  Regression aims at estimating the conditional mean of output given input.
However, regression is not informative enough if the conditional density is
multimodal, heteroscedastic, and asymmetric. In such a case, estimating the
conditional density itself is preferable, but conditional density estimation
(CDE) is challenging in high-dimensional space. A naive approach to coping with
high-dimensionality is to first perform dimensionality reduction (DR) and then
execute CDE. However, such a two-step process does not perform well in practice
because the error incurred in the first DR step can be magnified in the second
CDE step. In this paper, we propose a novel single-shot procedure that performs
CDE and DR simultaneously in an integrated way. Our key idea is to formulate DR
as the problem of minimizing a squared-loss variant of conditional entropy, and
this is solved via CDE. Thus, an additional CDE step is not needed after DR. We
demonstrate the usefulness of the proposed method through extensive experiments
on various datasets including humanoid robot transition and computer art.
