let $x$ be $n$ by matrix and $Y$ be $n$ by $r$ matrix.

$Y=x{W}^T$

Let $R$ be a cost function to be minimized:

$R(W)=||x{W}^T-Y|{|}^2$

$W=(W_N,W_{N-1},$dots.,$W_3,W_2)=W_N{W}_{N-1}dots{W}_3{W}_2$

where vector: $S=(W_N,W_{N-1},..,W_1)in{R}^{d_{}}$

whered ${?}_{}=d_N{d}_{N-1}+d_{N-1}{d}_{N-2}+$dots$+d_1{d}_0$

$()=(W_N,W_{N-1},$dots,$W_1)$

$$ is a column vector

find the vector $$ that minimizes the following equation:

$J()=R(())$

Please use the gradient descent algorithm to find $$ and derive $gra{d}_{}J$

gradient $H=gra{d}_{W}R(W)$ with matrix entries are $H_{ij}=$delS$\frac{W}{d}el{W}_{ij}$

where it is defined by equation $2$

write $2$ explicit formulas for $gra{d}_{W}S(W).$

derive formula for each $H_{ij}$ using summations

and use the result to find $gra{d}_{W}S(W)$ in matrix notation without summation

write the directional derivatives $S_W^{\text{'}}(W)$ as a frobenius inner product of

two matrices

directional derivative ${}_{}^{\text{'}}()$ where $=(W_N,{}_{N-1},$dots,$W_1)$

show that: $W_{?}>N{d}_{N}x{d}_N{d}_{}x{d}_{}{J}_{}^{\text{'}}(){|}_{}={}_{0}S\frac{d}{dx}f(g(x)){|}_x=x_0=\frac{d}{dx}f(y){|}_y=g(x_0)*\frac{d}{dx}g(x){|}_x=x_0{J}_{}^{\text{'}}()gra{d}_{}J()=(2(()x^T-Y^T)x{W}_j^T(()x^T-Y^T)x{W}_1^T(()x^T-Y^T)x).W_{?}>j$