Papers › A Canonical Transform for Strengthening the Local Lᵖ-Type Universal Approximation Property
A Canonical Transform for Strengthening the Local Lᵖ-Type Universal Approximation Property
Anastasis Kratsios, Behnoosh Zamanlooy
Most Lᵖ-type universal approximation theorems guarantee that a given machine learning model class ℱ⊆C(ℝᵈ,ℝᴰ) is dense in Lᵖ_μ(ℝᵈ,ℝᴰ) for any suitable finite Borel measure μ on ℝᵈ. Unfortunately, this means that the model's approximation quality can rapidly degenerate outside some compact subset of ℝᵈ, as any such measure is largely concentrated on some bounded subset of ℝᵈ. This paper proposes a generic solution to this approximation theoretic problem by introducing a canonical transformation which "upgrades ℱ's approximation property" in the following sense. The transformed model class, denoted by ℱ-tope, is shown to be dense in Lᵖ_(μ,strict)(ℝᵈ,ℝᴰ) which is a topological space whose elements are locally p-integrable functions and whose topology is much finer than usual norm topology on Lᵖ_μ(ℝᵈ,ℝᴰ); here μ is any suitable σ-finite Borel measure μ on ℝᵈ. Next, we show that if ℱ is any family of analytic functions then there is always a strict "gap" between ℱ-tope's expressibility and that of ℱ, since we find that ℱ can never dense in Lᵖ_(μ,strict)(ℝᵈ,ℝᴰ). In the general case, where ℱ may contain non-analytic functions, we provide an abstract form of these results guaranteeing that there always exists some function space in which ℱ-tope is dense but ℱ is not, while, the converse is never possible. Applications to feedforward networks, convolutional neural networks, and polynomial bases are explored.
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