Mathematical models of functional extreme leaning machins: operator-algebraic and free-probabilistic approaches

In this paper, we establish mathematical models for an arbitrarily fixed functional extreme learning machine (FELM). From a FELM \({\mathfrak {M}}\) $\mathfrak{M}$ , we construct a direct graph G induced by \({\mathfrak {M}}\) $\mathfrak{M}$ , and then define the graph groupoid \({\mathbb {G}}\) $\mathbb{G}$ of G. Then the graph-groupoid \(C^{*}\) $C^{\ast }$ -algebra \(M_{G}\) $M_G$ of G generated by \({\mathbb {G}}\) $\mathbb{G}$ is well-determined. This \(C^{*}\) $C^{\ast }$ -algebra \(M_{G}\) $M_G$ is realized on a certain Hilbert space \(H_{G}\) $H_G$ up to a canonical representation. It means that the FELM \({\mathfrak {M}}\) $\mathfrak{M}$ is analyzed in a representation-depending structure in terms of operator algebra theory. By defining a natural free probability on \(M_{G}\) $M_G$ , one can have an assessment tool of the operator algebra on \(M_{G}\) $M_G$ , too.