ABSTRACT
In this chapter we outline the general theory for nonlinear sufficient dimension reduction as formulated in Li et al. (2011) and Lee et al. (2013). The sufficient dimension reduction problems studied so far rely on a set of linear functions of X. That is, Y ⊧ X | β T X $ Y \models X | \beta ^{\scriptscriptstyle \mathsf{T}}X $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119427/09dc902b-e49a-45e8-84d8-98b5c97c1e74/content/inline-math12_1.tif"/> . Nonlinear sufficient dimension reduction replaces the linear sufficient predictor β T X $ \beta ^{\scriptscriptstyle \mathsf{T}}X $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119427/09dc902b-e49a-45e8-84d8-98b5c97c1e74/content/inline-math12_2.tif"/> by a nonlinear predictor f(X). That is, we consider the following problem Y ⊧ X | f ( X ) , $$ \begin{aligned} Y \models X | f(X), \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119427/09dc902b-e49a-45e8-84d8-98b5c97c1e74/content/math12_1.tif"/>
