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-ETFs, 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 (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 k-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 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 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 Marčenko-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.deterministic frames | MANOVA | analog source coding | equiangular tight frames | restricted isometry property C onsider a frame {xi } n i=1 ⊂ R m or C m , and stack the vectors as rows to obtain the n-by-m frame matrix X . Assume that ||xi || 2 = 1 (deterministic frames) or limn→∞ xi = 1 almost surely (random frames). This paper studies properties of a random subframe {xi }i∈K , where K is chosen uniformly at random from [n] = {1, . . . , n} and |K | = k ≤ n. We let XK denote the k -by-m submatrix of X created by picking only the rows {xi }i∈K ; call this object a typical k submatrix of X . We consider a collection of well-known deterministic frames, listed in Table 1, which we denote by X . Most of the frames in X are equiangular tight frames (ETFs), and some are near-ETFs. This paper suggests that, for a frame in X , it is possible to calculate quantities of the form EK Ψ(λ(GK )), where λ(GK ) = (λ1(GK ), ..., λ k (GK )) is the vector of eigenvalues of the k -by-k Gram matrix GK = XK X K and Ψ is a functional of these eigenvalues. As discussed below, such quantities are of considerable interest in various applications where frames are used across a variety of domains, including compressed sensing, sparse recovery, and erasure coding.We present a simple and explicit formula for calculating EK Ψ(λ(GK )) for a given frame in X and a given spectral functional Ψ. Specifically, for the case k ≤ m,where β = k /m, γ = m/n, and f MANOVA β,γ is the density of Wachter's classical multivariate ANOVA limiting distributio...
The Welch Bound is a lower bound on the root mean square cross correlation between n unit-norm vectors f1, ..., fn in the m dimensional space (R m or C m ), for n ≥ m. Letting F = [f1|...|fn] denote the m-by-n frame matrix, the Welch bound can be viewed as a lower bound on the second moment of F , namely on the trace of the squared Gram matrix (F ′ F ) 2 . We consider an erasure setting, in which a reduced frame, composed of a random subset of Bernoulli selected vectors, is of interest. We extend the Welch bound to this setting and present the erasure Welch bound on the expected value of the Gram matrix of the reduced frame. Interestingly, this bound generalizes to the d-th order moment of F . We provide simple, explicit formulae for the generalized bound for d = 2, 3, 4, which is the sum of the d-th moment of Wachter's classical MANOVA distribution and a vanishing term (as n goes to infinity with m n held constant). The bound holds with equality if (and for d = 4 only if) F is an Equiangular Tight Frame (ETF). Our results offer a novel perspective on the superiority of ETFs over other frames in a variety of applications, including spread spectrum communications, compressed sensing and analog coding.
Analog coding decouples the tasks of protecting against erasures and noise. For erasure correction, it creates an "analog redundancy" by means of band-limited discrete Fourier transform (DFT) interpolation, or more generally, by an overcomplete expansion based on a frame. We examine the analog coding paradigm for the dual setup of a source with "erasure" side-information (SI) at the encoder. The excess rate of analog coding above the rate-distortion function (RDF) is associated with the energy of the inverse of submatrices of the frame, where each submatrix corresponds to a possible erasure pattern. We give a partial theoretical as well as numerical evidence that a variety of structured frames, in particular DFT frames with difference-set spectrum and more general equiangular tight frames (ETFs), with a common MANOVA limiting spectrum, minimize the excess rate over all possible frames. However, they do not achieve the RDF even in the limit as the dimension goes to infinity.
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This note outlines the steps for proving that the moments of a randomly-selected subset of a general ETF (complex, with aspect ratio 0 < γ < 1) converge to the corresponding MANOVA moments. We bring here an extension for the proof of the 'Kesten-Mckay' moments (real ETF, γ = 1/2) [1]. In particular, we establish a recursive computation of the rth moment, for r = 1, 2, . . ., and verify, using a symbolic program, that the recursion output coincides with the MANOVA moments.
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