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Suppose I'm given some bunch(say M) of matrices pairs (X,Y) where each is a square (n,n) matrix . I wonder if gradient descent algorithm can be applicable to minimize following MSE: $\sum_{i=1}^{M}(Y_i - C\dot X_i)^2$, namely find such C that minimize MSE above? I saw this question, but I want to understand if it is conceptually possible for matrices and especially in my setting above, where I need to find some specific matrix C? Put it simple: suppose that I want to use linear regression framework, but instead of scalar outcome I have matrix outcome? Does it have sense or this is complete nonsense?

Thanks in advance.

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    $\begingroup$ The question makes sense. Setting the gradient of the MSE to zero leads to $$C = \left(\sum_{k=1}^M Y_k X_k^T\right) \left(\sum_{j=1}^M X_j X_j^T\right)^{\bf +}$$ $\endgroup$
    – greg
    Aug 6 at 1:33
  • $\begingroup$ @greg: Thank you very much, + means inverse? $\endgroup$
    – Sharov
    Aug 6 at 12:25
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    $\begingroup$ The system may be under/over-determined, so $M^+$ is the pseudoinverse of $M.\;$ $\endgroup$
    – greg
    Aug 6 at 15:02

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