Riemannian Gradient Algorithm Research Articles

Energy efficiency optimization algorithm based on RIS and D2D for multi-user MIMO system

Reconfigurable Intelligent Surface (RIS) is composed of reconfigurable passive components, which can realize on-demand customization of wireless communication environment. Device-to-Device (D2D) communication is one of the essential technologies to improve the capacity and spectrum efficiency of network system. In this paper, we propose an energy efficiency optimization algorithm based on RIS and D2D assisted multi-user MIMO system. The purpose is to maximize the energy efficiency of the system by jointly optimizing subcarrier allocation, power allocation and passive beamforming at RIS. We first formulate the energy efficiency maximization problem with the constraints of different user rate, power, subcarrier allocation and phase. Since it is a non-convex problem, we use an alternating optimization (AO) algorithm to decouple it into three sub-problems. The solutions of these three sub-problems are solved by subcarrier allocation based on channel strength, SCA algorithm, Dinkelbach algorithm and Riemannian gradient algorithm respectively. Simulation results show that the proposed scheme can significantly improve the energy efficiency.

Hybrid Analog and Digital Beamforming for RIS-Assisted mmWave Communications

Reconfigurable intelligent surface (RIS) assisted millimeter wave (mmWave) communications has been envisioned as a prominent technology for future wireless networks, since it is capable of simultaneously providing abundant spectrum resources and favorable propagation environments. The small wavelength at mmWave bands also enables the widespread use of large antenna arrays, of which the hybrid beamforming structure has emerged as a cost-effective solution. In this paper, we aim to minimize the sum-mean-square-error (sum-MSE) in the RIS-assisted mmWave multiuser multiple input multiple output (MU-MIMO) system by jointly optimizing the hybrid analog-digital precoders and the RIS reflection matrix. We demonstrate that the role of RIS in assisting mmWave communications can be completely replaced by a large-scale Kronecker-structured hybrid array. Moreover, an accelerated Riemannian gradient algorithm using majorization minimization technique is proposed to tackle the unit-modulus constrained analog precoder/RIS design. Under the assumption of perfect channel state information (CSI), we firstly consider the single-user MIMO (SU-MIMO) setup and propose an effective alternating minimization (AM) procedure to characterize the system performance limit. Moreover, a two-stage scheme is developed for low-complexity implementation. This AM procedure is then extended to the general MU-MIMO scenario. In addition, we develop a novel enhanced regularized zero-forcing (ERZF) scheme for simultaneously combating strong noise in the low-SNR regime and mitigating multi-user interference (MUI) in the high-SNR regime. The optimality of our proposed algorithms is validated for some simplified practical scenarios. Numerical results illustrate that the proposed algorithms outperform existing benchmark schemes in terms of the actual complexity and performance.

Numerical method based on fiber bundle for solving Lyapunov matrix equation

In this paper, we firstly introduce the origin of Lyapunov matrix equation, and then the geometric methods for solving Lyapunov equation are given by using the Log-Euclidean metric and the fiber bundle-based Riemannian metric based on the manifold of positive definite Hermitian matrices. Then, the solution of Lyapunov matrix equation is presented by providing a natural gradient descent algorithm (NGDA), a Log-Euclidean descent algorithm (LGDA) and a Riemannian gradient algorithm based on fiber bundle (RGA). At last, the convergence speeds of the RGA, the NGDA and the LGDA are compared via two simulation examples. Simulation results show that the convergence speed of the RGA is faster than both of the LGDA and the NGDA.

Improvement and Assessment of a Blind Image Deblurring Algorithm Based on Independent Component Analysis

The aim of the present paper is to improve an existing blind image deblurring algorithm, based on an independent component learning paradigm, by manifold calculus. The original technique is based on an independent component analysis algorithm applied to a set of pseudo-images obtained by Gabor-filtering a blurred image and is based on an adapt-and-project paradigm. A comparison between the original technique and the improved method shows that independent component learning on the unit hypersphere by a Riemannian-gradient algorithm outperforms the adapt-and-project strategy. A comprehensive set of numerical tests evidenced the strengths and weaknesses of the discussed deblurring technique.

Application of gradient descent algorithms based on geodesic distances

In this paper, the Riemannian gradient algorithm and the natural gradient algorithm are applied to solve descent direction problems on the manifold of positive definite Hermitian matrices, where the geodesic distance is considered as the cost function. The first proposed problem is control for positive definite Hermitian matrix systems whose outputs only depend on their inputs. The geodesic distance is adopted as the difference of the output matrix and the target matrix. The controller to adjust the input is obtained such that the output matrix is as close as possible to the target matrix. We show the trajectory of the control input on the manifold using the Riemannian gradient algorithm. The second application is to compute the Karcher mean of a finite set of given Toeplitz positive definite Hermitian matrices, which is defined as the minimizer of the sum of geodesic distances. To obtain more efficient iterative algorithm compared with traditional ones, a natural gradient algorithm is proposed to compute the Karcher mean. Illustrative simulations are provided to show the computational behavior of the proposed algorithms.

Extended Hamiltonian algorithm for the solution of discrete algebraic Lyapunov equations

In this paper, we use a second-order learning algorithm for solving the numerical solution of the discrete algebraic Lyapunov equation. Specifically, Extended Hamiltonian algorithm based on the manifold of positive definite symmetric matrices is provided. Furthermore, this algorithm is compared with the Euclidean gradient algorithm, the Riemannian gradient algorithm and the two traditional iteration methods. Simulation examples show that the convergence speed of the Extended Hamiltonian algorithm is the fastest one among these algorithms.

Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations

A Riemannian gradient algorithm based on geometric structures of a manifold consisting of all positive definite matrices is proposed to calculate the numerical solution of the linear matrix equationQ=X+∑i=1mAiTXAi. In this algorithm, the geodesic distance on the curved Riemannian manifold is taken as an objective function and the geodesic curve is treated as the convergence path. Also the optimal variable step sizes corresponding to the minimum value of the objective function are provided in order to improve the convergence speed. Furthermore, the convergence speed of the Riemannian gradient algorithm is compared with that of the traditional conjugate gradient method in two simulation examples. It is found that the convergence speed of the provided algorithm is faster than that of the conjugate gradient method.

The Extended Hamiltonian Algorithm for the Solution of the Algebraic Riccati Equation

We use a second-order learning algorithm for numerically solving a class of the algebraic Riccati equations. Specifically, the extended Hamiltonian algorithm based on manifold of positive definite symmetric matrices is provided. Furthermore, this algorithm is compared with the Euclidean gradient algorithm, the Riemannian gradient algorithm, and the new subspace iteration method. Simulation examples show that the convergence speed of the extended Hamiltonian algorithm is the fastest one among these algorithms.

Riemannian Means on Special Euclidean Group and Unipotent Matrices Group

Among the noncompact matrix Lie groups, the special Euclidean group and the unipotent matrix group play important roles in both theoretic and applied studies. The Riemannian means of a finite set of the given points on the two matrix groups are investigated, respectively. Based on the left invariant metric on the matrix Lie groups, the geodesic between any two points is gotten. And the sum of the geodesic distances is taken as the cost function, whose minimizer is the Riemannian mean. Moreover, a Riemannian gradient algorithm for computing the Riemannian mean on the special Euclidean group and an iterative formula for that on the unipotent matrix group are proposed, respectively. Finally, several numerical simulations in the 3-dimensional case are given to illustrate our results.