ON THE CONDITIONAL DISTRIBUTIONS OF LOW-DIMENSIONAL PROJECTIONS FROM HIGH-DIMENSIONAL DATA

We study the conditional distribution of low-dimensional projections from high-dimensional data, where the conditioning is on other low-dimensional projections. To fix ideas, consider a random d-vector Z that has a Lebesgue density and that is standardized so that EZ = 0 and EZZ' = I-d. Moreover, consider two projections defined by unit-vectors alpha and beta, namely a response y = alpha'Z and an explanatory variable x = beta'Z. It has long been known that the conditional mean of y given x is approximately linear in x, under some regularity conditions; cf. Hall and Li [Ann. Statist. 21 (1993) 867-889]. However, a corresponding result for the conditional variance has not been available so far. We here show that the conditional variance of y given x is approximately constant in x (again, under some regularity conditions). These results hold uniformly in alpha and for most beta's, provided only that the dimension of Z is large. In that sense, we see that most linear submodels of a high-dimensional overall model are approximately correct. Our findings provide new insights in a variety of modeling scenarios. We discuss several examples, including sliced inverse regression, sliced average variance estimation, generalized linear models under potential link violation, and sparse linear modeling.