The Equivalence between Orthogonal Iterations and Alternating Least Squares

This note explores the relations between two different methods. The first one is the Alternating Least Squares (ALS) method for calculating a rank-k approximation of a real m×n matrix, A. This method has important applications in nonnegative matrix factorizations, in matrix completion problems, and in tensor approximations. The second method is called Orthogonal Iterations. Other names of this method are Subspace Iterations, Simultaneous Iterations, and block-Power method. Given a real symmetric matrix, G, this method computes k dominant eigenvectors of G. To see the relation between these methods we assume that G = AT A. It is shown that in this case the two methods generate the same sequence of subspaces, and the same sequence of low-rank approximations. This equivalence provides new insight into the convergence properties of both methods.

Monocular Vision Based Relative Localization For Fixed⁃wing Unmanned Aerial Vehicle Landing

Autonomous landing has become a core technology of unmanned aerial vehicle(UAV)guidance,navigation,and control system in recent years.This paper discusses the vision⁃based relative position and attitude estimation between fixed⁃wing UAV and runway,which is a key issue in autonomous landing.Images taken by a airborne camera was used and a runway detection method based on long⁃line feature and gradient projection is proposed,which solved the problem that the traditional Hough transform requires much calculation time and easily detects end points by mistake.Under the premise of the known width and length of the runway,position and attitude estimation algorithm used the image processing results and adopted an estimation algorithm based on orthogonal iteration.The method took the objective space error as error function and effectively improved the accuracy of linear algorithm through iteration.The experimental results verified the effectiveness of the proposed algorithms.

Vision-based Estimation of Relative Pose in Autonomous Aerial Refueling

The lack of autonomous aerial refueling capabilities is one of the greatest limitations of unmanned aerial vehicles. This paper discusses the vision-based estimation of the relative pose of a tanker and unmanned aerial vehicle, which is a key issue in autonomous aerial refueling. The main task of this paper is to study the relative pose estimation for a tanker and unmanned aerial vehicle in the phase of commencing refueling and during refueling. The employed algorithm includes the initialization of the orientation parameters and an orthogonal iteration algorithm to estimate the optimal solution of rotation matrix and translation vector. In simulation experiments, because of the small variation in the rotation angle in aerial refueling, the method in which the initial rotation matrix is the identity matrix is found to be the most stable and accurate among methods. Finally, the paper discusses the effects of the number and configuration of feature points on the accuracy of the estimation results when using this method.

Limited Memory BFGS Method for Least Squares Semidefinite Programming with Banded Structure

This work is intended to solve the least squares semidefinite program with a banded structure. A limited memory BFGS method is presented to solve this structured program of high dimension.In the algorithm, the inverse power iteration and orthogonal iteration are employed to calculate partial eigenvectors instead of full decomposition of n × n matrices. One key feature of the algorithm is that it is proved to be globally convergent under inexact gradient information. Preliminary numerical results indicate that the proposed algorithm is comparable with the inexact smoothing Newton method on some large instances of the structured problem.

FAST ALGORITHMS FOR HIGHER-ORDER SINGULAR VALUE DECOMPOSITION FROM INCOMPLETE DATA

Higher-order singular value decomposition （HOSVD） is an efficient way for data reduction and also eliciting intrinsic structure of multi-dimensional array data. It has been used in many applications, and some of them involve incomplete data. To obtain HOSVD of the data with missing values, one can first impute the missing entries through a certain tensor completion method and then perform HOSVD to the reconstructed data. However, the two-step procedure can be inefficient and does not make reliable decomposition. In this paper, we formulate an incomplete HOSVD problem and combine the two steps into solving a single optimization problem, which simultaneously achieves imputation of missing values and also tensor decomposition. We also present one algorithm for solving the problem based on block coordinate update （BCU）. Global convergence of the algorithm is shown under mild assumptions and implies that of the popular higher-order orthogonality iteration （HOOI） method, and thus we, for the first time, give global convergence of HOOI. In addition, we compare the proposed method to state-of-the-art ones for solving incom- plete HOSVD and also low-rank tensor completion problems and demonstrate the superior performance of our method over other compared ones. Furthermore, we apply it to face recognition and MRI image reconstruction to show its practical performance.