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I'm trying to expand this Frobenius form $||C \circ (A-XB)||_F^2$ (here $\circ$ is the Hadamard point-wise multiplication). I want to find the minimum value with respect to X.

$$ \frac{\partial}{\partial X}||C \circ (A-XB)||_F^2 = 0$$

I've being trying to develop using the fact that the Frobenius form is $||A||_F^2=trace(AA^*)$ but the Hadamard product is always on my way.

How would you approach this?

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You can put $g(X)=C\circ (A-XB)$ where $\circ$ is the Hadamard product, and use the chain rule. –  Davide Giraudo Jan 30 '12 at 16:41

1 Answer 1

You need to use the chain rule. $$\partial (\mathbf{X} \circ\mathbf{Y}) = \partial \mathbf{X} \circ \mathbf{Y} + \mathbf{X} \circ \partial \mathbf{Y} $$

Also, the fact that $$||\mathbf{A}||_2^2 = trace(\mathbf{A}^T\mathbf{A})$$

Look the following reference for matrix calculus it has all the stuff you would need.

Matrix Cookbook: [http://orion.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf][1]

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