Hi guys suppose a I have a family of functions
$$ \vec{f}(x_0,\ldots,x_{l-1} ; a_0,\ldots,a_{l-1}) = \vec{f}(\vec{x};\vec{a}) = (f_0(\vec{x};\vec{a}),\ldots,f_{n-1}(\vec{x};\vec{a}))^T $$
And I have samples
$$ \left\{(\vec{x}_0,\vec{f}_0),\ldots,(\vec{x}_{m-1},\vec{f}_{m-1})\right\} $$
And I wanted to find the vector $\vec{a}$ such that the error
$$ \epsilon^2(\vec{a}) = \sum_{k=0}^{m-1} \left\lVert \vec{f}(\vec{x}_k;\vec{a})-\vec{f}_k \right\rVert^2 $$
Is minimum, assuming $\vec{f}$ is differentiable w.r.t. $\vec{a}$. I get
$$ \nabla_{\vec{a}}\epsilon^2 = \sum_{k=0}^{m-1} \nabla_{\vec{a}} \left\lVert \vec{f}(\vec{x}_k;\vec{a})-\vec{f}_k \right\rVert^2= 2 \sum_{k=0}^{m-1} \left\langle \nabla_{\vec{a}} \vec{f}(\vec{x}_k;\vec{a}),\vec{f}(\vec{x}_k;\vec{a})-\vec{f}_k \right\rangle = {2 \sum_{k=0}^{m-1} \textbf{J}_{\vec{x}_k}(\vec{a})^T\left( \vec{f}(\vec{x}_k;\vec{a})-\vec{f}_k \right)} $$
Where with $\textbf{J}_{\vec{x}_k}(\vec{a})$ I denote $$ \textbf{J}_{\vec{x}_k}(\vec{a}) = \begin{pmatrix} \frac{\partial \vec{f}(\vec{x}_k;\vec{a})}{\partial a_0} & \ldots & \frac{\partial \vec{f}(\vec{x}_k;\vec{a})}{\partial a_{l-1}} \end{pmatrix} = \begin{pmatrix} \nabla_{\vec{a}}f_0(\vec{x}_k;\vec{a})^T \\ \vdots \\ \nabla_{\vec{a}}f_{n-1}(\vec{x}_k;\vec{a})^T \\ \end{pmatrix} $$
I'm asking If I have messed up anything in this process.