Superfast Tikhonov Regularization of Toeplitz Systems

Abstract

Toeplitz-structured linear systems arise often in practical engineering problems. Correspondingly, a number of algorithms have been developed that exploit Toeplitz structure to gain computational efficiency when solving these systems. The earliest "fast" algorithms for Toeplitz systems required O(n^2) operations, while more recent "superfast" algorithms reduce the cost to O(n (log n)^2) or below. In this work, we present a superfast algorithm for Tikhonov regularization of Toeplitz systems. Using an "extension-and-transformation" technique, our algorithm translates a Tikhonov-regularized Toeplitz system into a type of specialized polynomial problem known as tangential interpolation. Under this formulation, we can compute the solution in only O(n (log n)^2) operations. We use numerical simulations to demonstrate our algorithm's complexity and verify that it returns stable solutions.Comment: 26 pages, 6 figures, 3 tables, 2 algorithms Submitted to IEEE Transactions on Signal Processing on 25 Feb. 201

Convergence of detection probability, computational gains, and asymptotic analysis of an algorithm for physical‐layer intrusion detection system

“…This Karhunen‐Loève expansion is used for whitening the received signal y ( t ) that is colored Gaussian distributed and is modeled as a wide‐sense stationary random process. The closed‐form solution of PLIDS is given as The threshold γ is equal to where τ is a constant in and h 0 ( t ) and h 1 ( t ) are found via Fredholm integral equation of the first kind using Tikhonov regularization . In aforementioned equations, R Y Y ( t 1 , t 2 ) is the autocorrelation function of signal y ( t ).…”

“…Inherently, the main task of regularization is to make R Y Y a well‐conditioned matrix. Using Tikhonov regularization, the solution of is given by as Here, Λ o is an optimum regularizer selected from set of regularizers Λ={Λ 1 ,Λ 2 …,Λ ψ } using L‐Curve criterion, where ψ is displacement rank of R Y Y given in where L S and U S are lower‐shift matrix and upper‐shift matrix, respectively. Moreover, Λ is discrete approximation of smoothing operator and related to generalized singular values of R Y Y .…”

“…However, by exploiting the Toeplitz nature of matrix like in , it can be solved in O ( n 2 ) complexity via Levinson‐Durbin algorithm . Algorithms for solving linear system involving Toeplitz matrix are generally classified as fast algorithms or superfast algorithms . Levinson algorithm is a fast algorithm and some other examples of fast algorithms are mentioned in our other work and in the work of Heinig and Rost .…”

“…For large values of n , superfast algorithms like those in the works of Van Barel et al and Chandrasekaran et al can be used, but they are not suitable for rectangular ill‐conditioned systems. Superfast algorithms for generic solution of linear system can be found in other works . Superfast algorithms for ill‐conditioned system involving a Toeplitz structure can be found in the works of Li et al and Turnes et al Computational complexity of the technique described in the work of Turnes et al is .…”

“…Superfast algorithms for generic solution of linear system can be found in other works . Superfast algorithms for ill‐conditioned system involving a Toeplitz structure can be found in the works of Li et al and Turnes et al Computational complexity of the technique described in the work of Turnes et al is . Here, k is number of regularizers, which is equivalent to displacement rank ψ , and n is the number of parameters or length of vector .…”