Triangular factorization of inverse data covariance matrices

Abstract

A new Cholesky factorization of the inverse covariance matrix can be performed with fully parallel matrix-vector operations, instead of more costly back substitutions. This factorization reformulates the Sherman-Morrison-Woodbury matrix inverse identity as a downdating problem. Givens rotations provide triangularized factors of the inverse data covariance matrix, and the final adaptive solution is obtained after two triangular matrix-vector products. This new factorization algorithm operates in the voltage (or square root) domain to maintain the numerical robustness of earlier factorization techniques. Simulation results show that this algorithm is highly stable and can yield more accurate results at lower processor precisions.

An Improved Square-Root Algorithm for V-BLAST Based on Efficient Inverse Cholesky Factorization

“…Now from (13), (7) and (4), it can be seen that (9) (proposed in [4]) and (11) actually reveal the relation between the m th and the (m − 1) th order inverse Cholesky factor of the matrix R. This relation is also utilized to implement adaptive filters in [15], [16], where the m th order inverse Cholesky factor is obtained from the m th order Cholesky factor [15, equation (12)], [16, equation (16)]. Thus the algorithms in [15], [16] are still similar to the conventional matrix inversion algorithm [17] using Cholesky factorization, where the inverse Cholesky factor is computed from the Cholesky factor by the back-substitution (for triangular matrix inversion), an inherent serial process unsuitable for the parallel implementation [18]. Contrarily, the proposed algorithm computes the inverse Cholesky factor of April 2, 2020 DRAFT R m from R m directly, as shown in (17) and (11).…”

Contents

“…implementation [18], and then requires only half divisions. Usually the expected, rather than worst-case, complexities of various algorithms are studied [20].…”

“…while a satisfying (18) can be computed by (17a). We can use (17) and (11) to compute F from F −1 iteratively till we get F .…”

“…Now from (13), (7) and (4), it can be seen that (9) (proposed in [4]) and (11) actually reveal the relation between the ℎ and the ( − 1) ℎ order inverse Cholesky factor of the matrix R. This relation is also utilized to implement adaptive filters in [15], [16], where the ℎ order inverse Cholesky factor is obtained from the ℎ order Cholesky factor [15, equation (12)], [16, equation (16)]. Thus the algorithms in [15], [16] are still similar to the conventional matrix inversion algorithm [17] using Cholesky factorization, where the inverse Cholesky factor is computed from the Cholesky factor by the backsubstitution (for triangular matrix inversion), an inherent serial process unsuitable for the parallel implementation [18]. Contrarily, the proposed algorithm computes the inverse Cholesky factor of R from R directly by (17) and (11).…”

“…(9) and (11)) has been mentioned [4], [15], [16], our contributions in this letter include substituting this relation into (14) to find (18) and (17). Specifically, to compute the ℎ order inverse Cholesky factor, the conventional matrix inversion algorithm using Cholesky factorization [17] usually requires 2 divisions (i.e.…”