NA↗

arXiv2025

An improved error analysis of CholeskyQR with the randomized model

This work is about an improved error analysis of CholeskyQR with the randomized model for the tall-skinny $X \in \mathbb{R}^{m\times n}$. Due to the structure of CholeskyQR, we utilize the randomized model in the first step of CholeskyQR with a weak assumption. We receive a better sufficient condition of $κ_{2}(X)$ and a tighter upper bound of residual for CholeskyQR2, together with a probabilistic shifted item $s$ for Shifted CholeskyQR3 based on $\norm{X}_{F}$ after improved error analysis. Numerical experiments demonstrate the effectiveness of our new theoretical results. The probabilistic $s$ for Shifted CholeskyQR3 can enhance the applicability of Shifted CholeskyQR3 while maintaining numerical stability. It is also robust enough after numerous experiments.

math.NA↗

arXiv2025

An improved Shifted CholeskyQR based on columns

Among all the deterministic CholeskyQR-type algorithms, Shifted CholeskyQR3 is specifically designed to address the QR factorization of ill-conditioned matrices. This algorithm introduces a shift parameter $s$ to prevent failure during the initial Cholesky factorization step, making the choice of this parameter critical for the algorithm's effectiveness. Our goal is to identify a smaller $s$ compared to the traditional selection based on $\norm{X}_{2}$. In this research, we propose a new definition for the input matrix $X$ called $[X]_{g}$, which is based on the column properties of $X$. $[X]_{g}$ allows us to obtain a reduced shift parameter $s$ for the Shifted CholeskyQR3 algorithm, thereby improving the sufficient condition of $κ_{2}(X)$ for this method. We provide rigorous proofs of orthogonality and residuals for the improved algorithm using our proposed $s$. Numerical experiments confirm the enhanced numerical stability of orthogonality and residuals with the reduced $s$. We find that Shifted CholeskyQR3 can effectively handle ill-conditioned $X$ with a larger $κ_{2}(X)$ when using our reduced $s$ compared to the original $s$. Furthermore, we compare CPU times with other algorithms to assess performance improvements.

math.NA↗

arXiv2024

A new randomized CholeskyQR based on LU decomposition with partial pivoting

CholeskyQR has received considerable attention in recent years for its efficiency and simplicity in computing QR decomposition of the tall-skinny $X \in \mathbb{R}^{m\times n}$ with $m \ge n$ and $\mbox{rank}(X)=n$. Leveraging matrix sketching from randomized linear algebra, randomized CholeskyQR (RCholeskyQR) has been proposed to accelerate the computation by reducing the dimension of the problems. In this work, we propose RCLUPP, a new randomized CholeskyQR-type algorithm based on LU decomposition with partial pivoting (LUP decomposition). By taking LUP decomposition and the thin HouseholderQR on the sketched matrix, RCLUPP significantly improves the applicability and efficiency compared with LU-CholeskyQR2 (LC2). We present a rigorous rounding error analysis of RCLUPP, with a sharper bound of residual compared with those in the existing works. Comparative studies demonstrate that RCLUPP outperforms CholeskyQR2, Shifted CholeskyQR3 (SCholeskyQR3), and LC2 in terms of applicability while maintaining competitive accuracy and efficiency. A variant, RCLUPPr, performs LUP decomposition directly on $X \in \mathbb{R}^{m\times n}$, offering exceptional robustness and numerical stability for the ill-conditioned scenarios, which exceeds that of RCLUPP and RCholeskyQR. Numerical experiments on the synthetic and real-world matrices validate the theoretical results.

math.NA↗

arXiv2023

Self-Inspection Method of Unmanned Aerial Vehicles in Power Plants Using Deep Q-Network Reinforcement Learning

For the purpose of inspecting power plants, autonomous robots can be built using reinforcement learning techniques. The method replicates the environment and employs a simple reinforcement learning (RL) algorithm. This strategy might be applied in several sectors, including the electricity generation sector. A pre-trained model with perception, planning, and action is suggested by the research. To address optimization problems, such as the Unmanned Aerial Vehicle (UAV) navigation problem, Deep Q-network (DQN), a reinforcement learning-based framework that Deepmind launched in 2015, incorporates both deep learning and Q-learning. To overcome problems with current procedures, the research proposes a power plant inspection system incorporating UAV autonomous navigation and DQN reinforcement learning. These training processes set reward functions with reference to states and consider both internal and external effect factors, which distinguishes them from other reinforcement learning training techniques now in use. The key components of the reinforcement learning segment of the technique, for instance, introduce states such as the simulation of a wind field, the battery charge level of an unmanned aerial vehicle, the height the UAV reached, etc. The trained model makes it more likely that the inspection strategy will be applied in practice by enabling the UAV to move around on its own in difficult environments. The average score of the model converges to 9,000. The trained model allowed the UAV to make the fewest number of rotations necessary to go to the target point.

cs.RO↗