Re: Does the Group Invariance Theorem for all linear threshold functions?
Actually, I suspect this is a limitation only for linear functions.
Linear threshold functions which are invariant under the action of some mathematical group can be mapped to functions whose coefficients depend only on that group.
So, the only linear functions invariant under groups that are transitive (scaling, for example) end up being a measurement of size or area. But size and area do not necessarily preserve such transitive conditions, so building a distinguishing hyperplane invariant under such conditions is impossible, unless one of your features depends on all of the points.
So as long as your function is linear, this will hold. Right? If this is wrong please speak up. If it's right, the question is now: how do we learn order2 (or, generically, orderp) functions, where this would not be a limitation? Neural nets are one obvious solution, but I'm having trouble mapping this orderp function fitting problem to the framework of neural networks. (2)
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