Comparing condition numbers

Question

I have two matrices, $A$ and $B$ where $A,B \in R_{mxn}$ ($m>>n$). I need to answer the question: Which one is better-conditioned among $A$ and $B$ ? However, I will be making these comparisons for tons of large matrices and therefore, I am reluctant to explicitly calculate condition numbers for all of them. Is there any way of making this comparison without a computationally expensive process?