Matrix Inversion Approximation Research Articles

Hyperpower least squares progressive iterative approximation

Geometric iterative methods (GIMs) generate curves or surfaces that fit (interpolate or approximate) given sets of data points. Standard GIMs use the Richardson iteration or gradient descent method, so they converge relatively slowly. This paper investigates the GIMs acceleration method, which is based on the use of more efficient methods for the matrix inverse approximation. That is, the possibility to use hyperpower iterative methods is being explored. As a result, a novel GIM named hyperpower least squares progressive iterative approximation (HPLSPIA) is proposed. The convergence of the HPLSPIA method is analyzed, and the computational complexity is briefly discussed. The method is then applied to tensor product B-spline surface fitting. Several HPLSPIA methods utilizing different hyperpower methods for the matrix inverse approximation are experimentally compared. It is shown that polynomial factorizations used in the hyperpower iterative methods significantly affect the efficiency of the HPLSPIA methods.

Decentralised signal processing on graphs via matrix inverse approximation

In the processing of signals defined over graph domains, it is highly desirable to have algorithms that can be implemented in a decentralised manner, whereby each node only needs to exchange information within a localized subgraph of nodes. The most common example is when the subgraph consist of immediate neighbors and this allows for distributed processing. Many graph signal operators, in their original forms, are not amenable to distributed processing. To overcome this deficiency, a polynomial approximation is usually applied to the original operator to yield a distributed operator which is a matrix polynomial of the graph shift matrix. This approach however is only applicable when the original operator is a function of the graph shift matrix. In this paper, we propose a generalized approach to approximate graph signal operators that are not necessary functions of the shift matrix. The key idea here is to restrict the approximated matrix inverse to have small geodesic-width so that multiplication with the small geodesic-width matrix can be implemented in a decentralised manner. Furthermore, to increase the accuracy of the approximated operator, an iterative algorithm which also has the decentralised property and low computational complexity, is proposed. We apply the proposed approach to signal inpainting and signal reconstruction in filter banks. Numerical results verify the effectiveness of the proposed algorithm. The proposed algorithms can also be used to efficiently solve linear system of equations.

On the Low-Complexity, Hardware-Friendly Tridiagonal Matrix Inversion for Correlated Massive MIMO Systems

In massive multiple-input and multiple-output (M-MIMO) systems, one of the key challenges in the implementation is the large-scale matrix inversion operation, as widely used in channel estimation, equalization, detection, and decoding procedures. Traditionally, to handle this complexity issue, several low-complexity matrix inversion approximation methods have been proposed, including the classic Cholesky decomposition and the Neumann series expansion (NSE). However, the conventional approaches failed to exploit neither the special structure of channel matrices nor the critical issues in the hardware implementation, which results in poorer throughput performance and longer processing delay. In this paper, by targeting at the correlated M-MIMO systems, we propose a modified NSE based on tridiagonal matrix inversion approximation (TMA) to accommodate the complexity as well as the performance issue in the conventional hardware implementation, and analyze the corresponding approximation errors. Meanwhile, we investigate the very-large-scale integration implementation for the proposed detection algorithm based on a Xilinx Virtex-7 XC7VX690T FPGA platform. It is shown that for correlated massive MIMO systems, it can achieve near minimum mean square error performance and 630 Mb/s throughput. Compared with other benchmark systems, the proposed pipelined TMA detector can get high throughput-to-hardware ratio. Finally, we also propose a fast iteration structure for further research.

An Eigen-Based Approach for Enhancing Matrix Inversion Approximation in Massive MIMO Systems

This correspondence presents a new matrix inversion approximation (MIA) method for massive multiple-input multiple-output systems (MIMO). In contrast to the existing methods that are mostly derived from the Neumann series expansion framework, additional coefficients have been introduced in our proposed method to enhance the precision of approximation. We propose an efficient algorithm for the coefficient design, which consists of an eigenvalue estimation procedure derived from random matrix theory, and a least-squares fitting procedure that solves a low-dimensional overdetermined system of linear equations. Complexity analysis and simulation results show that our eigen-based MIA method exhibits practically comparable computational complexity, while achieving substantial performance enhancement compared to other benchmark methods.

Approaches of approximating matrix inversion for zero-forcing pre-coding in downlink massive MIMO systems

Several approximation approaches including the Gauss–Seidel (GS) method have been proposed to reduce the complexity of matrix inversion for zero-forcing pre-coding in massive multiple-input–multiple-output systems. However, extra computation is required to obtain the matrix inversion from the iteration result of the GS method. In this paper, we propose a new GS-based matrix inversion approximation (GSBMIA) approach. Unlike the traditional GS method, the GSBMIA approach approximates the matrix inversion, which will simplify further calculations. Furthermore, in order to speed up convergence, we propose a joint algorithm based on the GSBMIA and Newton iteration method where the GSBMIA approach is employed to provide an efficient searching direction for the following Newton iterations. Compared with other approximation methods, the joint algorithm can accommodate more single antenna users for the same base station antenna number. Simulation results demonstrate that the joint algorithm and the GSBMIA approach converge faster than the Neumann series and Newton iteration method.

Joint Newton Iteration and Neumann Series Method of Convergence-Accelerating Matrix Inversion Approximation in Linear Precoding for Massive MIMO Systems

Due to large numbers of antennas and users, matrix inversion is complicated in linear precoding techniques for massive MIMO systems. Several approximated matrix inversion methods, including the Neumann series, have been proposed to reduce the complexity. However, the Neumann series does not converge fast enough. In this paper, to speed up convergence, a new joint Newton iteration and Neumann series method is proposed, with the first iteration result of Newton iteration method being employed to reconstruct the Neumann series. Then, a high probability convergence condition is established, which can offer useful guidelines for practical massive MIMO systems. Finally, simulation examples are given to demonstrate that the new joint Newton iteration and Neumann series method has a faster convergence rate compared to the previous Neumann series, with almost no increase in complexity when the iteration number is greater than or equal to 2.