  The expected gradient outerproduct (EGOP) of an unknown regression function
is an operator that arises in the theory of multi-index regression, and is
known to recover those directions that are most relevant to predicting the
output. However, work on the EGOP, including that on its cheap estimators, is
restricted to the regression setting. In this work, we adapt this operator to
the multi-class setting, which we dub the expected Jacobian outerproduct
(EJOP). Moreover, we propose a simple rough estimator of the EJOP and show that
somewhat surprisingly, it remains statistically consistent under mild
assumptions. Furthermore, we show that the eigenvalues and eigenspaces also
remain consistent. Finally, we show that the estimated EJOP can be used as a
metric to yield improvements in real-world non-parametric classification tasks:
both by its use as a metric, and also as cheap initialization in metric
learning tasks.
