An Algebraic Translation of Cayley-Dickson Linear Systems and Its Applications to Online Learning

The
$m$
-dimensional Cayley–Dickson number system
${\BBA}_m$
is a standard extension of real
$(m=1)$
, complex
$(m=2)$
, quaternion
$(m=2^2)$
, octonion
$(m=2^3)$
and sedenion
$(m=2^4)$
etc. In this paper, we present a systematic algebraic translation of the Cayley–Dickson hypercomplex valued linear systems into a real vector valued linear model. This translation is designed by using jointly two new isomorphisms between real vector spaces and enables us to straightforwardly apply the well established schemes in real domain to problems for the hypercomplex linear model. We also clarify useful algebraic properties of the proposed translation. As an example of many potential algorithms through the proposed algebraic translation, we present
${\BBA}_m$
-adaptive projected subgradient method (
${\BBA}_m$
-APSM) for
${\BBA}_m$
valued adaptive system identification, and show that many hypercomplex adaptive filtering algorithms can be viewed as special cases of this algorithm. Moreover, we also apply the
${\BBA}_m$
-APSM to nonlinear adaptive filtering by using the kernel trick. Numerical examples show that the effectiveness of the
${\BBA}_m$
-APSM in many Cayley–Dickson valued linear system identification and nonlinear channel equalization problems.