Fast computation of Frobenius norm under memory limits Given a large dense matrix $A^{n \times n}$, that does not fit to the memory (RAM). Is there any fast way to compute the exact Frobenius norm of the matrix or its accurate approximation (perhaps, via sampling with reasonable bounds provided)? What if this matrix is a subtraction of two $n \times n$ matrices: $\| A - B \|_F$, where $A$ is sparse, and B is dense?
Some of the results depend on the model under which the matrix is presented, but there's a large collection of results that use sampling or sketching techniques to estimate norms accurately. Among some of the key results are:


sketching methods based on the use of 2-stable distributions: these work even if the entries are "streamed in" in some order, and also work if you are presented instead with the A and the B entries. 
column sampling methods.


Again, it would be unfair to single out a single paper, also because it's not exactly clear which model you're working in: best to google 'norm estimation' or 'column sampling' to find some of the related results. This survey by Mahoney is a useful reference for sampling methods.