Uncertainty Quantification for Matrix Compressed Sensing and Quantum Tomography Problems
Uncertainty Quantification for Matrix Compressed Sensing and Quantum Tomography Problems
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DOI：
10.1007/978-3-030-26391-1_18
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发表时间：
2015-04
期刊：
影响因子：
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通讯作者：
A. Carpentier;J. Eisert;D. Gross;Richard Nickl
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文献类型：
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作者：
A. Carpentier;J. Eisert;D. Gross;Richard Nickl
我们构造极小极大最优非渐近置信集的低秩矩阵恢复算法，如矩阵Lasso或Dantzig选择器。这些都是用来设计自适应顺序采样程序，保证恢复的真实矩阵Frobenius规范后，数据驱动的停止时间的测量，必须采取的数量。在很大的概率下，这个停时是极大极小最优的。我们详细的应用程序，量子层析成像的问题，测量产生的泡利观测。我们还给出了具有最佳核模直径的量子态密度矩阵的置信集的理论构造。我们的信心集的非渐近性质进一步研究在模拟研究。
We construct minimax optimal non-asymptotic confidence sets for low rank matrix recovery algorithms such as the Matrix Lasso or Dantzig selector. These are employed to devise adaptive sequential sampling procedures that guarantee recovery of the true matrix in Frobenius norm after a data-driven stopping timefor the number of measurements that have to be taken. With high probability, this stopping time is minimax optimal. We detail applications to quantum tomography problems where measurements arise from Pauli observables. We also give a theoretical construction of a confidence set for the density matrix of a quantum state that has optimal diameter in nuclear norm. The non-asymptotic properties of our confidence sets are further investigated in a simulation study.