Let $f_i,b_1,\cdots b_k$ be column vectors of length $d$, and let $[c_{i1}, \cdots c_{ik}]$ be a row vector.
Let $O = ||f_i -\sum_{r=1}^{k} c_{ir}(b_r) ||^2_2$. The second term is a linear combination of the $b_r$.
I am interested in calculating the partial derivative with respect to $c$, i.e $[\frac{\partial O}{\partial c_{i1}} \cdots \frac{\partial O}{\partial c_{ik}}]$.
When I use the norm definition and calculate the partial derivative, the expression looks very long for each $c_{ir}$. Is there any way to represent the above partial derivative in a very compact manner?