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The problem of recovering a matrix of low rank from an incomplete and possibly noisy set of linear measurements arises in a number of areas. In order to derive rigorous recovery results, the measurement map is usually modelled probabilistically. We derive sufficient conditions on the minimal amount of measurements ensuring recovery via convex optimization. We establish our results via certain properties of the null space of the measurement map. In the setting where the measurements are realized as Frobenius inner products with independent standard Gaussian random matrices, we show that 10r(n1 + n2) measurements are enough to uniformly and stably recover an n1 x n2 matrix of rank at most r. We then significantly generalize this result by only requiring independent mean zero, variance one entries with four finite moments at the cost of replacing 10 by some universal constant. We also study the case of recovering Hermitian rank-r matrices from measurement matrices proportional to rank-one projectors. For m ≥ Crn rank-one projective measurements onto independent standard Gaussian vectors, we show that nuclear norm minimization uniformly and stably reconstructs Hermitian rank-r matrices with high probability. Next, we partially de-randomize this by establishing an analogous statement for projectors onto independent elements of a complex projective 4-designs at the cost of a slightly higher sampling rate m ≥ Crn log n. Moreover, if the Hermitian matrix to be recovered is known to be positive semidefinite, then we show that the nuclear norm minimization approach may be replaced by minimizing the lq-norm of the residual subject to the positive semidefinite constraint (e.g. by a positive semidefinite least squares problem). Then no estimate of the noise level is required a priori. We discuss applications in quantum physics and the phase retrieval problem. [ABSTRACT FROM AUTHOR]

We study the recovery of Hermitian low rank matrices X ∈ C n × 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 , … , a m , i.e., the measurements are given by b j = 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 = | 〈 x , a j 〉 | 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 , … , 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 ≥ C r n measurements, while in the case of 4-designs we need m ≥ Cr n 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 Gross, Krahmer and Kueng. 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. [ABSTRACT FROM AUTHOR]