Minimizing $\|A−XB\|^2_F$

Question

Find a closed form solution for $$\underset{X}{\operatorname{argmin}} \|A−XB\|^2_F$$ where $\| \cdot \|_F$ denotes the Frobenius norm.

I have found this thread. Not sure if it helps here though. Any ideas? Do I just start analyzing the above given the Frobenius norm and then simply calculate the derivatives? Is there a better way to go?

Please learn how to write formulas in Latex , here $$\operatorname{argmin}_X \|A−XB\|^2_F$$ (and take note of use the name "argmin") — Jean Marie, Mar 27 '22 at 11:32
It helps! Since your problems is convex (linear function composed with Frobenius norm), only knowing the function's critical points is enough to solve the convex problem. Actually, I would say to you to vectorize the problem by using the Frobenius norm definition. This is a least-square problem — R. W. Prado, Mar 27 '22 at 11:48
$X=AB^+$, where $B^+$ denotes the Moore-Penrose pseudoinverse of $B$, is a minimiser. — user1551, Mar 27 '22 at 12:04
Why don't you start by using the definition of the Frobenius norm? — Rodrigo de Azevedo, Mar 27 '22 at 12:05
@JeanMarie Rather than defining a new operator, you can cheat and combine two existing ones, i.e. $$\arg\!\min_X$$ — greg, Mar 27 '22 at 15:48

score 1 · Answer 1 · answered Nov 15 '24 at 19:55

1

$X=AB^+$, where $B^+$ denotes the Moore Penrose Pseudo Inverse of $B$, is a minimiser.

answered Nov 15 '24 at 19:55

Royi

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Minimizing $\|A−XB\|^2_F$

1 Answers1