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**1 - 3**of**3**### STABLE LOW-RANK MATRIX RECOVERY VIA NULL SPACE PROPERTIES

"... Abstract. 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 recove ..."

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Abstract. 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(n1 + n2) measurements are enough to uniformly and stably recover an n1 × n2 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

### Efficient Compressive Phase Retrieval with Constrained Sensing Vectors

"... Abstract We propose a robust and efficient approach to the problem of compressive phase retrieval in which the goal is to reconstruct a sparse vector from the magnitude of a number of its linear measurements. The proposed framework relies on constrained sensing vectors and a two-stage reconstructio ..."

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Abstract We propose a robust and efficient approach to the problem of compressive phase retrieval in which the goal is to reconstruct a sparse vector from the magnitude of a number of its linear measurements. The proposed framework relies on constrained sensing vectors and a two-stage reconstruction method that consists of two standard convex programs that are solved sequentially. In recent years, various methods are proposed for compressive phase retrieval, but they have suboptimal sample complexity or lack robustness guarantees. The main obstacle has been that there is no straightforward convex relaxations for the type of structure in the target. Given a set of underdetermined measurements, there is a standard framework for recovering a sparse matrix, and a standard framework for recovering a low-rank matrix. However, a general, efficient method for recovering a jointly sparse and low-rank matrix has remained elusive. Deviating from the models with generic measurements, in this paper we show that if the sensing vectors are chosen at random from an incoherent subspace, then the low-rank and sparse structures of the target signal can be effectively decoupled. We show that a recovery algorithm that consists of a low-rank recovery stage followed by a sparse recovery stage will produce an accurate estimate of the target when the number of measurements is O(k log d k ), where k and d denote the sparsity level and the dimension of the input signal. We also evaluate the algorithm through numerical simulation.

### Efficient quantum tomography

, 2015

"... In the quantum state tomography problem, one wishes to estimate an unknown d-dimensional mixed quantum state ρ, given few copies. We show that O(d/) copies suffice to obtain an estimate ρ ̂ that satisfies ‖ρ̂−ρ‖2F ≤ (with high probability). An immediate consequence is that O(rank(ρ)·d/2) ≤ O(d2/2 ..."

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In the quantum state tomography problem, one wishes to estimate an unknown d-dimensional mixed quantum state ρ, given few copies. We show that O(d/) copies suffice to obtain an estimate ρ ̂ that satisfies ‖ρ̂−ρ‖2F ≤ (with high probability). An immediate consequence is that O(rank(ρ)·d/2) ≤ O(d2/2) copies suffice to obtain an -accurate estimate in the standard trace distance. This improves on the best known prior result of O(d3/2) copies for full tomography, and even on the best known prior result of O(d2 log(d/)/2) copies for spectrum estimation. Our result is the first to show that nontrivial tomography can be obtained using a number of copies that is just linear in the dimension. Next, we generalize these results to show that one can perform efficient principal component analysis on ρ. Our main result is that O(kd/2) copies suffice to output a rank-k approximation ρ ̂ whose trace distance error is at most more than that of the best rank-k approximator to ρ. This subsumes our above trace distance tomography result and generalizes it to the case when ρ is not guaranteed to be of low rank. A key part of the proof is the analogous generalization of our spectrum-learning results: we show that the largest k eigenvalues of ρ can be estimated to trace-distance error using O(k2/2) copies. In turn, this result relies on a new coupling theorem concerning the Robinson–Schensted–Knuth algorithm that should be of independent combinatorial interest. 1