Kernel dimension reduction in regression

We present a new methodology for sufficient dimension reduction (SDR). Our methodology derives directly from the formulation of SDR in terms of the conditional independence of the covariate $X$ from the response $Y$, given the projection of $X$ on the central subspace [cf. J. Amer. Statist. Assoc. 86 (1991) 316--342 and Regression Graphics (1998) Wiley]. 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.Comment: Published in at http://dx.doi.org/10.1214/08-AOS637 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

"Feeling" others' painful actions: the sensorimotor integration of pain and action information.

Sensorimotor regions of the brain have been implicated in simulation processes such as action understanding and empathy, but their functional role in these processes remains unspecified. We used functional magnetic resonance imaging (fMRI) to demonstrate that postcentral sensorimotor cortex integrates action and object information to derive the sensory outcomes of observed hand-object interactions. When subjects viewed others' hands grasping or withdrawing from objects that were either painful or nonpainful, distinct sensorimotor subregions emerged as showing preferential responses to different aspects of the stimuli: object information (noxious vs. innocuous), action information (grasps vs. withdrawals), and painful action outcomes (painful grasps vs. all other conditions). Activation in the latter region correlated with subjects' ratings of how painful each object would be to touch and their previous experience with the object. Viewing others' painful grasps also biased behavioral responses to actual tactile stimulation, a novel effect not seen for auditory control stimuli. Somatosensory cortices, including primary somatosensory areas 1/3b and 2 and parietal area PF, may therefore subserve somatomotor simulation processes by integrating action and object information to anticipate the sensory consequences of observed hand-object interactions

A finite difference solution for the cylindrical expansion of a gas cloud into vacuum

Finite difference method for solution of cylindrical expansion of gas cloud into vacuu

Ground States in the Spin Boson Model

We prove that the Hamiltonian of the model describing a spin which is linearly coupled to a field of relativistic and massless bosons, also known as the spin-boson model, admits a ground state for small values of the coupling constant lambda. We show that the ground state energy is an analytic function of lambda and that the corresponding ground state can also be chosen to be an analytic function of lambda. No infrared regularization is imposed. Our proof is based on a modified version of the BFS operator theoretic renormalization analysis. Moreover, using a positivity argument we prove that the ground state of the spin-boson model is unique. We show that the expansion coefficients of the ground state and the ground state energy can be calculated using regular analytic perturbation theory