Approximation Of Matrices Research Articles

Overapproximating Reachable Tubes of Linear Time-Varying Systems

We present a method to over-approximate reachable tubes over compact time-intervals, for linear continuous-time, time-varying control systems whose initial states and inputs are subject to compact convex uncertainty. The method uses numerical approximations of transition matrices, is convergent of first order, and assumes the ability to compute with compact convex sets in finite dimension. We also present a variant that applies to the case of zonotopic uncertainties, uses only linear algebraic operations, and yields zonotopic over-approximations. The performance of the latter variant is demonstrated on an example.

Bayesian Robust Tensor Factorization for Angle Estimation in Bistatic MIMO Radar With Unknown Spatially Colored Noise

In this paper, we propose a robust tensor-based scheme for joint direction-of-departure (DOD) and direction-of-arrival (DOA) estimation in bistatic multiple-input multiple-output (MIMO) radar with unknown spatially colored noise. To eliminate the colored noise, the proposed algorithm first denoises the received signal from the temporal cross-correlation approach and constructs a third-order complex tensor model. Then, a real-valued compressed tensor model is developed by utilizing the characteristic of radar antenna arrays. Finally, Bayesian tensor factorization is derived to fit the constructed real-valued model and approximate factor matrices, from which DOD and DOA can be extracted. We also derive Cramér-Rao bound (CRB) results for angle estimation. The proposed algorithm is suitable for both uniform linear array (ULA) and uniform planar array (UPA) manifolds. Compared with existing angle estimation methods, the proposed one has better estimation accuracy and more stable performance. In addition, the proposed algorithm works well even if the number of targets is unknown. Simulation results demonstrate the effectiveness and improvement of the proposed angle estimation algorithm.

A New Algorithm for Reducing Dimensionality of L1-CSVM Use Augmented Lagrange Method

Principal component analysis and generalized low rank approximation of matrices are two different dimensionality reduction methods. Two different dimensionality reduction algorithms are applied to the L1-CSVM model based on augmented Lagrange method to explore the variation of running time and accuracy of the model in dimensionality reduction space. The results show that the improved algorithm can greatly reduce the running time and improve the accuracy of the algorithm.

Fast approximation of orthogonal matrices and application to PCA

Orthogonal projections are a standard technique of dimensionality reduction in machine learning applications. We study the problem of approximating orthogonal matrices so that their application is numerically fast and yet accurate. We find an approximation by solving an optimization problem over a set of structured matrices, that we call extended orthogonal Givens transformations, including Givens rotations as a special case. We propose an efficient greedy algorithm to solve such a problem and show that it strikes a balance between approximation accuracy and speed of computation. The approach is relevant to spectral methods and we illustrate its application to PCA.

Stabilization of a class of underactuated Euler Lagrange system using an approximate model

The energy shaping method, Controlled Lagrangian, is a well-known approach to stabilize the underactuated Euler Lagrange (EL) systems. In this approach, to construct a control rule, some nonlinear and nonhomogeneous partial differential equations (PDEs), which are called matching conditions, must be solved. In this paper, a method is proposed to obtain an approximate solution of these matching conditions for a class of underactuated EL systems. To develop this method, the potential energy matching condition is transformed to a set of linear PDEs using an approximation of inertia matrices. Hence, the assignable potential energy function and the controlled inertia matrix both are constructed as a common solution of these PDEs. Subsequently, the gyroscopic and dissipative forces are determined as the solution for kinetic energy matching condition. Conclusively, the control rule is constructed by adding energy shaping rule and additional dissipation injection to provide asymptotic stability. The stability analysis of the closed-loop system which used the control rule derived with the proposed method is also provided. To demonstrate the success of the proposed method, the stability problem of the inverted pendulum on a cart is considered.

Optimal Rank‐1 Hankel Approximation in the Spectral Norm for Matrices with Multiple Largest Eigenvalue

AbstractWe extend a result from [2, 3] on the optimal rank‐1 Hankel approximation of real symmetric matrices with isolated largest eigenvalue to the case where the largest eigenvalue is not isolated. To illustrate our findings, we give an example where the optimal rank‐1 Hankel approximation is easily obtained and one where it does not exist.

