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Random subsets of structured deterministic frames have MANOVA spectra.

Marina Haikin1, Ram Zamir1, Matan Gavish2

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Randomly selecting subsets of frame vectors reveals universal spectral distributions, specifically Wachter

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Area of Science:

  • Linear Algebra
  • Random Matrix Theory
  • Signal Processing

Background:

  • Equiangular tight frames (ETFs) and tight non-ETFs are crucial in signal processing and data analysis.
  • Understanding the spectral properties of random subframes is essential for applications like compressed sensing.
  • Previous research primarily focused on random frames or specific deterministic frames for spectral analysis.

Purpose of the Study:

  • To investigate the spectral distribution of singular values for random subsets of deterministic and non-deterministic frames.
  • To determine if a universal spectral distribution applies to these subsets, even for deterministic frames.
  • To establish the connection between frame subset spectra and established random matrix ensembles.

Main Methods:

  • Randomly selecting subsets of rows (vectors) from various deterministic and non-deterministic frames.
  • Analyzing the singular value distribution of the resulting subset matrices.
  • Comparing the observed distributions with known probability distributions like Wachter's MANOVA and Marčenko-Pastur.

Main Results:

  • For large subset sizes, singular value distributions converge to Wachter's MANOVA spectral distribution for diverse frames.
  • The subset matrices are empirically indistinguishable from the classical MANOVA (Jacobi) random matrix ensemble.
  • Universality holds for both deterministic and notable random frames, with convergence to Marčenko-Pastur for small aspect ratios.

Conclusions:

  • The MANOVA ensemble provides a universal description for the spectra of random subframes, extending beyond random frames.
  • This finding enables precise calculations for systems of linear equations and has implications for coding and recovery.
  • The empirical results are exhaustive, precise, and reproducible, validating the universality of the observed spectral phenomena.