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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 such as quantum state tomography, machine learning and the PhaseLift approach to phaseless reconstruction problems. In order to derive rigorous recovery results, the measurement map is usually modeled probabilistically and convex optimization approaches including nuclear norm minimization are often used as recovery method. In this article, we derive sufficient conditions on the minimal amount of measurements that ensure 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 m > 10r(n 1 + n 2 ) measurements are enough to uniformly and stably recover an n 1 × n 2 matrix of rank at most r. Stability is meant both with respect to passing from exactly rank-r matrices to approximately rank-r matrices and with respect to adding noise on the measurements. 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 particular case of recovering Hermitian rank-r matrices from measurement matrices proportional to rank-one projectors. For r = 1, such a problem reduces to the PhaseLift approach to phaseless recovery, while the case of higher rank is relevant for quantum state tomography. 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. Subsequently, we partially de-randomize this result 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. Complex projective t-designs are discrete sets of vectors whose uniform distribution reproduces the first t moments of the uniform distribution on the sphere. 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 the simpler optimization program of minimizing the ℓ 2 -norm of the residual subject to the positive semidefinite constraint. This has the additional advantage that no estimate of the noise level is required a priori. We discuss applications of such a result in quantum physics and the phase retrieval problem. Apart from the case of independent Gaussian measurements, the analysis exploits Mendelson's small ball method.Keywords. low rank matrix recovery, quantum state tomography, phase retrieval, convex optimization, nuclear norm minimization, positive semidefinite least squares problem, complex projective designs, random measurements MSC 2010. 94A20, 94A12, 60B20, 90C25, 81P50 Date: October 16, 2018.
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 such as quantum state tomography, machine learning and the PhaseLift approach to phaseless reconstruction problems. In order to derive rigorous recovery results, the measurement map is usually modeled probabilistically and convex optimization approaches including nuclear norm minimization are often used as recovery method. In this article, we derive sufficient conditions on the minimal amount of measurements that ensure 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 m > 10r(n 1 + n 2 ) measurements are enough to uniformly and stably recover an n 1 × n 2 matrix of rank at most r. Stability is meant both with respect to passing from exactly rank-r matrices to approximately rank-r matrices and with respect to adding noise on the measurements. 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 particular case of recovering Hermitian rank-r matrices from measurement matrices proportional to rank-one projectors. For r = 1, such a problem reduces to the PhaseLift approach to phaseless recovery, while the case of higher rank is relevant for quantum state tomography. 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. Subsequently, we partially de-randomize this result 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. Complex projective t-designs are discrete sets of vectors whose uniform distribution reproduces the first t moments of the uniform distribution on the sphere. 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 the simpler optimization program of minimizing the ℓ 2 -norm of the residual subject to the positive semidefinite constraint. This has the additional advantage that no estimate of the noise level is required a priori. We discuss applications of such a result in quantum physics and the phase retrieval problem. Apart from the case of independent Gaussian measurements, the analysis exploits Mendelson's small ball method.Keywords. low rank matrix recovery, quantum state tomography, phase retrieval, convex optimization, nuclear norm minimization, positive semidefinite least squares problem, complex projective designs, random measurements MSC 2010. 94A20, 94A12, 60B20, 90C25, 81P50 Date: October 16, 2018.
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