HOW MUCH DAMPING?

We often need to damp the solutions to least squares problems. We have a fitting problem (regression) with the two goals:

$$\begin{eqnarray} 0 \quad\approx\quad \bold r_d &=& \bold F \bold m - \bold d \\ 0 \quad\approx\quad \bold r_m &=& \lambda \bold R \bold m\end{eqnarray}$$ (5) (6)

where $\bold R$ is a roughening operator. How big should $\lambda$ be? Suppose in human terms we'd like ``half'' the properties of the solution to come from the fitting and ``half'' to come from the damping. How might we define what we mean by ``half''? We can start by considering balancing the two residual vectors $\bold r_d$ and $\bold r_m$.

$$\lambda_0 \quad=\quad{ \vert\vert \bold F \bold m - \bold d \vert\vert \over \vert\vert \bold R \bold m \vert\vert }$$ (7)

Another approach is to balance the gradients. The gradient is the ``force'' on the solution m.

$$\lambda_1 \quad=\quad{ \vert\vert \bold F'(\bold F \bold m ... ...t\vert \over \vert\vert \bold R'\bold R \bold m \vert\vert }$$ (8)

where $\bold F'$ is the transpose of $\bold F$and likewise for $\bold R'$.I suspect that $\lambda_1$ is the better choice for $\lambda$but a little more experience would add confidence.