Random Subsets of Structured Deterministic Frames have MANOVA Spectra

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.