Massive MIMO is a variant of multiuser MIMO where the number of base-station antennas M is very large (typically ≈ 100), and generally much larger than the number of spatially multiplexed data streams (typically ≈ 10). The benefits of such approach have been intensively investigated in the past few years, and all-digital experimental implementations have also been demonstrated. Unfortunately, the front-end A D conversion necessary to drive hundreds of antennas, with a signal bandwidth of the order of 10 to 100 MHz, requires very large sampling bitrate and power consumption. In order to reduce such implementation requirements, Hybrid Digital-Analog architectures have been proposed. In particular, our work in this paper is motivated by one of such schemes named Joint Spatial Division and Multiplexing (JSDM), where the downlink precoder (resp., uplink linear receiver) is split into the product of a baseband linear projection (digital) and an RF reconfigurable beamforming network (analog), such that only a reduced number m M of A D converters and RF modulation demodulation chains is needed. In JSDM, users are grouped according to similarity of their channel dominant subspaces, and these groups are separated by the analog beamforming stage, where multiplexing gain in each group is achieved using the digital precoder. Therefore, it is apparent that extracting the channel subspace information of the M -dim channel vectors from snapshots of m-dim projections, with m M , plays a fundamental role in JSDM implementation. In this paper, we develop novel efficient algorithms that require sampling only m = O(2 √ M) specific array elements according to a coprime sampling scheme, and for a given p M , return a p-dim beamformer that has a performance comparable with the best p-dim beamformer that can be designed from the full knowledge of the exact channel covariance matrix. We assess the performance of our proposed estimators both analytically and empirically via numerical simulations. We also demonstrate by simulation that the proposed subspace estimation methods provide near-ideal performance for a massive MIMO JSDM system, by comparing with the case where the user channel covariances are perfectly known.
For @math and for a high signal-to-noise ratio (SNR), the covariance matrix @math in ) is nearly low-rank. Recovery of low-rank matrices from a collection of a few possibly noisy samples is of great importance in signal processing and machine learning. Recently, it has been shown that this can be achieved via nuclear-norm minimization, which is a convex problem and can be efficiently solved @cite_47 . For a symmetric matrix @math , the nuclear norm @math is given by the sum of the absolute values of the eigen-values of @math , and reduces to @math when @math is positive semi-definite (PSD). In our case, we have only a collection of @math snapshots @math as defined before. Let be the sample covariance of the full and projected signal. A natural extension of the matrix completion by nuclear-norm minimization to our case is readily give by: where @math is an estimate of the @math -norm of the error.
@cite_47
2952716509 1604604340 2047071281 2462232592
On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample of its entries, and comes up in many areas of science and engineering including collaborative filtering, machine learning, control, remote sensing, and computer vision to name a few. This paper surveys the novel literature on matrix completion, which shows that under some suitable conditions, one can recover an unknown low-rank matrix from a nearly minimal set of entries by solving a simple convex optimization problem, namely, nuclear-norm minimization subject to data constraints. Further, this paper introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown n x n matrix of low rank r from just about nr log^2 n noisy samples with an error which is proportional to the noise level. We present numerical results which complement our quantitative analysis and show that, in practice, nuclear norm minimization accurately fills in the many missing entries of large low-rank matrices from just a few noisy samples. Some analogies between matrix completion and compressed sensing are discussed throughout. Matrix completion concerns the recovery of a low-rank matrix from a subset of its revealed entries, and nuclear norm minimization has emerged as an effective surrogate for this combinatorial problem. Here, we show that nuclear norm minimization can recover an arbitrary n×n matrix of rank r from O(nr log2(n)) revealed entries, provided that revealed entries are drawn proportionally to the local row and column coherences (closely related to leverage scores) of the underlying matrix. Our results are order-optimal up to logarithmic factors, and extend existing results for nuclear norm minimization which require strong incoherence conditions on the types of matrices that can be recovered, due to assumed uniformly distributed revealed entries. We further provide extensive numerical evidence that a proposed two-phase sampling algorithm can perform nearly as well as local-coherence sampling and without requiring a priori knowledge of the matrix coherence structure. Finally, we apply our results to quantify how weighted nuclear norm minimization can improve on unweighted minimization given an arbitrary set of sampled entries. On the heels of compressed sensing, a new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample of its entries. It comes up in many areas of science and engineering, including collaborative filtering, machine learning, control, remote sensing, and computer vision, to name a few. This paper surveys the novel literature on matrix completion, which shows that under some suitable conditions, one can recover an unknown low-rank matrix from a nearly minimal set of entries by solving a simple convex optimization problem, namely, nuclear-norm minimization subject to data constraints. Further, this paper introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown matrix of low rank from just about log noisy samples with an error that is proportional to the noise level. We present numerical results that complement our quantitative analysis and show that, in practice, nuclear-norm minimization accurately fills in the many missing entries of large low-rank matrices from just a few noisy samples. Some analogies between matrix completion and compressed sensing are discussed throughout. Matrix completion is a rank minimization problem to recover a low-rank data matrix from a small subset of its entries. Since the matrix rank is nonconvex and discrete, many existing approaches approximate the matrix rank as the nuclear norm. However, the truncated nuclear norm is known to be a better approximation to the matrix rank than the nuclear norm, exploiting a priori target rank information about the problem in rank minimization. In this paper, we propose a computationally efficient truncated nuclear norm minimization algorithm for matrix completion, which we call TNNM-ALM. We reformulate the original optimization problem by introducing slack variables and considering noise in the observation. The central contribution of this paper is to solve it efficiently via the augmented Lagrange multiplier (ALM) method, where the optimization variables are updated by closed-form solutions. We apply the proposed TNNM-ALM algorithm to ghost-free high dynamic range imaging by exploiting the low-rank structure of irradiance maps from low dynamic range images. Experimental results on both synthetic and real visual data show that the proposed algorithm achieves significantly lower reconstruction errors and superior robustness against noise than the conventional approaches, while providing substantial improvement in speed, thereby applicable to a wide range of imaging applications.