Uncertainty Quantification for Matrix Compressed Sensing and Quantum Tomography Problems

摘要

We construct minimax optimal non-asymptotic confidence sets for low rankmatrix recovery algorithms such as the Matrix Lasso or Dantzig selector. Theseare employed to devise adaptive sequential sampling procedures that guaranteerecovery of the true matrix in Frobenius norm after a data-driven stopping time$hat n$ for the number of measurements that have to be taken. With highprobability, this stopping time is minimax optimal. We detail applications toquantum tomography problems where measurements arise from Pauli observables. Wealso give a theoretical construction of a confidence set for the density matrixof a quantum state that has optimal diameter in nuclear norm. Thenon-asymptotic properties of our confidence sets are further investigated in asimulation study.