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### Covariant Differentiation

With the notion of a gradient and a notion of movement that respects the constraints of the manifold, one might wish to begin with an optimization routine of some sort. A steepest descent could be implemented from these alone. However, in order to carry out sophisticated optimizations, one usually wants some sort of second derivative information about the function.

In particular, one might wish to know by how much one can expect the gradient to change if one moves from to . This can actually be a difficult question to answer on a manifold. Technically, the gradient at is a member of , while the gradient at is a member of . While taking their difference would work fine in a flat space (see Figure 9.8), if this were done on a curved space, it could give a vector which is not a member of the tangent space of either point (see Figure 9.9).

A more sophisticated means of taking this difference is to first move the gradient at to in some manner which translates it in a parallel fashion from to , and then compare the two gradients within the same tangent space. One can check that for the rule

where is the Levi-Civita connection, takes to an element of and preserves inner product information (to first order in ). This is the standard rule for parallel transport which can found in the usual literature ([80,459,273,222], and others).

Using this rule to compare nearby vectors to each other, one then has the following rule for taking derivatives of vector fields:

where is any vector field (but we are only interested in derivatives of the gradient field). This is the function implemented by dgrad in the software.

In an unconstrained minimization, the second derivative of the gradient along a vector is the Hessian times . Covariantly, we then have the analogy,

Next: Inverting the Covariant Hessian Up: Geometric Technicalities Previous: Getting Around   Contents   Index
Susan Blackford 2000-11-20