An Accelerated Iterative Hard Thresholding Method for Matrix Completion

Geng, Juan; Yang, Xiaoping; Wang, Xiuyu; Wang, Lai-Sheng

doi:10.14257/ijsip.2015.8.7.13

Cited by 5 publications

(5 citation statements)

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“…However, experimentally, we found that APGD leads to a faster convergence and hence we employ APGD in the proposed BPR algorithm. Similar observations were made in the context of low-rank matrix completion [46] and PhaseLift [24].…”

Section: The Binary Phase Retrieval (Bpr) Problemsupporting

confidence: 72%

Phase Retrieval From Binary Measurements

Mukherjee

Seelamantula

2018

IEEE Signal Process. Lett.

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We consider the problem of signal reconstruction from quadratic measurements that are encoded as +1 or −1 depending on whether they exceed a predetermined positive threshold or not. Binary measurements are fast to acquire and inexpensive in terms of hardware. We formulate the problem of signal reconstruction using a consistency criterion, wherein one seeks to find a signal that is in agreement with the measurements. To enforce consistency, we construct a convex cost using a one-sided quadratic penalty and minimize it using an iterative accelerated projected gradient-descent (APGD) technique. The PGD scheme reduces the cost function in each iteration, whereas incorporating momentum into PGD, notwithstanding the lack of such a descent property, exhibits faster convergence than PGD empirically. We refer to the resulting algorithm as binary phase retrieval (BPR). Considering additive white noise contamination prior to quantization, we also derive the Cramér-Rao Bound (CRB) for the binary encoding model. Experimental results demonstrate that the BPR algorithm yields a signal-toreconstruction error ratio (SRER) of approximately 25 dB in the absence of noise. In the presence of noise prior to quantization, the SRER is within 2 to 3 dB of the CRB.

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Section: The Binary Phase Retrieval (Bpr) Problemsupporting

confidence: 72%

Phase Retrieval From Binary Measurements

Mukherjee

Seelamantula

2018

IEEE Signal Process. Lett.

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“…For other related numerical results, we refer to papers [13,20], where they have considered a slightly different versions of the tensor iterative hard thresholding algorithm and compared it with NTIHT.…”

Section: Numerical Resultsmentioning

confidence: 99%

Low rank tensor recovery via iterative hard thresholding

Rauhut

Schneider

Stojanac

2017

Linear Algebra and its Applications

122

234

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We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measurements. While the theoretical understanding of low rank matrix recovery is already well-developed, only few contributions on the low rank tensor recovery problem are available so far. In this paper, we introduce versions of the iterative hard thresholding algorithm for several tensor decompositions, namely the higher order singular value decomposition (HOSVD), the tensor train format (TT), and the general hierarchical Tucker decomposition (HT). We provide a partial convergence result for these algorithms which is based on a variant of the restricted isometry property of the measurement operator adapted to the tensor decomposition at hand that induces a corresponding notion of tensor rank. We show that subgaussian measurement ensembles satisfy the tensor restricted isometry property with high probability under a certain almost optimal bound on the number of measurements which depends on the corresponding tensor format. These bounds are extended to partial Fourier maps combined with random sign flips of the tensor entries. Finally, we illustrate the performance of iterative hard thresholding methods for tensor recovery via numerical experiments where we consider recovery from Gaussian random measurements, tensor completion (recovery of missing entries), and Fourier measurements for third order tensors.

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“…In general, an acceleration of the PGD using Nesterov's scheme [49] is not guaranteed in this setting. Nonetheless, motivated by the accelerated singularvalue hard-thresholding strategy adopted in [50] for low-rank matrix completion problems, we go ahead with incorporating a momentum factor in the QPR and SQPR algorithms and investigate empirically if it would result in acceleration. It turns out, from the simulation results, that incorporating the momentum factor indeed results in accelerated convergence (Section V-A contains the simulation results).…”

Section: B Reconstruction Algorithms For Qprmentioning

confidence: 99%

“…Defining ϕ i (τ ) = Φ τ − u 2 i , ϕ i (τ ) = Φ τ − u 2 i , and ϕ i (τ ) = Φ τ − u 2i , for τ ∈ R, we write βi in(50) asβi = k j=1 2 [ϕ i (τ j−1 ) − ϕ i (τ j )] + 4u 2 i [ϕ i (τ j ) − ϕ i (τ j−1 )] − 4u [ϕ i (τ j ) − ϕ i (ϕ j−1 )] 2 ϕ i (τ j ) − ϕ i (τ j−1 ).…”

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Quantization-Aware Phase Retrieval

Mukherjee¹,

Seelamantula²

2018

Preprint

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We address the problem of phase retrieval (PR) from quantized measurements. The goal is to reconstruct a signal from quadratic measurements encoded with a finite precision, which is indeed the case in many practical applications. We develop a rank-1 projection algorithm that recovers the signal subject to ensuring consistency with the measurement, that is, the recovered signal when encoded must yield the same set of measurements that one started with. The rank-1 projection stems from the idea of lifting, originally proposed in the context of PhaseLift. The consistency criterion is enforced using a one-sided quadratic cost. We also determine the probability with which different vectors lead to the same set of quantized measurements, which makes it impossible to resolve them. Naturally, this probability depends on how correlated such vectors are, and how coarsely/finely the measurements get quantized. The proposed algorithm is also capable of incorporating a sparsity constraint on the signal. An analysis of the cost function reveals that it is bounded, both above and below, by functions that are dependent on how well correlated the estimate is with the ground truth. We also derive the Cramér-Rao lower bound (CRB) on the achievable reconstruction accuracy. A comparison with the state-of-theart algorithms shows that the proposed algorithm has a higher reconstruction accuracy and is about 2 to 3 dB away from the CRB. The edge, in terms of the reconstruction signal-to-noise ratio, over the competing algorithms is higher (about 5 to 6 dB) when the quantization is coarse.

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An Accelerated Iterative Hard Thresholding Method for Matrix Completion

Cited by 5 publications

References 0 publications

Phase Retrieval From Binary Measurements

Phase Retrieval From Binary Measurements

Low rank tensor recovery via iterative hard thresholding

Quantization-Aware Phase Retrieval

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