Kernel-based contrast functions for sufficient dimension reduction
Seminar Room 1, Newton Institute
We present a new methodology for sufficient dimension reduction (the problem of finding a subspace $S$ such that the projection of the covariate vector $X$ onto $S$ captures the statistical dependency of the response $Y$ on $X$). Our methodology derives directly from a formulation of sufficient dimension reduction in terms of the conditional independence of the covariate $X$ from the response $Y$, given the projection of $X$ on the central subspace (cf. Li, 1991; Cook, 1998). We show that this conditional independence assertion can be characterized in terms of conditional covariance operators on reproducing kernel Hilbert spaces and we show how this characterization leads to an M-estimator for the central subspace. The resulting estimator is shown to be consistent under weak conditions; in particular, we do not have to impose linearity or ellipticity conditions of the kinds that are generally invoked for SDR methods. We also present empirical results showing that the new methodology is competitive in practice.
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