Nonparametric approximation of conditional expectation operators
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Given the joint distribution of two random variables X,Y on some second countable locally compact Hausdorff space, we investigate the statistical approximation of the L^2-operator defined by [Pf](x) := 𝔼[ f(Y) | X = x ] under minimal assumptions. By modifying its domain, we prove that P can be arbitrarily well approximated in operator norm by Hilbert–Schmidt operators acting on a reproducing kernel Hilbert space. This fact allows to estimate P uniformly by finite-rank operators over a dense subspace even when P is not compact. In terms of modes of convergence, we thereby obtain the superiority of kernel-based techniques over classically used parametric projection approaches such as Galerkin methods. This also provides a novel perspective on which limiting object the nonparametric estimate of P converges to. As an application, we show that these results are particularly important for a large family of spectral analysis techniques for Markov transition operators. Our investigation also gives a new asymptotic perspective on the so-called kernel conditional mean embedding, which is the theoretical foundation of a wide variety of techniques in kernel-based nonparametric inference.
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