Low rank approximation of positive semi-definite symmetric matrices using Gaussian elimination and volume sampling

Positive semi-definite matrices commonly occur as normal matrices of least squares problems in statistics or as kernel matrices in machine learning and approximation theory. They are typically large and dense. Thus algorithms to solve systems with such a matrix can be very costly. A core idea to reduce computational complexity is to approximate the matrix by one with a low rank. The optimal and well understood choice is based on the eigenvalue decomposition of the matrix. Unfortunately, this is computationally very expensive. Cheaper methods are based on Gaussian elimination but they require pivoting. We show how invariant matrix theory provides explicit error formulas for an averaged error based on volume sampling. The formula leads to ratios of elementary symmetric polynomials on the eigenvalues. We discuss several bounds for the expected norm of the approximation error and include examples where this expected error norm can be computed exactly.

Joint nonnegative matrix factorization and network embedding for graph co-clustering

Graph co-clustering aims to simultaneously group heterogeneous vertices in bipartite networks. The current algorithms measure similarity of vertices by either topology or latent feature of networks, which is insufficient to fully characterize the structure of bipartite graphs. To overcome this problem, we propose a novel co-clustering algorithm by jointly integrating network embedding and NMF (called NENMF) based on the fact that graph representation learning implicitly implies matrix factorizations, where multiple views of bipartite networks are integrated for graph co-clustering. Specifically, the equivalence between nonnegative matrix factorization (NMF) graph embedding for co-clustering is proven, which serves as the theoretical foundation for the proposed algorithm. Then, two auxiliary graphs are generated to fully characterize the topology structure of bipartite networks. Finally, NENMF jointly learns low-rank approximation matrices for bipartite networks and network embedding of auxiliary graphs, where network embedding is regularized into objective function of NMF. The main advantage of the proposed algorithm is to boost the accuracy by combining the low-dimensional approximation and graph representation of bipartite networks without increasing time complexity. The experimental results demonstrate that NENMF outperforms state-of-the-art approaches in terms of accuracy.

Averaged Adaptive Cross Approximation for Structured Matrices

A modified form of the adaptive cross approximation (ACA) is considered for certain structured matrices for which the original ACA fails to provide controllably accurate matrix representations of low-rank submatrices. The modified ACA algorithm considered here is based on forming an averaged matrix, obtained via left- and right- multiplication of the original matrix block by sparse matrices that have LU factorizations. The performance of the modified ACA algorithm is examined for several sub-matrices for which the original ACA fails to provide controllably accurate representations, and significantly improved error control is observed.

Adaptive Cross Approximation for E-field-Guided Noninvasive Magnetic Brain Stimulation

Transcranial magnetic stimulation (TMS) is a noninvasive method that uses coils placed on the scalp to stimulate brain regions via magnetic induction. TMS is known to stimulate regions that are exposed to a large E-field magnitude. As a result, computational E-field dosimetry is increasingly being used to inform TMS administration. Existing methods for computational E-field dosimetry require substantial run-times that limit their applicability for many practical applications. We have developed a computational method based on adaptive cross approximations of matrices for fast computational E-field dosimetry of TMS. These methods will enable the development of improved TMS paradigms.

Optimal rank-1 Hankel approximation of matrices: Frobenius norm and spectral norm and Cadzow's algorithm

We characterize optimal rank-1 matrix approximations with Hankel or Toeplitz structure with regard to two different norms, the Frobenius norm and the spectral norm, in a new way. More precisely, we show that these rank-1 matrix approximation problems can be solved by maximizing special rational functions. Our approach enables us to show that the optimal solutions with respect to these two norms have completely different structure and only coincide in the trivial case when the singular value decomposition already provides an optimal rank-1 approximation with the desired Hankel or Toeplitz structure. We also prove that the Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case. However, it usually does not converge to the optimal solution, neither with regard to the Frobenius norm nor the spectral norm.

