Structured Least Squares Problems and Robust Estimators

Abstract

Abstract-A novel approach is proposed to provide robust and accurate estimates for linear regression problems when both the measurement vector and the coefficient matrix are structured and subject to errors or uncertainty. A new analytic formulation is developed in terms of the gradient flow of the residual norm to analyze and provide estimates to the regression. The presented analysis enables us to establish theoretical performance guarantees to compare with existing methods and also offers a criterion to choose the regularization parameter autonomously. Theoretical results and simulations in applications such as blind identification, multiple frequency estimation and deconvolution show that the proposed technique outperforms alternative methods in mean-squared error for a significant range of signal-to-noise ratio values.

Perturbed Orthogonal Matching Pursuit

“…At the th iteration, new vector is obtained from the solution of the following optimization problem: (5) where matrix is the projection operator to the column space of perturbed and is the set of all basis vectors that are not contained in . For each , this perturbation problem can be solved by using the technique given in [35]. However, due to its associated gradient descent based iterations, the complexity of solution is still a practical limitation for large .…”

“…Since the norm of a partitioned vector is the sum of the norms of each partition: (34) which can be written as: (35) where and . To have , we need the following inequality to hold true: (36) Using the triangle inequality on (35), the following bound can be obtained, (37) Since , (37) becomes: (38) By adding to both sides of (38), we obtain:…”

Robust estimation in flat fading channels under bounded channel uncertainties

“…is formed by using the equalizer impulse response w 1,t and the bias term l 1,t that minimize the worst case error, i.e., the error under the worst possible channel impulse response [5,3,4]. Instead of this conservative approach, another useful method to estimate the desired signal is the minimin approach, where the equalizer impulse response and the bias term are given by…”

“…The first approach we investigate is the affine minimax equalization method [4,8,9], which minimizes the estimation error for the worst case channel perturbation. The second approach we study is the affine minimin equalization method [5,10], which minimizes the estimation error for the most favorable perturbation. The third approach is the affine minimax regret equalization method [3,4,11,6], which minimizes a certain "regret" as defined in Section 2 and further detailed in Section 3.…”

“…Although the minimax equalization method is able to minimize the estimation error for the worst case channel perturbation, however, it usually provides unsatisfactory results on the average [5]. An alternative approach to the channel equalization problem is the minimin equalization method [5,10]. In this approach, equalizer parameters are selected to minimize the MSE with respect to the most favorable channel over the set of allowed perturbations.…”