In Latent Semantic Analysis one uses the SVD to perform a dimensional reduction of the term-document matrix, via the Eckart-Young theorem.
Now, the rank $k$ approximation obtained by E-Y is proven to be the best approximation in the Euclidean norm as well as in the Frobenius norm.
The whole literature I'm reading about LSA says something like:
"For the purposes of LSA, we consider the best approximation in the Frobenius norm, provided by Eckart-Young."
But nobody says why. I mean, why not in the Euclidean norm?