PaTux: An Authoring Tool for Level Design through Pattern Customisation Using Non-Negative Matrix Factorization

We present a demonstration of PaTux, an authoring tool for designing levels in SuperTux game through combining patterns. PaTux allows game designers to specify the design of their levels using patterns extracted from training level samples. The Non-negative Matrix Factorisation (NMF) method is utilised to approximate pattern and weight matrices from the training data. The patterns are visualised for designers to choose from and the changes made on the level structure are visualised in realtime. The designer can also specify the weight of each pattern permitting exploration of a wider variety. The data used to train the model can also be specified by the designer resulting in learning a new set of patterns. The system also suggests variations for a given design. When the designer is satisfied with the design, the system allows loading the resultant level in the game to be played.

A simple and efficient pseudo-inverse approximation for real symmetric matrices and applications to elasticity problems

This study proposes an efficient approximation method for the Moore–Penrose pseudo-inverse when, for a matrix, the eigenvectors associated to the eigenvalues zero (referred to herein as zeros eigenvectors) are known in advance. The method reduces the computational cost by several orders of magnitude. The approximation is performed by the addition of a small-amplitude diagonal matrix to regularise the matrix and multiplication with a projection matrix after its regular inversion. The projection removes the components of the zeros eigenvectors. The condition for obtaining a good approximation, is that the amplitude of the small-amplitude matrix should be sufficiently smaller than the smallest non-zero eigenvalue of the matrix. When the matrix is a stiffness matrix in a support-free elasticity problem (a problem whereby the elastic body is unsupported), the zeros eigenvectors indicate rigid-body motions. The method was applied to robust support-free topology optimization, revealing its excellent accuracy and efficiency. The observed computational time was found to be proportional to the size of the stiffness matrix. Furthermore, conducting robust topology optimization for fine-mesh problems resulted in structures that exhibited biological features.

Some algorithms for maximum volume and cross approximation of symmetric semidefinite matrices

Various applications in numerical linear algebra and computer science are related to selecting the rtimes r submatrix of maximum volume contained in a given matrix Ain mathbb R^{ntimes n}. We propose a new greedy algorithm of cost mathcal O(n), for the case A symmetric positive semidefinite (SPSD) and we discuss its extension to related optimization problems such as the maximum ratio of volumes. In the second part of the paper we prove that any SPSD matrix admits a cross approximation built on a principal submatrix whose approximation error is bounded by (r+1) times the error of the best rank r approximation in the nuclear norm. In the spirit of recent work by Cortinovis and Kressner we derive some deterministic algorithms, which are capable to retrieve a quasi optimal cross approximation with cost mathcal O(n^3).

QuicK-means: accelerating inference for K-means by learning fast transforms

K-means—and the celebrated Lloyd’s algorithm—is more than the clustering method it was originally designed to be. It has indeed proven pivotal to help increase the speed of many machine learning, data analysis techniques such as indexing, nearest-neighbor search and prediction, data compression and, lately, inference with kernel machines. Here, we introduce an efficient extension of K-means, dubbed QuicK-means, that rests on the idea of expressing the matrix of the \(K\) cluster centroids as a product of sparse matrices, a feat made possible by recent results devoted to find approximations of matrices as a product of sparse factors. Using such a decomposition squashes the complexity of the matrix-vector product between the factorized \(K\times D\) centroid matrix \({\mathbf {U}}\) and any vector from \({\mathcal {O}}\left( KD\right)\) to \({\mathcal {O}}\left( A \log B~ +B\right)\), with \(A=\min \left( K,D\right)\) and \(B=\max \left( K,D\right)\), where \(D\) is the dimension of the data. This drastic computational saving has a direct impact in the assignment process of a point to a cluster. We propose to learn such a factorization during the Lloyd’s training procedure. We show that resorting to a factorization step at each iteration does not impair the convergence of the optimization scheme, and demonstrate the benefits of our approach experimentally.

Recent advances in multi-frequency analysis with the boundary element method

The boundary element discretization of the acoustic Helmholtz equation yields frequency-dependent linear systems, which require efficient solution strategies for multi-frequency analysis. We briefly review recent advances in this field covering frequency approximation and extraction of coefficient matrices, frequency sampling strategies, iterative solvers for frequency-dependent linear systems and model order reduction techniques.

