I have just started to learn IA and I’m trying to build an autoencoder for data covariance denoising. I consider here a data covariance matrix (~30/40 rows/columns).
In particular the reconstruct eigenvectors matter,but not the matrix by itself; for that reason, as far I know, my problem is a kind of regression problems.

I know the MSE is adapted when we aim at reconstructing “accuratily” each coefficient of a vector (or a matrix, after a good reshape), but other loss functions seem interesting, like:

• the cosine similarity, which could be interesting because it relaxes the constrain to have “perfect coefficient matching”(I would like the reconstructed vectors to have the same direction as the training vectors, but not necessarily the same norm)
• the Frobenius norm could be an option but I don’t know the real benefits of using it.
  Thank you by advance for your help and sorry if the question has already been asked before