It turns out the inverse of the Hessian of a deep net is easy to apply to a vector. Doing this naively takes cubically many operations in the number of layers (so impractical), but it's possible to do this in time linear in the number of layers (so very practical)!

This is possible because the Hessian of a deep net has a matrix polynomial structure that factorizes nicely. The Hessian-inverse-product algorithm that takes advantage of this is similar to running backprop on a dual version of the deep net. It echoes an old idea of Pearlmutter's for computing Hessian-vector products.

Maybe this idea is useful as a preconditioner for stochastic gradient descent?