Separating capacity of analytic neurons

This paper extends the classical results of T. Cover (1965) and others on separating capacity of families of nonlinear neurons of the form x→sgn (Σ_i=1^d ω_iφ_i(x))=0, where w_i∈E are real coefficients (synaptic weights), φ_i:Eⁿ→E are functions (measurement transformation) and sgn is the signum function on E. We show that the capacity of such a system is 2dimφ input patterns, i.e. twice the number of linearly independent functions in the set φ₁,...φ_d, if the functions φ_i are analytic. This is achieved by showing that in such a case the Cover´s assumption of φ-general positions of input vectors is almost universally satisfied