低秩PSD矩阵的秩一测度具有小可行集
Rank-One Measurements of Low-Rank PSD Matrices Have Small Feasible Sets
摘要：
我们研究的约束集在确定低秩，半正定（PSD）矩阵传感问题的解决方案的作用。我们考虑的设置涉及秩一传感矩阵：特别是，给定一组近似低秩PSD矩阵的秩一投影，我们表征满足测量的PSD矩阵集的半径。这个结果产生一个采样率，以保证单例解集时，真正的矩阵是低秩，这样的目标函数或算法的选择是无关紧要的，在其恢复。我们讨论了这一贡献的应用，并比较它与最近的文献隐式正则化类似的问题。我们证明了这一结果的实际影响，通过应用圆锥投影方法PSD矩阵恢复不纳入低秩正则化。
英文摘要：
We study the role of the constraint set in determining the solution to low-rank, positive semidefinite (PSD) matrix sensing problems. The setting we consider involves rank-one sensing matrices: In particular, given a set of rank-one projections of an approximately low-rank PSD matrix, we characterize the radius of the set of PSD matrices that satisfy the measurements. This result yields a sampling rate to guarantee singleton solution sets when the true matrix is exactly low-rank, such that the choice of the objective function or the algorithm to be used is inconsequential in its recovery. We discuss applications of this contribution and compare it to recent literature regarding implicit regularization for similar problems. We demonstrate practical implications of this result by applying conic projection methods for PSD matrix recovery without incorporating low-rank regularization.
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