  Conditional modeling x \to y is a central problem in machine learning. A
substantial research effort is devoted to such modeling when x is high
dimensional. We consider, instead, the case of a high dimensional y, where x is
either low dimensional or high dimensional. Our approach is based on selecting
a small subset y_L of the dimensions of y, and proceed by modeling (i) x \to
y_L and (ii) y_L \to y. Composing these two models, we obtain a conditional
model x \to y that possesses convenient statistical properties. Multi-label
classification and multivariate regression experiments on several datasets show
that this model outperforms the one vs. all approach as well as several
sophisticated multiple output prediction methods.