Hybrid precoding for mmWave massive MU-MIMO systems with overlapped subarray: A modified GLRAM approach

Abstract Mmillimeter-wave (mmWave) massive multiuser multiple-input multiple-output (MU-MIMO) system usually utilizes hybrid precoding to reduce complex hardware and high energy consumption. However, most hybrid precoding scheme consider the fully connected architecture, which may result in a significant loss of energy efficiency. In this paper, we propose a hybrid precoding scheme based on the overlapped subarray architecture (OSA), where the analog part of the MU-MIMO system is realized by a OSA network. Based on the OSA architecture, a modified generalized low rank approximation of matrices (GLRAM) based hybrid precoding scheme is designed to maximize the achievable sum-rate, where the analog precoding algorithm apply the phase alignment technology and GLRAM approach to obtain RF precoder/combiner with high array gain provided by massive MIMO system. Then, the block diagonalization (BD) technology is performed to obtain the digital precoding matrix for the multiplexing gain. Simulation results show that the modified GLRAM based hybrid precoding algorithm for OSA architecture can reduce the number of iterations. Compared with the existing hybrid precoding algorithm, it achieves higher spectral efficiency and lower complexity.

A new implicit-explicit local differential method for boundary value problems

In this study, an effective numerical method based on Taylor expansions is presented for boundary value problems. This method is arbitrary directional and called as implicit-explicit local differential transform method (IELDTM). With the completion of this study, a reliable numerical method is derived by optimizing the required degrees of freedom. It is shown that the order refinement procedure of the IELDTM does not affect the degrees of freedom. A priori error analysis of the current method is constructed and order conditions are presented in a detailed analysis. The theoretical order expectations are verified for nonlinear BVPs. Stability of the IELDTM is investigated by following the analysis of approximation matrices. To illustrate efficiency of the method, qualitative and quantitative results are presented for various challenging BVPs. It is tested that the current method is reliable and accurate for a broad range of problems even for strongly nonlinear and singularly perturbed BVPs. The produced results have revealed that the IELDTM is more accurate than the existing ones in literature.

Cell‐by‐cell approximate Schur complement technique in preconditioning of meshfree discretized piezoelectric equations

AbstractThe radial point interpolation meshfree discretization is a very efficient numerical framework for the analysis of piezoelectricity, in which the fundamental electrostatic equations governing piezoelectric media are solved without mesh generation. Due to the mechanical‐electrical coupling property and the piezoelectric constant, the discrete linear system is sparse, of generalized saddle point form and often very ill conditioned. In this work, we propose a technique for constructing a family of cell‐by‐cell approximate Schur complement matrices, to be used in preconditioning to accelerate the convergence of Krylov subspace iteration methods for such problems. The approximate Schur complement matrices are simply and cheaply constructed in the process of the meshfree discretization and have a sparse structure. It is proved that the so‐constructed approximate Schur complement matrices are spectrally equivalent to the exact Schur complement matrix, which leads to very fast convergence when used in preconditioning. In addition, nondimensionalization of the piezoelectric equations is considered to make the computations more stable. The robustness and the efficiency of the proposed preconditioners is illustrated numerically on two test problems, arising from a piezoelectric strip shear deformation problem and a piezoelectric strip bending problem. Numerical results show that the number of iterations to achieve a given tolerance is independent of the number of degrees of freedom as well as of the various problem parameters.

Minimum unbiased risk estimate based 2DPCA for color image denoising

Low-rank approximation of matrices plays an important role in many application scenarios, including image denoising. This paper introduces a new low-rank approximation method named minimum unbiased risk estimate formulation of 2DPCA (MURE-2DPCA). MURE-2DPCA starts by considering the problem of estimating noise-free matrices from observations, and can exhibit robustness to outliers. In the case of a single data matrix constructed with Gaussian vectors, we find that the optimal dimension of the principal subspace can be determined automatically from the data itself. Based on MURE-2DPCA, a three-step algorithm is developed for color image denoising. Experiments demonstrate the ability and efficiency of our algorithm in achieving the denoising task.