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Formulating a Method of Steepest Ascent on Lie Groups

  1. Jan 30, 2014 #1
    Suppose we have a compact Lie group ##G##, and a differentiable function ##f:G_0\to\mathbb{R}## from the identity component of ##G## to the real numbers. I'm looking to maximize the value of this function.

    Being something of a neophyte at optimization, especially of this kind, I decided to stick with something I thought I knew well: the method of steepest ascent. Long story short, I'm having trouble with generalizing the concept to Lie groups.

    My idea was fairly simple. We start with some point ##p## on the identity component of ##G##, and then we take the logarithm of this point (that is, we take the inverse of the exponential map) because we want to add something to it. As a note, I justified this by saying that, if the point had more than one value for the logarithm, I could just pick one (every point on ##G_0## has at least one logarithm). Then, I would add some multiple of ##\nabla f_p## to ##\log(p)##, and finish by exponentiating to get ##p'=\exp(c\nabla f_p + \log(p))=\exp(c\nabla f_p)p##. Repeat.

    The problem is, I can't figure out what I would use for the analogue of the gradient. Maybe I'm just not seeing something? I don't know. Any nudge in the right direction would be greatly appreciated. Thank you.
     
    Last edited: Jan 30, 2014
  2. jcsd
  3. Jan 30, 2014 #2

    jgens

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    Not sure if this helps (the question is way outside my area of study), but you can always fix a (bi-invariant) Riemannian metric and define gradients using that.
     
  4. Jan 30, 2014 #3
    I'm unfamiliar with this concept, but Wikipedia claims to know something about this. To fact-check, you're suggesting that I could introduce a Riemannian metric ##g## and define ##\nabla f_p## by ##g_p(\nabla f_p, X_p)=X_p(f)##?

    Wouldn't that put ##\nabla f_p## on ##T_p G## and not ##T_e G## (where ##e## is the identity on ##G##), though?
     
  5. Jan 30, 2014 #4

    jgens

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    Correct definition. You could also use pushforwards on the relevant left-multiplication map to map vectors TGp into vectors TGe. Invariance of the metric should ensure this is pretty well-behaved with respect to the gradient too.
     
    Last edited: Jan 30, 2014
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