Scalable frames are frames with the property that the frame vectors can be
rescaled resulting in tight frames. However, if a frame is not scalable, one
has to aim for an approximate procedure. For this, in this paper we introduce
three novel quantitative measures of the closeness to scalability for frames in
finite dimensional real Euclidean spaces. Besides the natural measure of
scalability given by the distance of a frame to the set of scalable frames,
another measure is obtained by optimizing a quadratic functional, while the
third is given by the volume of the ellipsoid of minimal volume containing the
symmetrized frame. After proving that these measures are equivalent in a
certain sense, we establish bounds on the probability of a randomly selected
frame to be scalable. In the process, we also derive new necessary and
sufficient conditions for a frame to be scalable.Comment: 27 pages, 5 figure
