Regularization-Free Estimation in Trace Regression with Symmetric Positive Semidefinite Matrices

Trace regression models have received considerable attent ion in the context of matrix completion, quantum state tomography, and compress ed sensing. Estimation of the underlying matrix from regularization-based approaches promoting low-rankedness, notably nuclear norm regularization, hav e enjoyed great popularity. In this paper, we argue that such regularization may no l onger be necessary if the underlying matrix is symmetric positive semidefinite ( spd) and the design satisfies certain conditions. In this situation, simple lea st squares estimation subject to an spd constraint may perform as well as regularization-based app roaches with a proper choice of regularization parameter, which ent ails knowledge of the noise level and/or tuning. By contrast, constrained least s quares estimation comes without any tuning parameter and may hence be preferred due t o its simplicity.