  We draw a random subset of $k$ rows from a frame with $n$ rows (vectors) and
$m$ columns (dimensions), where $k$ and $m$ are proportional to $n$. For a
variety of important deterministic equiangular tight frames (ETFs) and tight
non-ETF frames, we consider the distribution of singular values of the
$k$-subset matrix. We observe that for large $n$ they can be precisely
described by a known probability distribution -- Wachter's MANOVA spectral
distribution, a phenomenon that was previously known only for two types of
random frames. In terms of convergence to this limit, the $k$-subset matrix
from all these frames is shown to be empirically indistinguishable from the
classical MANOVA (Jacobi) random matrix ensemble. Thus empirically the MANOVA
ensemble offers a universal description of the spectra of randomly selected
$k$-subframes, even those taken from deterministic frames. The same
universality phenomena is shown to hold for notable random frames as well. This
description enables exact calculations of properties of solutions for systems
of linear equations based on a random choice of $k$ frame vectors out of $n$
possible vectors, and has a variety of implications for erasure coding,
compressed sensing, and sparse recovery. When the aspect ratio $m/n$ is small,
the MANOVA spectrum tends to the well known Marcenko-Pastur distribution of the
singular values of a Gaussian matrix, in agreement with previous work on highly
redundant frames. Our results are empirical, but they are exhaustive, precise
and fully reproducible.
