MHB Relation within Gauss-Newton method for minimization

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The Gauss-Newton method updates the parameter θ in nonlinear regression by minimizing the sum of squares of residuals. The update formula is derived from the first-order Taylor expansion of the function around the current estimate. Specifically, the update is expressed as θ^(t+1) = θ^(t) + [(A^(t))^TA^(t)]^(-1)(A^(t))^Tx^(t), where A^(t) contains the Jacobian of the model and x^(t) represents the residuals. This formulation allows for efficient computation of the parameter updates by leveraging matrix operations. Understanding these relationships is crucial for implementing the Gauss-Newton method effectively in optimization tasks.
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If we study model fit on a nonlinear regression model $Y_i=f(z_i,\theta)+\epsilon_i$, $i=1,...,n$, and in the Gauss-Newton method, the update on the parameter $\theta$ from step $t$ to $t+1$ is to minimize the sum of squares $\sum_{i=1}^{n}[Y_i-f(z_i,\theta^{(t)})-(\theta-\theta^{(t)})^Tf'(z_i,\theta^{(t)})]^2$. Can we prove that (why) (part 1) the update is given in the following form: $\theta^{(t+1)}=\theta^{(t)}+[(A^{(t)})^TA^{(t)}]^{-1}(A^{(t)})^Tx^{(t)}$,(part 2) where $A^{(t)}$ is a matrix whose $i$-th row is $f'(z_i,\theta^{(t)})^T$, and $x^{(t)}$ is a column vector whose $i$-th entry is $Y_i-f(z_i,\theta^{(t)})$.
Any solution or hints? How to derive those relationships?

Thanks in advance!
 
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ianchenmu said:
If we study model fit on a nonlinear regression model $Y_i=f(z_i,\theta)+\epsilon_i$, $i=1,...,n$, and in the Gauss-Newton method, the update on the parameter $\theta$ from step $t$ to $t+1$ is to minimize the sum of squares $\sum_{i=1}^{n}[Y_i-f(z_i,\theta^{(t)})-(\theta-\theta^{(t)})^Tf'(z_i,\theta^{(t)})]^2$. Can we prove that (why) (part 1) the update is given in the following form: $\theta^{(t+1)}=\theta^{(t)}+[(A^{(t)})^TA^{(t)}]^{-1}(A^{(t)})^Tx^{(t)}$,(part 2) where $A^{(t)}$ is a matrix whose $i$-th row is $f'(z_i,\theta^{(t)})^T$, and $x^{(t)}$ is a column vector whose $i$-th entry is $Y_i-f(z_i,\theta^{(t)})$.
Any solution or hints? How to derive those relationships?

Thanks in advance!

How are those iterations (or updates) defined in the Gauss-Newton method?
 
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