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let $f: \mathbb{R}^{n} \to \mathbb{R}^{m}$ be differentiable and $y \in \mathbb{R}^{m}$ be fixed

Define $g: \mathbb{R}^{n} \to \mathbb{R}$ as $g(x) = \|f(x) - y\|^2$

Now, I want to find $\nabla g$

Okay so, $g(x) = \langle f(x)- y, f(x) - y\rangle$ using properties of inner product we get

$g(x) = \langle f(x), f(x)\rangle - 2 \langle f(x), y\rangle + \langle y,y\rangle$

Now the derivative of $g$ at point $x\in \mathbb{R}^{n}$ is defined as :

$Dg_x = 2\langle Df_x, f\rangle - 2 \langle Df_x , y\rangle$

I am confused at this point and don't know how to get the gradient from here.

Any advice on how to proceed?

Thank you.

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Take $p(x) = f(x) - y$ and $h(x) = \|x\|^2$. Then, $g = h \circ p$. $$Dg_x = D(h\circ p)_x = Dh_{p(x)} \circ Dp_x$$ We know that $Dh_{p(x)}(h) = 2\langle p(x),h \rangle$. $Dp_{x} = Df_x$ since $y$ is constant.

$$Dg_x(h) = Dh_{p(x)} \circ Dp_x(h) = 2\langle f(x) - y, f'(x)h\rangle = \langle \nabla g(x), h \rangle$$ for all $h$.

Can you find $\nabla g(x)$ now?

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  • $\begingroup$ Yeah sorry that was a typo, we need to take that $f'(x)$ out of the inner product to get final expression. $\endgroup$
    – zeroflank
    Sep 5 '21 at 17:04
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HINT

If you are interested in the particular case where we are dealing with the Euclidian norm, we have: \begin{align*} g(x) & = \|f(x) - y\|^{2}\\\\ & = \|(f_{1}(x),f_{2}(x),\ldots,f_{m}(x)) - (y_{1},y_{2},\ldots,y_{m})\|^{2}\\\\ & = \|(f_{1}(x) - y_{1},f_{2}(x) - y_{2},\ldots,f_{m}(x) - y_{m})\|^{2}\\\\ & = (f_{1}(x) - y_{1})^{2} + (f_{2}(x) - y_{2})^{2} + \ldots + (f_{m}(x) -y_{m})^{2} \end{align*}

Can you take it from here?

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