Optimal approximation of C^k-functions using shallow complex-valued neural networks
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We prove a quantitative result for the approximation of functions of regularity C^k (in the sense of real variables) defined on the complex cube Ω_n := [-1,1]^n +i[-1,1]^n⊆ℂ^n using shallow complex-valued neural networks. Precisely, we consider neural networks with a single hidden layer and m neurons, i.e., networks of the form z ↦∑_j=1^m σ_j ·ϕ(ρ_j^T z + b_j) and show that one can approximate every function in C^k ( Ω_n; ℂ) using a function of that form with error of the order m^-k/(2n) as m →∞, provided that the activation function ϕ: ℂ→ℂ is smooth but not polyharmonic on some non-empty open set. Furthermore, we show that the selection of the weights σ_j, b_j ∈ℂ and ρ_j ∈ℂ^n is continuous with respect to f and prove that the derived rate of approximation is optimal under this continuity assumption. We also discuss the optimality of the result for a possibly discontinuous choice of the weights.
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