Is there a 'soft' version of the matrix rank where the rank of a matrix is a real number rather than an integer?
Suppose, for example, we have a matrix with $2$ row vectors, where one of the vectors is equal to the other one plus some small amount of 'noise'. In this case, using the usual, strict rank definition, the matrix will be of rank $2$ almost surely, although the matrix could be interpreted as less 'rankish' compared to a matrix consisting of perpendicular row vectors (which would be of rank $2$ too). However, it does not seem quite fair to treat both matrices in the same way.
Any help on this problem is appreciated.
Not sure if standard, but some texts use a "stable rank" defined as the ratio between squared Frobenius norm and the squared spectral norm of a matrix (equivalently the ratio of the sum of squares of all the singular values to the square of the max singular value).
For example, https://nickhar.wordpress.com/2012/02/29/lecture-15-low-rank-approximation-of-matrices/.