Low-rank Solution Research Articles

A new method for low-rank transform domain adaptive filtering

This paper introduces a least squares, matrix-based framework for adaptive filtering that includes normalized least mean squares (NLMS), affine projection (AP) and recursive least squares (RLS) as special cases. We then introduce a method for extracting a low-rank underdetermined solution from an overdetermined or a high-rank underdetermined least squares problem using a part of a unitary transformation. We show how to create optimal, low-rank transformations within this framework. For obtaining computationally competitive versions of our approach, we use the discrete Fourier transform (DFT). We convert the complex-valued DFT-based solution into a real solution. The most significant bottleneck in the optimal version of the algorithm lies in having to calculate the full-length transform domain error vector. We overcome this difficulty by using a statistical approach involving the transform of the signal rather than that of the error to estimate the best low-rank transform at each iteration. We also employ an innovative mixed domain approach, in which we jointly solve time and frequency domain equations. This allows us to achieve very good performance using a transform order that is lower than the length of the filter. Thus, we are able to achieve very fast convergence at low complexity. Using the acoustic echo cancellation problem, we show that our algorithm performs better than NLMS and AP and competes well with FTF-RLS for low SNR conditions. The algorithm lies in between affine projection and FTF-RLS, both in terms of its complexity and its performance.

Numerical solution of the Lyapunov equation by approximate power iteration

We present the approximate power iteration (API) algorithm for the computation of the dominant invariant subspace of the solution X of large-order Lyapunov equations AX + XA T + Q = 0 without first computing the matrix X itself. The API algorithm is an iterative procedure that uses Krylov subspace bases in computing estimates of matrix-vector products X v in a power iteration sequence. Application of the API algorithm requires that A + A T < 0; numberical experiments indicate that, if the matrix X admits a good low-rank solution, then API provides an orthogonal basis of a subspace that closely approximates the dominant X-invariant subspace of corresponding dimension. Analytical convergence results are also presented.

Krylov Subspace Methods for Solving Large Lyapunov Equations

This paper considers several methods for calculating low-rank approximate solutions to large-scale Lyapunov equations of the form $AP + PA' + BB' = 0$. The interest in this problem stems from model reduction where the task is to approximate high-dimensional models by ones of lower order. The two recently developed Krylov subspace methods exploited in this paper are the Arnoldi method [Saad, Math. Comput., 37 (1981), pp. 105–126] and the Generalised Minimum Residual method (GMRES) [Saad and Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869]. Exact expressions for the approximation errors incurred are derived in both cases. The numerical solution of the low-dimensional linear matrix equation arising from the GMRES method is discussed and an algorithm for its solution is proposed. Low rank solutions of discrete time Lyapunov equations and continuous time algebraic Riccati equations are also considered. Throughout this paper, the authors tackle problems in which B has more than one column with the use of block Krylov schemes.

Explicit Solutions of Linear Matrix Equations

Explicit Solutions of Linear Matrix Equations

Matrix Quadratic Solutions

Matrix quadratic equation solution derivation applied in finding steady state solutions of Riccati differential equations with constant coefficients