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
1657130172 1968843188 1485629353
This paper establishes information-theoretic limits for estimating a finite-field low-rank matrix given random linear measurements of it. These linear measurements are obtained by taking inner products of the low-rank matrix with random sensing matrices. Necessary and sufficient conditions on the number of measurements required are provided. It is shown that these conditions are sharp and the minimum-rank decoder is asymptotically optimal. The reliability function of this decoder is also derived by appealing to de Caen's lower bound on the probability of a union. The sufficient condition also holds when the sensing matrices are sparse-a scenario that may be amenable to efficient decoding. More precisely, it is shown that if the n × n-sensing matrices contain, on average, Ω(nlog n) entries, the number of measurements required is the same as that when the sensing matrices are dense and contain entries drawn uniformly at random from the field. Analogies are drawn between the aforementioned results and rank-metric codes in the coding theory literature. In fact, we are also strongly motivated by understanding when minimum rank distance decoding of random rank-metric codes succeeds. To this end, we derive minimum distance properties of equiprobable and sparse rank-metric codes. These distance properties provide a precise geometric interpretation of the fact that the sparse ensemble requires as few measurements as the dense one. 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. Nuclear norm minimization (NNM) has recently gained significant attention for its use in rank minimization problems. Similar to compressed sensing, using null space characterizations, recovery thresholds for NNM have been studied in. However simulations show that the thresholds are far from optimal, especially in the low rank region. In this paper we apply the recent analysis of Stojnic for compressed sensing to the null space conditions of NNM. The resulting thresholds are significantly better and in particular our weak threshold appears to match with simulation results. Further our curves suggest for any rank growing linearly with matrix size n we need only three times of oversampling (the model complexity) for weak recovery. Similar to we analyze the conditions for weak, sectional and strong thresholds. Additionally a separate analysis is given for special case of positive semidefinite matrices. We conclude by discussing simulation results and future research directions.