  It is common in machine learning to estimate a response $y$ given covariate
information $x$. However, these predictions alone do not quantify any
uncertainty associated with said predictions. One way to overcome this
deficiency is with conformal inference methods, which construct a set
containing the unobserved response $y$ with a prescribed probability.
Unfortunately, even with a one-dimensional response, conformal inference is
computationally expensive despite recent encouraging advances. In this paper,
we explore multi-output regression, delivering exact derivations of conformal
inference $p$-values when the predictive model can be described as a linear
function of $y$. Additionally, we propose \texttt{unionCP} and a multivariate
extension of \texttt{rootCP} as efficient ways of approximating the conformal
prediction region for a wide array of multi-output predictors, both linear and
nonlinear, while preserving computational advantages. We also provide both
theoretical and empirical evidence of the effectiveness of these methods using
both real-world and simulated data.
