Asymptotic MSE of the minimal nuclear norm estimator in matrix sensing/BSR

Develop an asymptotically exact analytical characterization of the mean-squared estimation error of the minimal nuclear norm estimator (MNNE) in the high-dimensional limit for the matrix sensing setting corresponding to the bilinear sequence regression model, as a function of the sample ratio, aspect ratio, and width ratio.

Background

The paper compares Bayes-optimal performance with the MNNE, for which strong recovery thresholds are known, and presents finite-size simulations indicating that MNNE is close to optimal below the Bayes threshold but suboptimal between the Bayes and MNNE strong recovery thresholds.

However, a general closed-form prediction for the MNNE mean-squared error curve in the high-dimensional limit appears to be missing, and providing it would quantify precisely MNNE’s suboptimality relative to Bayes-optimal inference across regimes.