  Empirical neural tangent kernels (eNTKs) can provide a good understanding of
a given network's representation: they are often far less expensive to compute
and applicable more broadly than infinite width NTKs. For networks with O
output units (e.g. an O-class classifier), however, the eNTK on N inputs is of
size $NO \times NO$, taking $O((NO)^2)$ memory and up to $O((NO)^3)$
computation. Most existing applications have therefore used one of a handful of
approximations yielding $N \times N$ kernel matrices, saving orders of
magnitude of computation, but with limited to no justification. We prove that
one such approximation, which we call "sum of logits", converges to the true
eNTK at initialization for any network with a wide final "readout" layer. Our
experiments demonstrate the quality of this approximation for various uses
across a range of settings.
