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Topic: Least-square optimization with a complex residual function
Replies: 4   Last Post: Oct 30, 2010 11:16 AM

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elgen

Posts: 7
Registered: 8/27/10
Least-square optimization with a complex residual function
Posted: Oct 28, 2010 9:10 PM
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I have a question on the least-square optimization with a complex
residual function. The residual function is r(z_1, z_2), in which z_1
and z_2 are complex variables.

The LS cost function is

\sum_i ||r(z_1, z_2||^2

where i is the number of samples.


Many textbooks discusses this problem in the context of a real residual
function and || \cdot || denotes the euclidean norm.

In my case r(z_1, z_2) is a complex function. If I use the Euclidean
norm (conjugated inner product), the cost function becomes

\sum_i conj(r)r

I am stuck on how to calculate the gradient of this cost function as
conj(r) is not an analytic function and the gradient needs to take the
derivative with respect to z_1 and z_2.

Any feedback is welcome. Thank you.


elgen




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