  We address the problem of making Conformal Prediction (CP) intervals locally
adaptive. Most existing methods focus on approximating the object-conditional
validity of the intervals by partitioning or re-weighting the calibration set.
Our strategy is new and conceptually different. Instead of re-weighting the
calibration data, we redefine the conformity measure through a trainable change
of variables, $A \to \phi_X(A)$, that depends explicitly on the object
attributes, $X$. Under certain conditions and if $\phi_X$ is monotonic in $A$
for any $X$, the transformations produce prediction intervals that are
guaranteed to be marginally valid and have $X$-dependent sizes. We describe how
to parameterize and train $\phi_X$ to maximize the interval efficiency.
Contrary to other CP-aware training methods, the objective function is smooth
and can be minimized through standard gradient methods without approximations.
