Random subsets of structured deterministic frames have MANOVA spectra.

ABSTRACT

We draw a random subset of [Formula see text] rows from a frame with [Formula see text] rows (vectors) and [Formula see text] columns (dimensions), where [Formula see text] and [Formula see text] are proportional to [Formula see text] For a variety of important deterministic equiangular tight frames (ETFs) and tight non-ETFs, we consider the distribution of singular values of the [Formula see text]-subset matrix. We observe that, for large [Formula see text], they can be precisely described by a known probability distribution-Wachter's MANOVA (multivariate ANOVA) spectral distribution, a phenomenon that was previously known only for two types of random frames. In terms of convergence to this limit, the [Formula see text]-subset matrix from all of 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 [Formula see text] 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 [Formula see text] frame vectors of [Formula see text] possible vectors and has a variety of implications for erasure coding, compressed sensing, and sparse recovery. When the aspect ratio [Formula see text] 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.