Stabilized Spectral Proximal Updates for Distributed Low-Rank Regression

In many applications, several responses are predicted from common predictors while the observations are stored on computers that cannot pool them. Estimating a low-rank coefficient matrix from such data remains a difficult task when local samples have fewer observations than predictors. Corrected local fitting problems can be unbounded, and nuclear-norm penalties shrink strong effects. In this article, we propose a stabilized spectral proximal method that exchanges model estimates and gradients. A majorizing metric computed from one scalar per worker, with the penalty on the matching transformed coefficient, makes every round an exact singular-value update: a nuclear-norm stage followed by a nonconvex refinement that leaves strong singular values unshrunk. Theoretical results show that the computed iterates satisfy finite-iteration prediction bounds, including designs with fewer combined observations than predictors, and that observed singular-value gaps certify the selected rank. Under signal and curvature conditions, the iterates recover the identifiable rank and converge to the least-squares fit of that rank. We also characterize when gradient exchange transmits fewer scalars than sufficient statistics. Simulations show exact rank recovery, accuracy close to centralized estimation with the same nonconvex penalty, and fewer transmitted scalars when predictors greatly outnumber responses.