$$
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Frobenius Approximate Inverses
Given a matrix \(A\), we look for its approximate inverse in the sense
of the Frobenius norm, restricted to a given matrix subspace \(E\):
\[\argmin{D\in E} \quad \norm{AD - I}^2_F\]
where \(\norm{X}^2_F = \tr(X^TX)\). The problem is a linear
least-squares that can be solved by classical methods, but closed-form
formula exist for special cases.
Diagonal Matrices
We begin by expanding:
\[\block{AD - I}^T \block{AD-I} = D^T A^TAD - DA^T - AD - I\]
The objective function then simplifies to:
\[\begin{align}
f(D) &= \tr(D^T A^TAD) - 2 \tr(AD)\\
&= \tr(A^TAD^2) - 2 \tr(AD) \\
&= d^T \diag(A^TA) d - 2 \diag(A)^Td
\end{align}\]
where \(d = \diag(D)\). The solution is simply:
\[d = \frac{\diag(A)}{\diag(A^TA)} = \block{ \frac{A_{ii}}{ \norm{A_i}^2} }_i\]
This preconditioner is cheap to compute, and in practice gives much
better results than the diagonal preconditioner. Convergence proof for
the Jacobi iteration exist in the case of diagonally-dominant
matrices.
Appendix
Sub-multiplicativity of the Frobenius norm
\[\begin{align}
\norm{AB}^2 &= \sum_{i,j} \block{a_i^T b_j}^2 \\
&\leq \sum_{i,j}\norm{a_i}^2\norm{b_j}^2 \quad \text{(Cauchy-Schwarz)}\\
&= \sum_i \norm{a_i}^2 \sum_j \norm{b_j}^2 \\
&= \norm{A}^2 \norm{B}^2
\end{align}\]
References
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