Partial-update strictly linear, semi-widely linear, and widely linear geometric-algebra adaptive filters

Geometric-algebra based adaptive filters have been successfully employed in many applications such as computer vision, data fusion and linear prediction where the unknown parameters of interest are high-dimensional multivectors. However, conventional geometric-algebra adaptive filters, such as the strictly linear geometric-algebra least mean square (SL-GA-LMS) algorithm, are only applicable to circular multivector-valued inputs with rotation-invariant probability distribution functions. To remove this limitation, we propose new semi-widely linear and widely linear GA-LMS algorithms. As geometric-algebra adaptive filters can have extremely high computational complexity, partial-update variants of these algorithms with reduced complexity are also developed employing stochastic, sequential and M-max partial updating strategies. Steady-state and transient performances of the proposed partial-update algorithms are analysed. As an isomorphism to the partial-update GA-LMS algorithms, widely linear, semi-widely linear and strictly linear quaternion LMS algorithms with partial updates are proposed and analysed for noncircular quaternion inputs. Finally, numerical studies are carried out to confirm the advantages of the proposed methods and the convergence analysis results for multivector and quaternion-valued inputs.