Prove the asymptotic growth of the ADRT’s smallest singular value

Prove the asymptotic growth rate of the smallest singular value σ_min(R^n) of the full approximate discrete Radon transform, in particular whether the measured behavior σ_min(R^n)∼N^{1/4} is correct.

Background

The complexity and conditioning of the explicit oblique inverses depend on σ_min(Rn). Numerical experiments suggest growth approximately like N{1/4}, although the observed range also permits slower effective powers.

The paper attributes the poor conditioning to aliasing produced by integer-sampled digital lines, especially near offset frequency 2π/3. A rigorous asymptotic estimate for σ_min(Rn) would determine the true complexity exponent of the proposed inverses.

References

The growth $\sigma_{\min}(Rn)\sim N{1/4}$ is measured but not proved, and proving it remains open. The asymptotic exponent of both constructions depends on it.

— Explicit Oblique Inverses of the Approximate Discrete Radon Transform  (2610.23363 - Rim, 16 Oct 2026) in Section 6, subsection “Summary” and Section 7, Discussion