0
$\begingroup$

WHAT I DON'T KNOW:

MATRIX CASE: I have a matrix equation $Y=AB$, where Y and A are given and we would like to find $B$. Here are more information: $Y \in R^{m \times n}$ with $m >n$, $A \in R^{m \times k}$ and $B \in R^{k \times n}$ with $k <n$. Obviously the problem is over-determined. Let $\{ \textbf{b}_i\}_{i=1}^{n}$ denote the column vectors of $B$. Now here are the constraints on these vectors. $\textbf{b}_i^TR_i \textbf{b}_i,i=0,1,2,\cdots,n$. How to formulate this problem as an optimization equation and solve it minimizing the least squares or Frobenius norm?

WHAT I KNOW

VECTOR CASE: I know for $\textbf{y}=A\textbf{b}~~$ ($\mathbf{y,b}$ are vectors) and with the constrain $\textbf{b}^TR \textbf{b}$, the optimization formulation is

\begin{equation} \underset{\textbf{b}}{\operatorname{min}}~ \| \textbf{y} - A \textbf{b} \|^{2} + \lambda (\textbf{b}^T R \textbf{b}) \end{equation}

with the least squares solution

\begin{equation} \widehat{\textbf{b}}=\left(A^T A +\lambda R \right)^{-1}A^T \textbf{y}~ \end{equation}

Could anyone help me with the matrix case?

$\endgroup$

1 Answer 1

0
$\begingroup$

If you want to have in a form you are familiar with, you can exploit the fact that $\text{vec}(Y) = (I^{k\times k} \otimes A) \text{vec}(B)$ where $I^{k\times k}$ denotes the identity matrix and $\otimes$ is the kronecker product.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.