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Descent directions on manifolds

  1. Sep 15, 2010 #1

    I'm trying to do some optimization on a manifold. In particular, the manifold is [itex] \mathfrak U(N) [/itex], the NxN unitary matrices.

    Now currently, I'm looking at "descent directions" on the manifold. That is, let [itex] f: \mathfrak U(N) \to \mathbb R [/itex] be a function that we want to minimize, [itex] p \in \mathfrak U(N) [/itex] a point and [itex] X \in T_p \mathfrak U(n)[/itex] a point in the tangent manifold at p. If [itex] \gamma: \mathbb R\to \mathfrak U(N) [/itex] is a curve with [itex] \gamma(0) = p, \dot \gamma (0) = X [/itex] then the geodesic emanating from p in the direction X is [itex] \gamma_X(t) = \exp_p[tX] [/itex]. This geodesic is a descent direction if
    [tex] \dot \gamma_X(0)f = \left.\frac d{dt} f(\gamma_X(t))\right|_{t=0} < 0 [/tex]

    My question is: does the magnitude of [itex] \dot\gamma(0)f [/itex] mean anything? For example, say that for a fixed p and two different [itex] X_1,X_2 \in T_p\mathfrak U(N) [/itex] I get that
    [tex] \dot \gamma_{X_1}(0) f = -85, \qquad \dot \gamma_{X_2}(0)f = -4 [/tex]
    These are both descent directions, but is one a "better" descent direction than the other? That is, does [itex] X_1[/itex] result in a decrease in f faster than [itex] X_2 [/itex]? Or are the numbers fairly meaningless?

    The above numbers are not far from actual values taken. The problem is that the direction of steepest descent (the negative of the Riemannian gradient) gives a value of -7. This would lead me to believe that the numbers are fairly useless, and only the sign is important. Any insight would be useful.
  2. jcsd
  3. Sep 18, 2010 #2
    You are searching for "directions". To compare the effectiveness of choosing one direction rather than other you should first normalize your vectors [tex]X_1,X_2[/tex].
  4. Sep 19, 2010 #3
    Also you're only going to get a local decent rate - decent rate will change as you move away from p.
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