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Let A, B, C and D be some matrix and x be a vector, I want to solve the following optimization problem:

$$ \min_x \| Ax -B\|^2_2 + \| Cx-D\|^2_2 $$ my solution:

$$ J = \| Ax -B\|^2_2 + \| Cx-D\|^2_2 $$

$$ \frac{\partial J}{\partial x} = 2A^T(Ax -B) + 2C^T(Cx-D) = 0 $$

so, $$ x = (A^TA+C^TC)^{-1}(A^TB+C^TD) $$

But in some literature, I see the authors use FFT to solve such problems.

My questions are:
1) How one can use FFT to solve the above problem?
2) What is the advantage to use FFT?

Thanks.

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