Yuehua Feng

Yuehua Feng, Jianwei Xiao, Ming Gu

The Bunch-Kaufman algorithm and Aasen's algorithm are two of the most widely used methods for solving symmetric indefinite linear systems, yet they both are known to suffer from occasional numerical instability due to potentially exponential element growth or unbounded entries in the matrix factorization. In this work, we develop a randomized complete pivoting (RCP) algorithm for solving symmetric indefinite linear systems. RCP is comparable to the Bunch-Kaufman algorithm and Aasen's algorithm in computational efficiency, yet enjoys theoretical element growth and bounded entries in the factorization comparable to that of complete-pivoting, up to a theoretical failure probability that exponentially decays with an oversampling parameter. Our finite precision analysis shows that RCP is as numerically stable as Gaussian elimination with complete pivoting, and RCP has been observed to be numerically stable in our extensive numerical experiments.

Yuehua Feng, Linzhang Lu

Aasen's algorithm factorizes a symmetric indefinite matrix $A$ as $A = P^TLTL^TP$, where $P$ is a permutation matrix, $L$ is unit lower triangular with its first column being the first column of the identity matrix, and $T$ is tridiagonal. In this note, we provide a growth factor upper bound for Aasen's algorithm which is much smaller than that given by Higham. We also show that the upper bound we have given is not tight when the matrix dimension is greater than or equal to $6$.

Hongxia Li, Ying Ji, Yongxin Dong et al.

Low-rank sparse regression models have been widely adopted in face recognition due to their robustness against occlusion and illumination variations. However, existing methods often suffer from insufficient feature representation and limited modeling of structured corruption across samples. To address these issues, this paper proposes a Hybrid second-order gradient Histogram based Global Low-Rank Sparse Regression (H2H-GLRSR) model. First, we propose the Histogram of Oriented Hessian (HOH) to capture second-order geometric characteristics such as curvature and ridge patterns. By fusing HOH and first-order gradient histograms, we construct a unified local descriptor, termed the Hybrid second-order gradient Histogram (H2H), which enhances structural discriminability under challenging conditions. Subsequently, the H2H features are incorporated into an extended version of the Sparse Regularized Nuclear Norm based Matrix Regression (SR\_NMR) model, where a global low-rank constraint is imposed on the residual matrix to exploit cross-sample correlations in structured noise. The resulting H2H-GLRSR model achieves superior discrimination and robustness. Experimental results on benchmark datasets demonstrate that the proposed method significantly outperforms state-of-the-art regression-based classifiers in both recognition accuracy and computational efficiency.