Large-p-small-n data, in which the number of recorded variables (p) exceeds the number of independent observational units (n), are becoming the norm in a variety of scientific fields. Sufficient dimension reduction provides a meaningful and theoretically motivated way to handle large-p-small-n regressions, by restricting attention to d p, because they rely on the inversion of the predictor sample covariance matrix. In this article we propose an iterative method that eliminates the need for such inversion, using instead powers of the covariance matrix. We illustrate our method with a genomics application; the discrimination of human regulatory elements from a background of â€œnon-functionalâ€ DNA, based on their alignment patterns with the genomes of other mammalian species. We also investigate the performance of the iterative method by simulation, obtaining excellent results when n < p or . We speculate that powers of the covariance matrix may allow us to effectively exploit available information on the predictor structure in identifying directions relevant to the regression.