Consistency of trace norm minimization

Regularization by the sum of singular values, also referred to as the trace norm, is a popular technique for estimating low rank rectangular matrices. In this paper, we extend some of the consistency results of the Lasso to provide necessary and sufficient conditions for rank consistency of trace norm minimization with the square loss. We also provide an adaptive version that is rank consistent even when the necessary condition for the non adaptive version is not fulfilled.

💡 Research Summary

The paper investigates the statistical consistency of trace‑norm (nuclear‑norm) regularization for estimating low‑rank rectangular matrices under the squared‑loss setting. While the ℓ₁‑penalized Lasso is known to enjoy variable‑selection consistency given certain irrepresentable‑type conditions, analogous results for matrix‑valued parameters have been less developed. This work bridges that gap by extending Lasso‑style consistency theory to the trace‑norm case, deriving both necessary and sufficient conditions for rank consistency—the property that the estimator recovers the exact rank of the true matrix with probability tending to one as the sample size grows.

Problem formulation.
The authors consider observations generated by a linear operator (\mathcal{X}) acting on an unknown matrix (W^{\star}\in\mathbb{R}^{p\times q}): \