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Low rank matrix recovery from rank one measurements. (English) Zbl 1393.94310

Summary: We study the recovery of Hermitian low rank matrices \(X \in \mathbb{C}^{n \times n}\) from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form \(a_j a_j^*\) for some measurement vectors \(a_1,\ldots, a_m\), i.e., the measurements are given by \(b_j = \text{tr}(X a_j a_j^*)\). The case where the matrix \(X = x x^*\) to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements \(b_j = | \langle x, a_j \rangle |^2\)) via the PhaseLift approach, which has been introduced recently. We derive bounds for the number \(m\) of measurements that guarantee successful uniform recovery of Hermitian rank \(r\) matrices, either for the vectors \(a_j\), \(j = 1, \ldots, m\), being chosen independently at random according to a standard Gaussian distribution, or \(a_j\) being sampled independently from an (approximate) complex projective \(t\)-design with \(t = 4\). In the Gaussian case, we require \(m \geq Crn\) measurements, while in the case of 4-designs we need \(m \geq Crn \log(n)\). Our results are uniform in the sense that one random choice of the measurement vectors \(a_j\) guarantees recovery of all rank \(r\)-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to D. Gross et al. [Appl. Comput. Harmon. Anal. 42, No. 1, 37–64 (2017; Zbl 1393.94250)]. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.

MSC:

94A12 Signal theory (characterization, reconstruction, filtering, etc.)
94A20 Sampling theory in information and communication theory
60B20 Random matrices (probabilistic aspects)
90C25 Convex programming
81P50 Quantum state estimation, approximate cloning

Citations:

Zbl 1393.94250
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References:

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