Projection pursuit algorithms approximate a function of p variables by a sum of nonlinear functions of linear combinations: \[ (1) f( {x_1 , ⋯ ,x_p } ) ∑i = 1^n {g_i ( {aᵢ₁ x_1 + ⋯ + aᵢₚ x_p } )} . \] We develop some approximation theory, give a necessary and sufficient condition for equality in (1), and discuss nonuniqueness of the representation.
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Diaconis et al. (1984) studied this question.
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