The Hessian of tall-skinny networks is easy to invert
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…
In plain words
This is a computational method for efficiently applying the inverse Hessian to vectors in deep neural networks. Rather than the impractical cubic-time approach, the algorithm exploits the matrix polynomial structure of the Hessian to achieve linear time complexity in the number of layers. The method works similarly to running backpropagation on a dual network representation and builds on classical ideas for computing Hessian-vector products. Researchers working on neural network optimization may find this useful as a preconditioner for stochastic gradient descent training.
written from the facts on this page · September 2026
From the sources
In the maker’s words, at launch
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?
Does the same job
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More life & fun this month
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Launched alongside, January 2026
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