An RKHS formulation of the inverse regression dimension-reduction problem

By T. Hsing, Haobo Ren. 2009

59 citations recorded by the source corpus.

Suppose that $Y$ is a scalar and $X$ is a second-order stochastic process, where $Y$ and $X$ are conditionally independent given the random variables $\xi_1,...,\xi_p$ which belong to the closed span $L_X^2$ of $X$. This paper investigates a unified framework for the inverse regression dimension-reduction problem. It is found that the identification of $L_X^2$ with the reproducing kernel Hilbert space of $X$ provides a platform for a seamless extension from the finite- to infinite-dimensional settings. It also facilitates convenient computational algorithms that can be applied to a variety of models.

Subfields: Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/961a2087229f4589c99b7ff836aa7ee83b4aaae9

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Citations in the corpus

Gradient-Based Kernel Dimension Reduction for Regression (2014, methodology): “Additionally, with RKHSs, Hsing and Ren (2009) discuss an extension of inverse regression from ﬁnite-dimensional to inﬁnite-dimensional problems.”
A general theory for nonlinear sufficient dimension reduction: Formulation and estimation (2013, methodology): “(5) This generalizes linear SDR to the infinite-dimensional case, but not to the nonlinear case, because 〈X,h1〉L2(λ), . . . , 〈X,hd〉L2(λ) are linear in X. Hsing and Ren (2009) considered a more general setting where the sample paths {Xt(ω) : t ∈ J } need not lie within L2(λ).”
Nonlinear sufficient dimension reduction for functional data (2017, methodology): “In particular, the function-valued predictor and response do not have to be related through linear indices, as assumed by the recently-developed sufficient dimension reduction methods for functional data [Müller and Stadtmüller (2005), Ferré and Yao (2003, 2005), Hsing and Ren (2009)].”