Relaxing prior tail assumptions beyond compact support

Determine whether the main universality and performance results for non-linear low-rank matrix estimation—specifically the Gaussian approximation of the free energy, universality of overlaps and MMSE, and AMP optimality—continue to hold when replacing the compact-support assumption on the prior distribution π with weaker tail conditions on the transformed prior π_{k_F}, such as sub-Gaussian or stretch-exponential decay, in line with spectral analyses of related models.

Background

The paper’s analysis assumes a compactly supported prior π and smoothness conditions on the log-likelihood to establish higher-order Fisher universality results, free-energy equivalence with a Gaussian model, and algorithmic guarantees via AMP and spectral methods.

In Section 2.4, the authors suggest that these assumptions may be stronger than necessary, and propose relaxing the prior’s compact support to weaker tail conditions (e.g., sub-Gaussian or stretch-exponential) as seen in recent spectral studies. They explicitly leave the task of proving the results under these weaker assumptions for future work.