# Multi-objective recursive least squares

1. May 2, 2010

### Superfish

Is this possible?

I've computed a multiobjective least squares solution and want to make it able to be updated recursively but I get stuck at applying the woodbury matrix identity since it's no longer a rank 1 udpate. Are there any derivations of this anywhere or is this not possible? Thanks

2. May 5, 2010

### jasonRF

Of course, if you have a rank 2 update, you can always use Woodbury's twice in a row, etc.

A generalization of Woodbury's identity is
$$(A+BCD)^{-1}=A^{-1} - A^{-1} B(D A^{-1} B + C^{-1} )^{-1} D A^{-1}$$
Where A is nxn, B is nxm, C is mxm, and D is mxn, so this can work for a rank m update. It isn't too suprising that you have to invert a rank m matrix to do it.

jason

EDIT: just realized that Woodbury's was never stated. In the above, if B is nx1 vector, D is the transpose of B and C is just the number 1, then you have Woodbury's identity. The generalization above is usually called the matrix inversion lemma

Last edited: May 5, 2010