Study of Sparsity Emanating from NKPD and its Utilization to Enhance NKPD based Adaptive Algorithms

Recently, the nearest Kronecker product (NKP) decomposition has become popular in several adaptive filtering (AF) applications owing to its fast convergence and tracking ability. In this paper, we study the nature of the smaller weight vectors resulting from NKP decomposition (NKPD) of a wide range of acoustic impulse responses (IRs). The study shows that the smaller weight vectors resulting from NKPD exhibit moderate to high degree of sparsity. To exploit this knowledge in AF problems, we propose a class of proportionate update based NKP normalized least-mean-square (NKP-NLMS) type algorithms: namely, the improved proportionate NKP-NLMS (NKP-IPNLMS) algorithm which uses the
$\ell_{1}$
-norm of the smaller weight vectors and the NKP- IPNLMS
$-\ell_{0}$
which uses an approximation of the
$\ell_{0}$
-norm. Further, we propose a new approximation of the
$\ell_{0}$
-norm with reduced computational complexity, using which we also propose the
$\text{NKP}-\text{IPNLMS}-\ell_{0}-2$
algorithm. Next, we present a comparison of computational complexity of the proposed algorithms. Simulation results show the improved performance achieved by the proposed algorithms, showing the advantage of exploiting sparsity in the smaller weight vectors in NKPD based adaptive algorithms.