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Register a new userMoments of Subsets of General Equiangular Tight Frames
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Added by
Marina Haikin
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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 $0<gamma<1$) converge to the corresponding MANOVA moments. We bring here an extension for the proof of the Kesten-Mckay moments (real ETF, $gamma=1/2$) cite{magsino2020kesten}. In particular, we establish a recursive computation of the $r$th moment, for $r = 1,2,ldots$, and verify, using a symbolic program, that the recursion output coincides with the MANOVA moments.
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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 -- Wachters 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.
We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzons bound is attained in each unitary geometry of dimension $d = 2^{2l+1}$ over the field $mathbb{F}_{3^2}$. We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.
We study tight projective 2-designs in three different settings. In the complex setting, Zauners conjecture predicts the existence of a tight projective 2-design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach to make quantitative progress on this conjecture in terms of the entanglement breaking rank of a certain quantum channel. We show that this quantity is equal to the size of the smallest weighted projective 2-design. Next, in the finite field setting, we introduce a notion of projective 2-designs, we characterize when such projective 2-designs are tight, and we provide a construction of such objects. Finally, in the quaternionic setting, we show that every tight projective 2-design for H^d determines an equi-isoclinic tight fusion frame of d(2d-1) subspaces of R^d(2d+1) of dimension 3.
We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but~18 known SRG parameters on $v leq 1300$ vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzons bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzons bound cannot be attained in a finite orthogonal geometry of dimension~5.
In the paper we obtain sufficient conditions for a trigonometric polynomial to be a refinement mask corresponding to a tight wavelet frame. The condition is formulated in terms of the roots of a mask. In particular, it is proved that any trigonometric polynomial can serve as a mask if its associated algebraic polynomial has only negative roots (at least one of them, of course, equals $-1$).
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