  Log-density gradient estimation is a fundamental statistical problem and
possesses various practical applications such as clustering and measuring
non-Gaussianity. A naive two-step approach of first estimating the density and
then taking its log-gradient is unreliable because an accurate density estimate
does not necessarily lead to an accurate log-density gradient estimate. To cope
with this problem, a method to directly estimate the log-density gradient
without density estimation has been explored, and demonstrated to work much
better than the two-step method. The objective of this paper is to further
improve the performance of this direct method in multi-dimensional cases. Our
idea is to regard the problem of log-density gradient estimation in each
dimension as a task, and apply regularized multi-task learning to the direct
log-density gradient estimator. We experimentally demonstrate the usefulness of
the proposed multi-task method in log-density gradient estimation and
mode-seeking clustering.
