Extend the Random Matrix Theory treatment to the underdetermined regime α < 1

Develop a Random Matrix Theory analysis for the underdetermined regime α = M/N < 1 of the Gaussian random linear system A x = b constrained to the sphere ||x||^2 = N, where A_ij ~ N(0,1/N) and b_k ~ N(0,σ^2), to compute the large-N limit of the average minimal loss and related quantities using the anti-Wishart spectral density, thereby completing the RMT treatment beyond the α > 1 case.

Background

The Lagrange multiplier and Random Matrix Theory (RMT) approach in these notes yields an explicit formula for the average minimal loss in the overdetermined regime α > 1. This relies on properties of the Wishart ensemble and the Marchenko–Pastur law.

The author notes that a parallel RMT calculation for the underdetermined regime α < 1 would require using the anti-Wishart ensemble and its spectral density, but this extension is not carried out here. Completing this analysis would provide a unified RMT account across both regimes and allow direct comparison with the replica-based results that already cover α < 1.

References

It is likely that the RMT treatment of the case α<1 could be carried out in full as well, however this is left for future work.

— Random Linear Systems with Quadratic Constraints: from Random Matrix Theory to replicas and back (2401.03209 - Vivo, 6 Jan 2024) in Section 2 (Lagrange multiplier method), following Eq. (finalEminLagrange), footnote