# Finding mapping between two spatial representations of the same objects

I have two matrices $U$ and $V$. $U$ is $n \times n$ and $V$ is $n \times m$. (Both are empirical results of an experiment.) I would like to find a linear transformation $A$, $m \times n$, such that $VA = \hat{U}$, where $\hat{U} = \underset{X}{\arg \min}$ $m(X, U)$ for some metric $m$.

Is there a closed form or iterative solution to this problem?

• What is the argmin taken over? If it is taken over all n x n matrices, then $\hat{U} = U$, so presumably you are imagining taking the argmin over a different set. What this set is and what the metric is seem potentially crucial to an answer. – Joshua Grochow Mar 17 '12 at 18:26
• Your formulation implies that any linear transformation A is acceptable to you as long as the product VA is close to U, but are you sure of this? – Tsuyoshi Ito Mar 17 '12 at 18:28
• @JoshuaGrochow even if the argmin is over all $n\times n$ matrices, it's not clear that it's possible to get $U$, so the question is still well defined (the implicit restriction on X is that it's the image of f(A) = VA) – Suresh Venkat Mar 17 '12 at 18:44
• @Suresh: That is what I first thought, but it is not correct. Note that if the argmin is really over all n×n matrices, then it has nothing to do with the product VA! – Tsuyoshi Ito Mar 17 '12 at 19:36
• Right. So X has to be restricted to matrices that can be formed as VA for a fixed V – Suresh Venkat Mar 17 '12 at 20:41

If your error norm is the Frobenius distance (that is, $m(X, U) = \|X - U \|_F$, then there is a closed form for your problem. $A = V^+ U$, where $V^+$ is the Moore-Penrose pseudoinverse of $V$ (this is a generalized inverse for non-square matrices).
• Thanks. This is great. I'm curious whether there is a closed form solution if (1) we take $m$ to be the Schatten $p$-norm of $X - U$ for any $p$ (not just the Frobenius, where $p = 2$); or (2) we take $m$ to be the 2-norm--i.e. the spectral radius of $X - U$. If there is a solution for (2), I wonder whether it is general for a matrix norm induced by a vector $p$-norm for any $p$. – aaronsteven Mar 21 '12 at 21:02