I repropose a question I have had no answer on

I am trying to calculate $∇_wMSE=0$ and $∇_mMSE=0$ with '$w$' and '$m$' being matrices of unknown parameters and $MSE=(X⋅m⋅w−Y)^2$ ($X$ and $Y$ are matrices of known values).

If I simply had one set of parameters, i.e., $∇_wMSE=0→∇_m(X⋅m−Y)^2=0$, the solution would be $m=(X^T⋅X)−1⋅X^T⋅Y$

With the two sets of parameters: $∇_wMSE=0→∇_w(X⋅m⋅w−Y)^2=0$


How to solve this last equation? I could apply the chain rule, multiplying by $∇_mX⋅m$, but I am not sure how to then multiply the equation by this result.

  • 1
    $\begingroup$ It would be easier to follow your post if you used math formatting: math.meta.stackexchange.com/questions/5020/… $\endgroup$
    – Sycorax
    Jun 14, 2020 at 15:22
  • $\begingroup$ You can not solve this as a linear equation (stats.stackexchange.com/questions/470818/…) and instead you need to find the minimum with some gradient method, starting with some m and w and improve the solution in small steps. $\endgroup$ Jun 14, 2020 at 15:44
  • $\begingroup$ You may have a problem that there'll be multiple solutions. If some m and w minimize the MSE then m/k and w*k give the same MSE and also minimize the MSE $\endgroup$ Jun 14, 2020 at 15:49


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