how can I differentiate the following wrt $\mathbf{d}_i$?
$\frac{|\mathbf{d}_i^T\mathbf{d}_j|}{\|\mathbf{d}_i\|_2\|\mathbf{d}_j\|_2}$
Thanks in advance.
2026-04-02 18:40:38.1775155238
how to differentiate this equation (contains absolute and norm)
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Hints: For $x,y \in \mathbb{R}^n$, $$\nabla_{x} |x^Ty|= \text{sign}(|x^Ty|)y,\qquad \nabla_{x}\|x\|_2 = \dfrac{ x }{ \|x\|_2 }, $$ So $$\nabla_x \left(\dfrac{|x^Ty|}{\|x\|_2\|y\|_2}\right) = \dfrac{\|x\|_2\nabla_{x}|x^Ty|+|x^Ty|\nabla_{x}\|x\|_2}{\|x\|_2^2\|y\|_2}=\ldots $$