TITLE: Minimize $ \arg \min_{A} {\| A x - b \|}_{2} $ (Least Squares Minimization on the Matrix)
QUESTION [9 upvotes]: I have a machine learning regression problem. I need to minimize
$$\sum_i \|Ax_i-b_i\|_2^2$$
However I am trying to find matrix $A$, not the usual $x$, and I have lots of example data for $x_i$ and $b_i$. In general $A$ has more rows than columns.
Additionally I would like a solution for minimizing the Mahalanobis distance version, where $C$ is the SPSD covariance matrix:
$$\sum_i(Ax_i-b_i)^TC^{-1}(Ax_i-b_i)$$
I'm thinking that my problem can be re-written into the usual least squares problem but not sure if I am doing it correctly.

REPLY [3 votes]: For the second problem, note $$\frac{\rm d}{\rm{d} A}((Ax-b)^TC^{-1}(Ax-b))=-2xb^T{C^{-1}}^T+{C^{-1}}^TAxx^T+C^{-1}Axx^T$$
so if $C^{-1}$ is symmetric (hence $C$ is also symmetric), we have the normal equation for $\sum_i(Ax_i-b_i)^TC^{-1}(Ax_i-b_i)$
$$\sum_iAx_ix_i^T=\sum_iCx_ib_i^TC^{-1}$$
or
$$\sum_ix_ix_i^TA^T=\sum_iC^{-1}b_ix_i^TC$$
So $A$ can be computed row by row, and let $a_j$ be the $j$-th column of $A^T$ ($j$-th row of $A$) and $d_j$ be the $j$-th column of $\sum_iC^{-1}b_ix_i^TC$, thus $a_j$ can be computed by solving $$\sum_ix_ix_i^Ta_j=d_j$$