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A Canonical Transform for Strengthening the Local $L^p$-Type Universal Approximation Property

Most $L^p$-type universal approximation theorems guarantee that a given machine learning model class $\mathscr{F}\subseteq C(\mathbb{R}^d,\mathbb{R}^D)$ is dense in $L^p_{\mu}(\mathbb{R}^d,\mathbb{R}^D)$ for any suitable finite Borel measure $\mu$ on $\mathbb{R}^d$. Unfortunately, this means that the model's approximation quality can rapidly degenerate outside some compact subset of $\mathbb{R}^d$, as any such measure is largely concentrated on some bounded subset of $\mathbb{R}^d$. This paper proposes a generic solution to this approximation theoretic problem by introducing a canonical transformation which "upgrades $\mathscr{F}$'s approximation property" in the following sense. The transformed model class, denoted by $\mathscr{F}\text{-tope}$, is shown to be dense in $L^p_{\mu,\text{strict}}(\mathbb{R}^d,\mathbb{R}^D)$ which is a topological space whose elements are locally $p$-integrable functions and whose topology is much finer than usual norm topology on $L^p_{\mu}(\mathbb{R}^d,\mathbb{R}^D)$; here $\mu$ is any suitable $\sigma$-finite Borel measure $\mu$ on $\mathbb{R}^d$. Next, we show that if $\mathscr{F}$ is any family of analytic functions then there is always a strict "gap" between $\mathscr{F}\text{-tope}$'s expressibility and that of $\mathscr{F}$, since we find that $\mathscr{F}$ can never dense in $L^p_{\mu,\text{strict}}(\mathbb{R}^d,\mathbb{R}^D)$. In the general case, where $\mathscr{F}$ may contain non-analytic functions, we provide an abstract form of these results guaranteeing that there always exists some function space in which $\mathscr{F}\text{-tope}$ is dense but $\mathscr{F}$ is not, while, the converse is never possible. Applications to feedforward networks, convolutional neural networks, and polynomial bases are explored.

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