Inverting the Covariant Hessian (Technical Considerations)

Since the tangent space of $\mbox{Stief}(n,k)$ is a subspace of ${\cal R}^{n \times k}$, the covariant Hessian must be inverted stably on this subspace. This requires any algorithms designed to solve $D_H G = V$ for $H$ to really be pseudoinverters in a least squares or some other sense.

A second consideration is that many useful functions $f(Y)$ have the property that $f(YQ) = f(Y)$ for all block-diagonal orthogonal $Q$ (i.e.,

$$Q = \left[ \begin{array}{cccc} Q_1 & 0 & \cdots & 0 \cr 0 & ... ...dotfill & \dotfill \cr 0 & 0 & \cdots & Q_p \end{array}\right],$$

where the $Q_i$ are orthogonal matrices). In this case, all tangent vectors of the form

$$H = Y \left[ \begin{array}{cccc} H_1 & 0 & \cdots & 0 \cr 0 &... ...dotfill & \dotfill \cr 0 & 0 & \cdots & H_p \end{array}\right],$$

where the $H_i$ are antisymmetric, have the property $D_H f = 0$. These extra symmetry vectors are then null vectors of the linear system $D_H G = V$. Because of these null directions, the effective dimension of the tangent space is reduced by the dimension of this null space (this is the dimension returned by the dimension function).

Thus, we see that in order to invert the covariant Hessian, we must take care to use a stable inversion scheme which will project out components of $H$ which do not satisfy the infinitesimal constraint equation and those which are in the direction of the additional symmetries of $f$. The invdgrad function carries out a stable inversion of the covariant Hessian by a conjugate gradient routine, with the dgrad function calling the function nosym to project out any extra symmetry components.