ABSTRACT Quaternion numbers are commonly used to represent color image pixels. Since color image pixels are nonnegative, nonnegativity constraint should be incorporated into the components of quaternion numbers. However, existing quaternion‐based matrix factorization models do not fully satisfy nonnegativity constraints. The main purpose of this paper is to propose and develop nonnegative quaternion matrix factorization models to address the relevant color image processing problems. Our main idea is to employ the product of reduced biquaternion (RB) matrices in the factorization such that nonnegativity constraint can be incorporated properly. The proposed nonnegative RB matrix factorization (NRBMF) model is solved by an alternating minimization algorithm, which alternately solves the RB nonnegative least squares problem of latent features and encoding factors. The theoretical convergence of the alternating minimization algorithm is also established. Experimental results on color face recognition demonstrate that the proposed NRBMF model achieves superior recognition performance compared with existing competing models.
ABSTRACT State‐of‐the‐art iterative refinement (IR)‐based methods mainly employ LU factorization to construct preconditioners, which severely limits their applicability to sparse linear systems due to excessive memory overhead. In order to address such issues, we exploit incomplete LU ( ILU ) factorization as a preconditioning technique for the mixed‐precision GMRES‐IR to develop a new method named as ILU ‐GMRES‐IR. A theoretical error bound of the residual matrix associated with ILU factors is established, which, combined with experimental observations, provides a reference for selecting appropriate ILU parameters to balance memory overhead and convergence rate. In addition, a bound analysis is made for the condition number of the preconditioned matrix to characterize the effectiveness of ILU preconditioning. In comparison with the popular LU ‐GMRES‐IR method, it is verified from numerical experiments that our ILU ‐GMRES‐IR provides a tunable trade‐off between memory consumption and convergence performance, allowing substantial memory savings while retaining satisfactory convergence.
This work presents a hybrid numerical approach for solving linear systems arising from the discretization of the two-dimensional parametric Helmholtz equation. A convolutional neural network based on the U-Net architecture is trained in an self-supervised manner (i.e., without explicitly generating reference solutions) to approximate the inverse of the discretized Helmholtz operator, using a loss function involving the residual norm of the linear system. The trained network is used as a nonlinear preconditioner within the flexible GMRES (FGMRES) algorithm. Numerical experiments show that while the neural network is not accurate enough to act as a standalone solver, it significantly improves the convergence of FGMRES when employed as a preconditioner. The neural preconditioner demonstrates robust performance and generalization capabilities with respect to variations in the velocity field and the domain size. Comparisons with algebraic preconditioners, based on LU factorizations of approximations of the original system matrices, indicate the superior efficiency of the neural approach under equivalent conditions. We believe that the proposed method is not tied to a specific neural architecture and can be extended to other parametric PDEs.
We address the problem of recovering highly oscillatory operators, represented as matrices with a fixed set of observed entries. Given that these matrices can be well compressed by butterfly matrix decomposition of L = & Oscr; ( log n ) levels requiring only degrees of freedom, we propose a novel reformulation of the butterfly structure as a compact tensor network. Specifically, we reshape the input matrix as an order dense tensor, and cast its butterfly decomposition as a tensor network consisting of order dense tensors. This enables efficient utilization of the existing software infrastructure for dense and sparse tensor computations. Next, we propose several tensor completion algorithms based on the tensor reformulation of butterfly format, and compare them against algorithms using the quantized tensor train (QTT) format. These algorithms leverage popular completion methods such as alternating least squares, gradient-based optimization, and alternating directions fitting. Moreover, we propose a novel strategy that uses low-rank matrix completion to efficiently generate an initial guess for the proposed algorithms. To demonstrate the efficiency and applicability of our proposed algorithms, we perform three numerical experiments using simulated oscillatory operators in seismic applications. In these experiments, we use & Oscr; ( n log n ) observed entries in the input matrix and demonstrate an & Oscr; ( n n ) computational cost of the proposed algorithms, leading to speedups of orders of magnitude and significant accuracy improvements for large matrices compared to state-of-the-art low-rank matrix and QTT tensor completion algorithms.
We consider a class of collocation matrices associated with the Newton basis of the space of polynomials of degree at most , evaluated at a set of nodes. In the most general setting, we allow of these nodes to either coincide with or differ from those defining the Newton basis. We establish necessary and sufficient conditions for to be strictly totally positive (STP), based solely on an appropriate ordering of the nodes. We derive explicit formulas for the bidiagonal decomposition of and present an efficient algorithm for computing it to high relative accuracy (HRA). This factorization enables accurate and fast computation of solutions to linear systems with coefficient matrix , as well as the evaluation of its eigenvalues, singular values, and inverse, all to HRA. Furthermore, we apply our framework to the least squares approximation problem in the Newton basis. We show that the STP property of guarantees both uniqueness of the solution and, under mild conditions, its computation to an accuracy comparable to HRA. Extensive numerical experiments confirm the theoretical properties and demonstrate the superior accuracy and performance of the proposed algorithms when compared to standard numerical methods, especially in the presence of severe ill-conditioning.
