The paper presents several results that address a fundamental question in low-rank matrices recovery: how many measurements are needed to recover low rank matrices? We begin by investigating the complex matrices case and show that @math generic measurements are both necessary and sufficient for the recovery of rank- @math matrices in @math by algebraic tools. Thus, we confirm a conjecture which is raised by Eldar, Needell and Plan for the complex case. We next consider the real case and prove that the bound @math is tight provided @math . Motivated by Vinzant's work, we construct @math matrices in @math by computer random search and prove they define injective measurements on rank- @math matrices in @math . This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than @math orthogonal projections are possible for the recovery of @math from the norm of them.
In context of low-rank matrix recovery, it is Eldar, Needell and Plan @cite_5 that show that @math Gaussian matrices @math has low-rank matrix recovery property for @math with probability 1 (see also @cite_0 @cite_14 @cite_4 ) provided @math . Naturally, one may be interested in whether the number @math is tight. In @cite_5 , the authors made the following conjecture:
@cite_0 @cite_5 @cite_14 @cite_4
2963848703 2963583445 1968843188 2153742431
Abstract The paper presents several results that address a fundamental question in low-rank matrix recovery: how many measurements are needed to recover low-rank matrices? We begin by investigating the complex matrices case and show that 4 n r − 4 r 2 generic measurements are both necessary and sufficient for the recovery of rank-r matrices in C n × n . Thus, we confirm a conjecture which is raised by Eldar, Needell and Plan for the complex case. We next consider the real case and prove that the bound 4 n r − 4 r 2 is tight provided n = 2 k + r , k ∈ Z + . Motivated by Vinzant's work [19] , we construct 11 matrices in R 4 × 4 by computer random search and prove they define injective measurements on rank-1 matrices in R 4 × 4 . This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than 2 n − 1 orthogonal projections are possible for the recovery of x ∈ R n from the norm of them, which gives a negative answer for a question raised in [1] . 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 , … , 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 Low-rank matrix recovery addresses the problem of recovering an unknown low-rank matrix from few linear measurements. There has been a large influx of literature deriving conditions under which certain tractable methods will succeed in recovery, demonstrating that m ⩾ C n r Gaussian measurements are often sufficient to recover any rank-r n × n matrix. In this paper we address the theoretical question of how many measurements are needed via any method whatsoever — tractable or not. We show that for a family of random measurement ensembles, m ⩾ 4 n r − 4 r 2 and m ⩾ 2 n r − r 2 + 1 measurements are sufficient to guarantee strong recovery and weak recovery, respectively, by rank minimization. These results give a benchmark to which we may compare the efficacy of tractable methods such as nuclear-norm minimization. Recovery of low-rank matrices has recently seen significant activity in many areas of science and engineering, motivated by recent theoretical results for exact reconstruction guarantees and interesting practical applications. A number of methods have been developed for this recovery problem. However, a principled method for choosing the unknown target rank is generally not provided. In this paper, we present novel recovery algorithms for estimating low-rank matrices in matrix completion and robust principal component analysis based on sparse Bayesian learning (SBL) principles. Starting from a matrix factorization formulation and enforcing the low-rank constraint in the estimates as a sparsity constraint, we develop an approach that is very effective in determining the correct rank while providing high recovery performance. We provide connections with existing methods in other similar problems and empirical results and comparisons with current state-of-the-art methods that illustrate the effectiveness of this approach.