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Assuming $A$ a set of vectors from a normal distribution, and $X$ a projection matrix and $B$ a set of projected vectors of $A$ using $X$:

$B=A*X$

Using an EM approach and by initializing X from random, I am interested in learning $X$ from training data. For the E step, I guess I should initialize $X$ from random, then using the current estimation of $X$, I should update $X$ using my likelihood function. how can I define my likelihood function? and then how can I update $X$ in each iteration?

here is a psudo code for the training data an a simple projection:

A=randn(13, 1000);
X=2*eye(13);
B=X*A;
X_hat=randn(13, 13);


for iter=1:1000
    Ex = E_step(X_hat,A,B);
    X_hat = M_step(Ex ,X_hat,A,B);
    %how to learn X initialized from random?
end
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  • $\begingroup$ I suggest you write down the observed and completed likelihoods in your question to make it clearer. $\endgroup$
    – Xi'an
    Jan 21, 2016 at 9:53
  • $\begingroup$ @Xi'an I have a bit of problem with defining the likelihood function myself. Not sure, but I know I want to minimize $B - A*X_hat$ as much as possible by optimizing $X_hat$ initiated from random. May I ask for your help, please? $\endgroup$
    – PickleRick
    Jan 21, 2016 at 11:05
  • $\begingroup$ What is your observable? your latent variable? $\endgroup$
    – Xi'an
    Jan 21, 2016 at 11:28
  • $\begingroup$ $A_i$ are the observations over time, and $X$ is the variable I want to learn. $B_i$ are the projected observations from $A_i$ via $X$. $\endgroup$
    – PickleRick
    Jan 21, 2016 at 12:21

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