Low rank matrix recovery from rank one measurements

Abstract

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 , . . . , am, i.e., the measurements are given by y j = tr(Xa j a * j ). The case where the matrix X = xx * to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements, y 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 ≥ Crn measurements, while in the case of 4-designs we need m ≥ 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 Gross, Kueng and Krahmer. 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.Date: October 25, 2014.

The Local Convexity of Solving Systems of Quadratic Equations

“…However, as such algorithms optimize over the "lifted" space of n × n positive semidefinite matrices, the computational complexity becomes quite high. In the more general rank-r setting, whereby the measurements are y i := tr(XX T a i a T i ) = X T a i 2 2 , the recent works [7,24] demonstrate that convex relaxation techniques based on nuclear norm minimization can solve such problems from an optimal number of measurements O(nr), but still require large computational cost.…”

Low rank matrix recovery from few orthonormal basis measurements

“…This recovery guarantee was partially derandomized (at the cost of a larger sampling rate) in [13] using the concept of spherical t-designs. Both results were improved by means of uniform counterparts [14], [15] getting by with lower sampling rates 3 .…”

“…Clearly, such a procedure requires one to be able to choose from a continuous, very generic union of bases. However, the results in [13], [15] suggest that such a requirement might not be necessary and that more structured, finite unions of bases may suffice to establish low rank matrix recovery guarantees by means of nuclear norm minimization. For going further into that direction -and, by doing so, partially derandomizing the recovery scheme proposed above -we rely on the concept of spherical designs.…”

“…It uses ideas presented in [13] which resulted in a similar non-optimal factor appearing in the sampling rate. There, employing different techniques allowed for eradicating such a factor and substantially strengthening the statement [15]. In turn, we believe that a more careful analysis will allow for establishing a recovery guarantee getting by with a sampling rate that already for t = 3, or t = 4, is optimal up to polylog-factors.…”

“…A first step in this direction was done in [15], where uniform recovery guarantees for inferring hermitian rank r-matrices from m = Crn projectors onto i.i.d Gauss-random vectors, or from m = Crn log(n) projectors onto randomly chosen elements of a spherical 4-design, were established.…”