Null space conditions and thresholds for rank minimization

Recht, Benjamin; Xu, Weiyu; Hassibi, Babak

doi:10.1007/s10107-010-0422-2

Cited by 122 publications

(113 citation statements)

References 29 publications

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“…To answer the raised question probabilistically, represent the linear map in the matrix form as , where is a matrix and is a vector obtained from by stacking up the columns of . It is shown in [14] that, as goes to infinity, the probability that optimizations (34) and (35) have the same solution is equal to one if the entries of matrix are independently sampled from a zero-mean unit-variance Gaussian distribution. In other words, the aforementioned heuristic method works almost always correctly for a standard rank-minimization problem whose linear constraints are randomly generated using a Gaussian probability distribution.…”

Section: Heuristic Methods For Rank Minimizationmentioning

confidence: 99%

“…Since matrix is nonsingular, applying the Sylvester's Law of Inertia to the relation yields (14) On the other hand, it can be concluded from the Hamiltonian structure of matrix that (15) Furthermore, since every eigenvalue of is an eigenvalue of and since all eigenvalues of are positive, the quantity is at least equal to . In light of the equalities (14) and (15), the relation is possible only if . Thus, matrix has negative eigenvalues.…”

Section: B Passive Control Unitmentioning

confidence: 99%

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Solving Large-Scale Hybrid Circuit-Antenna Problems

Lavaei

Babakhani

Hajimiri

et al. 2011

IEEE Trans. Circuits Syst. I

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Abstract-Motivated by different applications in circuits, electromagnetics, and optics, this paper is concerned with the synthesis of a particular type of linear circuit, where the circuit is associated with a control unit. The objective is to design a controller for this control unit such that certain specifications on the parameters of the circuit are satisfied. It is shown that designing a control unit in the form of a switching network is an NP-complete problem that can be formulated as a rank-minimization problem. It is then proven that the underlying design problem can be cast as a semidefinite optimization if a passive network is designed instead of a switching network. Since the implementation of a passive network may need too many components, the design of a decoupled (sparse) passive network is subsequently studied. This paper introduces a tradeoff between design simplicity and implementation complexity for an important class of linear circuits. The superiority of the developed techniques is demonstrated by different simulations. In particular, for the first time in the literature, a wavelength-size passive antenna is designed, which has an excellent beamforming capability and which can concurrently make a null in at least eight directions.

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Section: Heuristic Methods For Rank Minimizationmentioning

confidence: 99%

Section: B Passive Control Unitmentioning

confidence: 99%

Solving Large-Scale Hybrid Circuit-Antenna Problems

Lavaei

Babakhani

Hajimiri

et al. 2011

IEEE Trans. Circuits Syst. I

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“…We show that OptSpace is also capable of finding the missing nearby distances in our scenario and hence provide us with their corresponding ToFs. To the best of our knowledge, all the above work, as well as the recent matrix completion algorithms [17], [18], only deal with the random missing entries. However, in our case, we have structured missing entries in addition to random ones (see Section II-B), an aspect that was absent from the previous work.…”

Section: B Related Workmentioning

confidence: 99%

“…In this case, one can show that, (17) Case 2 : In this case, the minimum area is achieved when the center of the circle is on the exterior boundary as in Fig. 13(b), where Thus, we will have If we assume that , which is a reasonable assumption according to the problem statement, we will have (18) Combining (17) and (18), we can find the lower bound for as …”

Section: Upper Bound Onmentioning

confidence: 99%

Calibration Using Matrix Completion With Application to Ultrasound Tomography

Parhizkar

Karbasi

et al. 2013

IEEE Trans. Signal Process.

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Abstract-We study the application of matrix completion in the process of calibrating physical devices. In particular we propose an algorithm together with reconstruction bounds for calibrating circular ultrasound tomography devices. We use the time-of-flight (ToF) measurements between sensor pairs in a homogeneous medium to calibrate the system. The calibration process consists of a low-rank matrix completion algorithm to de-noise and estimate random and structured missing ToFs, and the classic multi-dimensional scaling method to estimate the sensor positions from the ToF measurements. We provide theoretical bounds on the calibration error. Several simulations are conducted to evaluate the theoretical results presented in this paper.

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“…but also for the bound itself. The proof is based partly on the methodology of [25], where the minimum nuclear norm relaxation of the rank minimization problem is analyzed under linear constraints on the unknown matrix. In contrast, related probabilistic analysis in [11] and [5] is based on a generalization of the restricted isometry property of A that serves only as a sufficient condition for the exactness of the convex relaxation; see also [8].…”

Section: B Probability Boundmentioning

confidence: 99%

From Sparse Signals to Sparse Residuals for Robust Sensing

Kekatos

Giannakis

2011

IEEE Trans. Signal Process.

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One of the key challenges in sensor networks is the extraction of information by fusing data from a multitude of distinct, but possibly unreliable sensors. Recovering information from the maximum number of dependable sensors while specifying the unreliable ones is critical for robust sensing. This sensing task is formulated here as that of finding the maximum number of feasible subsystems of linear equations, and proved to be NP-hard. Useful links are established with compressive sampling, which aims at recovering vectors that are sparse. In contrast, the signals here are not sparse, but give rise to sparse residuals. Capitalizing on this form of sparsity, four sensing schemes with complementary strengths are developed. The first scheme is a convex relaxation of the original problem expressed as a second-order cone program (SOCP). It is shown that when the involved sensing matrices are Gaussian and the reliable measurements are sufficiently many, the SOCP can recover the optimal solution with overwhelming probability. The second scheme is obtained by replacing the initial objective function with a concave one. The third and fourth schemes are tailored for noisy sensor data. The noisy case is cast as a combinatorial problem that is subsequently surrogated by a (weighted) SOCP. Interestingly, the derived cost functions fall into the framework of robust multivariate linear regression, while an efficient block-coordinate descent algorithm is developed for their minimization. The robust sensing capabilities of all schemes are verified by simulated tests.

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Null space conditions and thresholds for rank minimization

Cited by 122 publications

References 29 publications

Solving Large-Scale Hybrid Circuit-Antenna Problems

Solving Large-Scale Hybrid Circuit-Antenna Problems

Calibration Using Matrix Completion With Application to Ultrasound Tomography

From Sparse Signals to Sparse Residuals for Robust Sensing

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