I am having a difficult time visualizing this.
Is there a word for a function, f, that has the same critical points as another function, g, albeit with different critical values and where there isn't necessarily equivalence between minimum <-> minimum, maximum <-> maximum, and saddle point <-> saddle point?
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I guess the simplest example would be if, f = -g(x) such that a maximum of f was a minimum of g. My problem isn't that straight-forward, but if there is some word from analysis that's relevant, it can help me find the right vein to follow.
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To what derivative do you care about? Would you want critical points from different levels to be matched? e.g. x^2 matches x^3 because the min and inflection are at the same x-value?
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I'm mostly thinking out-loud. I think
@kldivergence tact will help. Basically, I want my f() to just be g() but it's not. Factoring to make it similar was convoluted. I'd just like to make the hand-wavy argument that under stochastic gradient descent there are *similar* dynamics.
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@JamesJohndrow thinks you might want to check out ideals/varieties, though this only corresponds to polynomial functions, not all functions. - 1 more reply
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