Calibration Using Matrix Completion With Application to Ultrasound Tomography

Parhizkar, Reza; Karbasi, Amin; Oh, Sewoong; Vetterli, Martin

doi:10.1109/tsp.2013.2272925

Cited by 17 publications

(20 citation statements)

References 32 publications

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“…This problem has many applications in various areas of engineering. For example, collaborative filtering [1], ultrasonic tomography [2], direction-of-arrival estimation [3], and machine learning [4] This work was supported in part by the Iran Telecommunication Research are some of these applications. For more comprehensive lists of applications, we refer the reader to [1], [5], [6].…”

Section: Introductionmentioning

confidence: 99%

“…The constraints are underdetermined meaning that m < n 1 n 2 or more often m n 1 n 2 . The above formulation has the so-called matrix completion (MC) problem as an important instant corresponding to min X rank(X) subject to [X] ij = [M] ij , ∀(i, j) ∈ Ω, (2) where M ∈ R n1×n2 is the matrix whose elements are partially known, Ω ⊂ {1, 2, ..., n 1 } × {1, 2, ..., n 2 } is the set of the indexes of known entries of M, and [X] ij designates the (i, j)th entry of X. When rank(X * ) is sufficiently low and A has some favorable properties, X * is a unique solution to (1) [5], [7].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Iterative Concave Rank Approximation for Recovering Low-Rank Matrices

Malek-Mohammadi¹,

Babaie‐Zadeh²,

Skoglund³

2014

IEEE Trans. Signal Process.

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In this paper, we propose a new algorithm for recovery of low-rank matrices from compressed linear measurements. The underlying idea of this algorithm is to closely approximate the rank function with a smooth function of singular values, and then minimize the resulting approximation subject to the linear constraints. The accuracy of the approximation is controlled via a scaling parameter δ, where a smaller δ corresponds to a more accurate fitting. The consequent optimization problem for any finite δ is nonconvex. Therefore, to decrease the risk of ending up in local minima, a series of optimizations is performed, starting with optimizing a rough approximation (a large δ) and followed by successively optimizing finer approximations of the rank with smaller δ's. To solve the optimization problem for any δ > 0, it is converted to a new program in which the cost is a function of two auxiliary positive semidefinite variables. The paper shows that this new program is concave and applies a majorize-minimize technique to solve it which, in turn, leads to a few convex optimization iterations. This optimization scheme is also equivalent to a reweighted Nuclear Norm Minimization (NNM). For any δ > 0, we derive a necessary and sufficient condition for the exact recovery which are weaker than those corresponding to NNM. On the numerical side, the proposed algorithm is compared to NNM and a reweighted NNM in solving affine rank minimization and matrix completion problems showing its considerable and consistent superiority in terms of success rate.

QC 20150417

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Iterative Concave Rank Approximation for Recovering Low-Rank Matrices

Malek-Mohammadi¹,

Babaie‐Zadeh²,

Skoglund³

2014

IEEE Trans. Signal Process.

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QC 20150417

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“…Here it is desired to determine the structure of a molecule ("molecular conformation") from information about interatomic distances, e.g., [48,103,267]. Recent applications include the use of acoustic echoes to reveal room shape, [67], and calibration of ultrasound tomography devices, [210]. Note that in many cases we seek the locations of points in R 2 or R 3 .…”

Section: Corollary 21 Any Row-centered Positions Matrix Enjoys the Mmentioning

confidence: 99%

Imputing Missing Entries of a Data Matrix: A Review

Dax¹

2014

JAC

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The problem of imputing missing entries of a data matrix is easy to state: Some entries of the matrix are unknown and we want to assign "appropriate values" to these entries. The need for solving such problems arises in several applications, ranging from traditional ields to modern ones. Typical examples of traditional ields are statistical analysis of incomplete survey data, business reports, operations management, psychometrika, meteorology and hydrology. Modern applications arise in machine learning, data mining, DNA microarrays data, chemometrics, computer vision, recommender systems and collaborative iltering. The problem is highly interesting and challenging. Many ingenious algorithms have been proposed, and there is vast literature on imputing techniques. Yet, most of the papers consider the imputing problem within the context of a speci ic application or a speci ic approach. The current survey provides a broad view of the problem, one that exposes the large variety of imputing methods. Old and new methods are examined with focus on the ideas behind the methods. It is illustrated how simple ideas evolve into highly sophisticated algorithms. The review enables us to identify a number of basic concepts and tools which are shared by several imputing methods. Equivalence theorems that we prove reveal surprising relations between apparently different methods.

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“…This group consists of algorithms that try first to estimate the missing arXiv:1811.12803v1 [cs.IT] 30 Nov 2018 DRAFT 2 distances by utilizing the rank property of the EDM and then use the classic MDS to find the positions from the reconstructed distance matrix. SVD-MDS [1] and OptSpace-MDS [9] are two examples of this class where SVD-Reconstruct [1] and OptSpace [10] are employed for EDM completion, respectively. The algorithms in the second group formulate the localization problem as a non-convex optimization problem and then employ different relaxation schemes to solve it.…”

Section: Introductionmentioning

confidence: 99%

Localization From Incomplete Euclidean Distance Matrix: Performance Analysis for the SVD–MDS Approach

Zhang

Liu

2019

IEEE Trans. Signal Process.

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Localizing a cloud of points from noisy measurements of a subset of pairwise distances has applications in various areas, such as sensor network localization and reconstruction of protein conformations from NMR measurements. In [1], Drineas et al. proposed a natural two-stage approach, named SVD-MDS, for this purpose. This approach consists of a low-rank matrix completion algorithm, named SVD-Reconstruct, to estimate random missing distances, and the classic multidimensional scaling (MDS) method to estimate the positions of nodes. In this paper, we present a detailed analysis for this method. More specifically, we first establish error bounds for Euclidean distance matrix (EDM) completion in both expectation and tail forms. Utilizing these results, we then derive the error bound for the recovered positions of nodes. In order to assess the performance of SVD-Reconstruct, we present the minimax lower bound of the zerodiagonal, symmetric, low-rank matrix completion problem by Fano's method. This result reveals that when the noise level is low, the SVD-Reconstruct approach for Euclidean distance matrix completion is suboptimal in the minimax sense; when the noise level is high, SVD-Reconstruct can achieve the optimal rate up to a constant factor.

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Calibration Using Matrix Completion With Application to Ultrasound Tomography

Cited by 17 publications

References 32 publications

Iterative Concave Rank Approximation for Recovering Low-Rank Matrices

Iterative Concave Rank Approximation for Recovering Low-Rank Matrices

Imputing Missing Entries of a Data Matrix: A Review

Localization From Incomplete Euclidean Distance Matrix: Performance Analysis for the SVD–MDS Approach

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