Now suppose you're doing higher order multivariate automatic differentiation (AD). The nth derivatives form a symmetric tensor. You want to pack these efficiently in an n-D pyramid structure because it's n! times smaller than a cube. But every row now has a different size and...
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...you can't use constant strides any more. The addresses are binomial functions/polynomials in the indices. So return to ordinary tensors: storing an address with n strides is itself analogous to AD. The strides are finite differences, similar to derivatives. Back to symmetric..
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...tensors: you can track how your strides vary as you walk around your pyramid by maintaining a small pyramid of higher order finite differences of your strides. In other words the machinery you need to walk through addresses of cells in a symmetric tensor turns out...
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...to exactly mimic the problem you needed symmetric tensors for in first place. It works out nicely and gives a way to walk round symmetric tensors updating strides using nothing more than addition and subtraction despite the fact that the address are polynomials in the indices.
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It's a surprising application of a kind of microcosm-macrocosm principle. I think it's also connected to zippers and differentiating data structures but I haven't worked out the details of that yet.
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Similar tricks are used to render splines. Eg. see the pyramids of finite differences here: https://www.cs.unm.edu/~angel/CS534/LECTURES/CS534_08.pdf …
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