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 \({\mathfrak {M}}\), we construct a direct graph G induced by \({\mathfrak {M}}\), and then define the graph groupoid \({\mathbb {G}}\) of G. Then the graph-groupoid \(C^{*}\)-algebra \(M_{G}\) of G generated by \({\mathbb {G}}\) is well-determined. This \(C^{*}\)-algebra \(M_{G}\) is realized on a certain Hilbert space \(H_{G}\) up to a canonical representation. It means that the FELM \({\mathfrak {M}}\) is analyzed in a representation-depending structure in terms of operator algebra theory. By defining a natural free probability on \(M_{G}\), one can have an assessment tool of the operator algebra on \(M_{G}\), too.