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首页> 外文期刊>IEEE Transactions on Signal Processing >Low-Rank Positive Semidefinite Matrix Recovery From Corrupted Rank-One Measurements
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Low-Rank Positive Semidefinite Matrix Recovery From Corrupted Rank-One Measurements

机译：低等级正半定矩阵从损坏的秩一测量中恢复

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摘要

We study the problem of estimating a low-rank positive semidefinite (PSD) matrix from a set of rank-one measurements using sensing vectors composed of i.i.d. standard Gaussian entries, which are possibly corrupted by arbitrary outliers. This problem arises from applications, such as phase retrieval, covariance sketching, quantum space tomography, and power spectrum estimation. We first propose a convex optimization algorithm that seeks the PSD matrix with the minimum ℓ1-norm of the observation residual. The advantage of our algorithm is that it is free of parameters, therefore, eliminating the need for tuning parameters and allowing easy implementations. We establish that with high probability, a low-rank PSD matrix can be exactly recovered as soon as the number of measurements is large enough, even when a fraction of the measurements are corrupted by outliers with arbitrary magnitudes. Moreover, the recovery is also stable against bounded noise. With the additional information of an upper bound of the rank of the PSD matrix, we propose another nonconvex algorithm based on subgradient descent that demonstrates excellent empirical performance in terms of computational efficiency and accuracy.
机译：我们研究了使用i.i.d组成的传感向量从一组秩一测量值估计低秩正半定（PSD）矩阵的问题。标准高斯条目，可能会被任意离群值破坏。这个问题源于诸如相位检索，协方差草图绘制，量子空间层析成像和功率谱估计之类的应用。我们首先提出一种凸优化算法，该算法以观测残差的最小ℓ1-范数寻找PSD矩阵。我们算法的优点是它没有参数，因此无需调整参数，并且易于实现。我们确定，即使测量的一部分被任意大小的异常值所破坏，只要测量的数量足够大，就可以准确地恢复低秩的PSD矩阵。而且，恢复对于边界噪声也是稳定的。借助PSD矩阵秩的上限的附加信息，我们提出了另一种基于次梯度下降的非凸算法，该算法在计算效率和准确性方面展示了出色的经验性能。

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