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

Generic MANOVA limit theorems for products of projections

We study the convergence of the empirical spectral distribution of $\mathbf{A} \mathbf{B} \mathbf{A}$ for $N \times N$ orthogonal projection matrices $\mathbf{A}$ and $\mathbf{B}$, where $\frac{1}{N}\mathrm{Tr}(\mathbf{A})$ and $\frac{1}{N}\mathrm{Tr}(\mathbf{B})$ converge as $N \to \infty$, to Wachter's MANOVA law. Using free probability, we show mild sufficient conditions for convergence in moments and in probability, and use this to prove a conjecture of Haikin, Zamir, and Gavish (2017) on random subsets of unit-norm tight frames. This result generalizes previous ones of Farrell (2011) and Magsino, Mixon, and Parshall (2021). We also derive an explicit recursion for the difference between the empirical moments $\frac{1}{N}\mathrm{Tr}((\mathbf{A} \mathbf{B} \mathbf{A})^k)$ and the limiting MANOVA moments, and use this to prove a sufficient condition for convergence in probability of the largest eigenvalue of $\mathbf{A} \mathbf{B} \mathbf{A}$ to the right edge of the support of the limiting law in the special case where that law belongs to the Kesten-McKay family. As an application, we give a new proof of convergence in probability of the largest eigenvalue when $\mathbf{B}$ is unitarily invariant; equivalently, this determines the limiting operator norm of a rectangular submatrix of size $\frac{1}{2}N \times \alpha N$ of a Haar-distributed $N \times N$ unitary matrix for any $\alpha \in (0, 1)$. Unlike previous proofs, we use only moment calculations and non-asymptotic bounds on the unitary Weingarten function, which we believe should pave the way to analyzing the largest eigenvalue for products of random projections having other distributions.Comment: 49 pages, 1 figur

Models and Analysis of Vocal Emissions for Biomedical Applications

The International Workshop on Models and Analysis of Vocal Emissions for Biomedical Applications (MAVEBA) came into being in 1999 from the particularly felt need of sharing know-how, objectives and results between areas that until then seemed quite distinct such as bioengineering, medicine and singing. MAVEBA deals with all aspects concerning the study of the human voice with applications ranging from the neonate to the adult and elderly. Over the years the initial issues have grown and spread also in other aspects of research such as occupational voice disorders, neurology, rehabilitation, image and video analysis. MAVEBA takes place every two years always in Firenze, Italy