  In this paper we introduce a general theory for nonlinear sufficient
dimension reduction, and explore its ramifications and scope. This theory
subsumes recent work employing reproducing kernel Hilbert spaces, and reveals
many parallels between linear and nonlinear sufficient dimension reduction.
Using these parallels we analyze the properties of existing methods and develop
new ones. We begin by characterizing dimension reduction at the general level
of $\sigma$-fields and proceed to that of classes of functions, leading to the
notions of sufficient, complete and central dimension reduction classes. We
show that, when it exists, the complete and sufficient class coincides with the
central class, and can be unbiasedly and exhaustively estimated by a
generalized sliced inverse regression estimator (GSIR). When completeness does
not hold, this estimator captures only part of the central class. However, in
these cases we show that a generalized sliced average variance estimator
(GSAVE) can capture a larger portion of the class. Both estimators require no
numerical optimization because they can be computed by spectral decomposition
of linear operators. Finally, we compare our estimators with existing methods
by simulation and on actual data sets.
